47721775 Ines Descriptive Statistics Level i Asta 2010

8/3/2019 47721775 Ines Descriptive Statistics Level i Asta 2010

1/81

INES- RUHENGERIFaculty of Fundamental Applied sciencesDepartment of Applied Statistics

LECTURE NOTES

DESCRIPTIVE STATISTICS I

LEVEL I APPLIED STATISTICS, 2010

BY Ir. DANCILLE NYIRARUGERO, Tutorial Assistant

1


2/81

DESCRITIVE STATISTICS I

COURSE OBJECTIVE

At the end of this Course students must be knowledgeable about vocabulary, concepts,

and statistical procedures used in these studies.

Students may be called on to conduct research in their fields, since statistical procedures

are basic to research. To accomplish this, they must also be able to collect, organize,

analyze, summarize data and present data and communicate the results of the study in

their own words. Students must be also able to determine measures of central tendency,

measures of dispersion and position.

COURSE CONTENTS

Chapter 1: Introduction, Definitions and statistics vocabulary ;

Chapter 2: Frequency distributions and graphs: organizing data, histograms,

frequency polygons and ogives, other types of graphs;

Chapter 3: Data description: measures of central tendency, measures of

dispersion ( variation), measures of position;

Chapter 4: Exploratory data analysis: Box plot, Moments, Skweness andKurtosis, Contingency table, presentation and charts.

2


3/81

Bibliographie indicative:

1. Marcel AVELANGE : Statistique Descriptive, classe de 3me Ed. Sciences et

Lettres ; Lige, 197

2. DAGNERIE, P., Statistiques thorique et applique, T.1, De Boeck & Larcier s.a,

Paris, Bruxelles 199

3. Allan G. Bluman, Elementary Statistics,2004

4. MURRAY R. SPIEGEL, Ph. D., SCHAUMS OUTLINE OF Theory and Problems

of STATISTICS, 3eme Ed, 2008

5. Cottrell M, Genon-Catalot V, Duhamel C, et Meyre T. Exercices de probabilits.

Licence-Master-coles d'ingnieurs. Cassini, 200

6. Foata D et Fuchs A. Calcul des probabilits. Cours, exercices et problmes corrigs.Dunod, 2003

7. DOMINICK SALVATORE, Ph. D. DERRICK REAGLE, Ph.D, SCHAUMS

OUTLINE OF Theory and Problems of Statistics and Econometrics, 2th Ed, New

York, 2001

8. Saporta G. Probabilits, analyse des donnes et statistique. Technip, 2006

9. ANDRE FRANCIS, Business Mathematics and Statistics, sixth edition, 2004

10. Douglas A. Lind, Statistical Techniques in Business & Economics, TwelfthEdition, 2005

11. GEORGE K. KINGORIAH, Fundamentals of Applied Statistics, Nairobi, 2004

12. P.S.S. Sundar Rao and J. Richard, Introduction to Biostatistics and Research

Methods, 4th Ed, 2006;

13. GIARD VINCENT. Statistique Applique la gestion, 2me Ed. Economica 2003 ;

14. Walder Masiri, Statistique et Calcul des Probabilits, 2001.

15. CB Gupta. Vijay Gupta, An Introduction to Statistical Methods, 23rd Revised

Edition , 2007;

16. Dr. P.K. Srimani & M. Vinayaka Moorthy, Probability & Statistics, 1st Edition,

Bangarore, 2000

3


4/81

CHAPTER 1: INTRODUCTION, DEFINITIONS AND STATISTICS

VOCABULARY

1.1 Introduction

Statistics refers to the collection, organizing, presentation, analyzing, and interpretation

of numerical data to make inferences and reach decisions in all branches of economics,

business, medicine, and other social and physical sciences.

A. Definition: the Meaning of Statistics

The word statistics has two meanings:

1. In plural sense, statistics is considered as a numerical description of quantitativeaspect of things. It stands for numerical facts pertaining to a collection of objects.

2. In singular sense, statistics means the science of collection, organization,

presentation, analysis and interpretation of numerical data to assist in making

more effective decisions.

The term statistics is used to mean either statistical data or statistical method.

When it used in the sense of statistical data it refers to quantitative aspect of things, and is

numerical description.

Every data is not statistics. It must fulfil certain essential characteristics to be called

statistics.

B. Branches of Statistics

Statistics is subdivided into two branches: descriptive and inductive or inferential.

(i) Descriptive statistics consists of the collection, organization, summarization,

and presentation of data in variousforms such as tables, graphs and diagramsor using a numerical summary. The purpose of descriptive statistics is to

display and pass on information from which conclusions can be drawn and

decisions made. Businesses, for example, use descriptive statistics when

presenting their annual accounts and reports.

4


5/81

(ii) Inferential or inductive statistics consists of generalizing from samples to

populations, performing estimations and hypothesis tests, determining

relationships among variables, and making decisions.

.

1.2 Characteristics of statistics are the following:

A. Statistics means an aggregate of facts.

Facts can be analyzed only when there are more than one fact. Single fact cannot be

analyzed.

Example: the weights of 60 students of a class can be statistically analyzed. But theweight of one student cannot be called statistics.

Hence, only a collection of many facts can be called statistics.

B. Statistics are affected to a marked extent by multiplicity of causes

The facts are the results of action and interaction of a number of factors.

C. Statistics are numerically expressed.

Only numerical facts can be statistically analyzed. Therefore, facts such as Pricedecrease with increasing production can not be called statistics.

5

Statistics

Describing data

Numerical summariesVisual display

Making inferences from samples

Estimating parameters Testing hypotheses


6/81

D. Statistics are enumerated or estimated according to reasonable standards of

accuracy.

The facts should be enumerated or estimated with required degree of accuracy. The

degree of accuracy differs from purpose to purpose.

E. Statistics are collected in a systematic manner.

The facts should be collected according to planned and scientific methods. Otherwise,

they are likely to be wrong and misleading.

F. Statistics are collected for a pre determined purpose

There must be a definite purpose for collecting facts. Otherwise, the facts become useless

and hence, they cannot be called statistics.

G. Statistics are placed in relation to each other

The facts must be placed in such a way that a comparative and analytical study becomes

possible. Thus, only related facts which are arranged in logical order can be statistics.

1.3 Functions of statistics

The following are the six important functions of the science of statistics:

i) To present facts in a precise and definite form (i.e., helps proper comprehension

and avoids ambiguity).

ii) To simplify mass of figures (i.e., condensing the mass of data).

iii) To facilitate comparison (by furnishing suitable devices). Statistics adds

precision to thinking.

iv) To help formulation and testing of hypothesis (by appropriate statistical tools).

Statistics helps in comparing different sets of figures. For example, the

imports and exports of a country may be compared among themselves or they

may be compared with those of another country.

v) To help in framing suitable policies and plans (i.e., in making predictions). It

guides in the formulation of policies and helps in planning. Planning and

policy making by the government is based on statistics of production, demand,

6


7/81

etc. it indicates trends and tendencies. Knowledge of trend and tendencies

helps future planning.

vi) To help in the formulation of policies (i.e., to provide the basic Material).

Statistics helps in studying relationship between different factors. Statistical

methods may be used for studying the relation between production and price

of commodities.

Limitations of statistics

Statistics deals with only those subjects of inquiry which are capable of being

quantitatively measured and numerically expressed.

This is an essential condition for the application of statistical methods.

1.4. Origin of statistics

The term statistics is linked to the notion of State from Latin STATUS which was

changed into Latin word statisticum. Statisticum was the activity of collecting data

which helped government to ensure knowledge about state income and possessions.

The history of statistics showed that the first census had been made in Sumerian

Kingdom (Babylone) around 3000 before J.C. In 2238 before Jesus-Christ,

agriculture survey had been done in Chine by King YAO. In 2500 before J.C. inEgypt they had to collect data for taxes.

Statistics originated from two quite dissimilar field, games of chance and political

states. These two different fields are also termed as two disciplines

10. Primarily analytical

20. Secondarily essentially descriptive.

Some of pioneers of statistics are: Pascal (1623-1662), Bernouilli (1654-1705),

As regards the descriptive side of statistics it may be stated that statistics is as old as

statecraft.

Since time immorial men must have been compiling information about wealth and

manpower for purpose of peace and war. This activity considerably expanded at each

upsurge of social and political development and received added impetus in periods of

war.

7


8/81

The development of statistics can be divided into three stages: the empirical stage

(down to 1600), the comparative stage (1600-1800), the modern stage (1800 up to

day).

It has now become a useful tool and statistical methods of analysis are now being

increasingly used in biology, psychology, education, economics and business.

1.5. Statistics vocabulary

subject or individual is : an item for study;

Population or universe: a population consists of all subjects (the totalities

of all observations) that are being studied;

Statistical units: the individual subjects or objects upon whom the data

are collected.

Raw data: are collected data have not been organized numerical;

ARRAY: An array is an arrangement of raw numerical data in ascending

or descending order of magnitude;

Frequency: the frequency is the number of values in a specific class of

the distribution.

Variable: is a characteristic of the subject or individual which varies from

unit to unit. Example: height, weight, age, etc.,

1.6. Types of variables

There are two main types of variables: qualitative and quantitative.

A. Qualitative variable

A qualitative variable is one that, generally, cannot be expressed in numbers. It is an

attribute, and is descriptive in nature.

Example sex (male or female); state of birth, cause of death, religious (Catholics,

protestants, ect).

When the data are qualitative, we are usually interested in how many or what proportion

in each category. Qualitative data are often summarized in charts and bar graphs.

8


9/81

B. Quantitative variable

A quantitative variableis numerical and can be ordered or ranked.

Example: level of hemoglobin in the blood;age; heights, weights; body temperatures;

the number of children in a family.A quantitative variable can be a discrete variable ora continuous variable.

Discrete variables assume values that can be counted and represented by an integer

such as 1, 2, 3, etc.

Example: number of children in a family, the number of rooms in a house, number of

patients in a hospital, etc.

Continuous variables can assume all values between any two specific values (within

an interval). They are obtained by measuring (ex: heights, weights, age, level of

protein in blood, etc.)

Figure 1.1: summary of the types of variables

9

Types of variables

Qualitative Quantitative

Gender Color Marital

Discrete Continuous

1. children in family2. cows in a farm3. patient in a

oAgeoWeightoHeightoTime


10/81

1.7 Levels of Measurement

Data can be classified according to levels of measurement. The level of measurement of

the data often dictates the calculations that can be done to summarize and present the

data. There are four levels of measurement: Nominal, Ordinal, Interval and Ratio.

a) Nominal-level data or nominal measurement

From Latin nomen meaning name, nominal data are the same as qualitative, attribute,

categorical, or classification.

With the nominal level, the data is classified into categories and cannot be arranged in

any particular order. EX: gender, eye color, Religions affiliation, marital status.

Nominal level variables must be: mutually exclusive and exhaustive.

- Mutually exclusive means an individual or object is included in only one category.

- Exhaustive means each individual or object must appear in a category.

To summarise, the nominal-level data have the following properties:

Data categories are mutually exclusive and exhaustive.

Data categories have no logical order.

Example: list of jobs in Rwanda, consumption in Rwanda

We usually code nominal data numerically. However, the codes are arbitrary placeholders with no numerical meaning, so it in improper to perform mathematical

analysis on them.

Example: yes as 1. No as 2.

b) Ordinal-level data involves data arranged in some order, but the differences

between data values cannot be determined.

Example 1: when appreciating student dissertation we can have:

Superior, good, average, poor, inferior.

The data classifications are mutually exclusive and exhaustive.

Data classifications are ranked or ordered according to the particular trait they

possess.

10


11/81

c) Interval-level data or interval measurement

This kind of data is acquired through process of measurement where equal measuring

units are employed. The movement in magnitude between one measure to the one above

it or below it is identical in the subject population under consideration.

The data contains all the characteristics of nominal and ordinal data; the only one

difference being the scale of measurement that moves uniformly in equal interval in

which real number form can show several decimal places.

Example: temperature, shoe size.

Data classifications are mutually exclusive and exhaustive.

Data classifications are ordered according to the amount of characteristic they

possess.

Equal differences in the characteristic are represented by equal differences in the

measurements.

d) Ratio-level data or ratio measurement: Practically all quantitative data are the ratio

level of measurement. The ratio level is the "highest level of measurement. It hasall the

characteristics of the interval level, but in addition, the o point is meaningful and the ratio

between two numbers is meaningful. Ex: Wages, Weight, etc.

Data classifications are mutually exclusive and exhaustive.

Data classifications are ordered according to the amount of the characteristics

they possess.

Equal differences in the characteristic are represented by equal differences in the

numbers assigned to classifications.

The zero point is the absence of the characteristic.

11

Levels of measurements


12/81

Figure 1.2: Summary of the characteristic for Levels of Measurement

1.8 Statistic Method

For the purpose the following, procedure may be adopted with advantages:

Collect data: information should be collected regarding

Organize the data obtained

Present this information by means of diagrams or other visual aids Analyze the data above to determine the average, the extent of disparities that

exist.

To have an understanding of the phenomenon (interpretation of facts)

All this lead to a policy decision for improvement of the existing situation.

1.9 Collection of data

Statistics is concerned with the analysis of numerical data, so the first stage in statisticalmethod must be the collection of the data to be analyzed. Data can be collected in two

ways: first as primary data and second as secondary data.

a) Primary data

12

Nominal Ordinal Interval Ratio

Data may onlybe classified

Data are ranked Meaningful differencebetween values

Meaningful o point andratio between values

Type of residence(rural, urban)

Rank in class Temperature Number of patients


13/81

Primary data is data which is collected by the investigator himself with a specific

objective. This means that primary data is original in character. Sources of primary data

are eithercensuses orsamples.

Census

A census is the name given toa survey which examines every item of the population

Three important official censuses are the population census, the census of distribution

and the census of production.

A census has the advantages of completeness and being accepted and as representative,

but of course must be paid for in terms of manpower, time and resources.

Sample

A sample is a relatively small subset of a population with advantages over a census that

costs, time and resources are much less. Sample is used when it is impossible or

impractical to observe the entire group or population. The main disadvantage is that of

acceptability by layman.

b) Secondary data

Secondary data is data that has already been collected by some other investigator or

agency, and used by an investigator for his purpose.

As far as the investigator is concerned, the data he uses is from a secondary source, that

is, he did not collect it.

The prime example of secondary data is the official statistics that are published by the

Government: Financial statistics, Economic trends, etc.

The advantages of using secondary data are savings in time, manpower and resources in

sampling and data collection.

The dangers of secondary Data

13


14/81

If we have to use secondary data, there are dangers to be aware of:

(i) The data available may not be very up-to-date.

(ii) We do not necessarily know how the data were collected and analyzed or for

what reason. They may be biased because of poor collection techniques or

simply because they were collected for a different purpose.

(iii) We may not be able to find a complete set of data for our purposes in one

place. This could mean we would have to collate data from several sources

with the chance of making errors while doing so. Obtaining the data from

more than one source may also compound the chances of bias discussed in.(iv) There is the distinct possibility of transcription or printing errors in published

data.

If you are using secondary data to support arguments in reports, articles or essays it is

advisable to try to find out more about how the data were collected and analyzed and why

they were collected. These mean that:

Before using secondary data it is necessary to scrutinize them in the light of the

following points:

(i) The type and purpose of the institution that publishes statistics as a routine;

(ii) The purpose for which the data are issued and the consumers to whom they

are addressed;

(iii) The nature of the data themselves. Are the data biased? Are the data samples

only or complete enumeration?

(iv) In what types of units are the data expressed? Are they the same at differenttimes, at different places, and for all cases at the same time or place?

(v) Are the data accurate?

(vi) Do the data refer to homogeneous condition?

(vii) Are the data germane to the problem under study?

14


15/81

1.10 Misuse of statistics

The figures themselves cannot mislead, but the statisticians who present the figures

certainly can.Data can be misused in the following ways:

(i) They can be used for the wrong purpose, that is, one that is different from the

purpose for which they were collected.

(ii) They can be collected incorrectly so that they are biased

(iii) They can be analyzed carelessly so that the results obtained from them are

misleading.

.

1.11. Data classification

The data collected from the sample is generally referred to as the raw data, because it is

not arranged and organized into any format. Raw data conveys very little information to

the investigator or to anyone interested in that investigation. Therefore, the mass of

numbers must be classified.

Classifications are the process of arranging the available facts into in groups or classesaccording to their resemblances, affinities and other relationships.

The main objectives of classifying data are:

1. To condense the mass of data into a concise format;

2. to bring out the relevant points of similarity and dissimilarity, and thus

facilitate comparison;

3. To make the statistical treatment of the data easy.

Types of classification

15


16/81

Generally, classification of data may be of the following types: spatial or geographical,

temporal or chronological, qualitative, and quantitative.

Spatial or geographical classification: this classification is based on space, that is,

geographical locations. For example, data on human population may be classified on thebasis of different continents or countries or states of a country or districts of a state or

towns and villages of a district.

Temporal or chronological: data are arranged on the basis of time (years, months, days,

hours, minutes and seconds).

Qualitative classification: data are classified on the basis of quality or attribute such as

sex, colour, behaviour, religion, marital status, literacy, etc.

Quantitative classification: the classification of data is done according to some variable

(characteristics) that may be measured, such as, height, weight etc., in this type of

classification there are two elements: the variable and frequency.

Classification of units on the basis of one variable is called simple or one-way

classification.

Simultaneous classification of units on the basis of two variables is called two way

classification. A table that presents the two way classification is called Contingency

table.

CHAPTER2: FREQUENCY DISTRIBUTIONS AND CHARTS

16


17/81

2.1 FREQUENCY DISTRIBUTIONS

2.1.1. Introduction

After collecting the data, the researcher must organize and present them so they can be

understood by those who will benefit from reading the study. The most convenient

method of organizing data is to construct a frequency distribution.

The most useful method of presenting the data is by constructing charts and graphs.

This chapter describes how to organize data by constructing frequency distributions and

how to present the data by constructing charts and graphs. The charts and graphs

illustrated here are histograms, frequency polygons, ogives, pie graphs.

2. 1.2 .Organizing Data

Before the data obtained from a statistical survey or investigations have been worked on,they are called raw data. Since little information can be obtained from looking at raw

data.

The following table gives an example of a set of raw data.

Table 2.1 Marks in Statistics obtained by 20 Students of Level I STEA in 2004

Data as originally collected

15 18 7 12 17 9 13 14 12 14

16 11 10 8 9 16 13 14 10 8

In order to make the data easily understandable, the first task of the researcher is to

prepare an array ". The array is prepared by arranging the values of the variable in an

ascending or descending order. Data array give a general idea of distribution.

Example: the raw data of table 2.1 have been arrayed and are shown in table 2.2.

Table 2.2 Raw data of Table 1 put into an array

7 8 8 9 9 10 10 11 12 12

13 13 14 14 14 15 16 16 17 18

From this table, the highest and lowest marks are immediately seen and the marks

which occur most frequently are readily identified.

17


18/81

After arranging the data, their bulk must be condensed, reduced, and simplified so

that the mind comprehends them easily. A first step in such a condensation would be

achieved by representing the repetitions of a particular value of observation by tallies

instead of rewriting the value itself.

The number of tallies corresponding to any given values is the frequency of that

value and usually represented by the letter f. Frequency means thus the number of

times a certain value of the variables is repeated in the given data. A table so formed

is known as frequency distribution

In other words a frequency distribution is the organization of raw data in table form,

using classes and frequencies.

Statistical table

A statistical table presents numerical data in columns and rows. The main object of

statistical table is to arrange the physical presentation of numerical facts that the attention

of the reader is automatically directed to the information. Some of advantages statistical

tables are:

Tabulated data can be easily understood than facts stated in the form of

descriptions; They facilitate quick comparison;

They leave a lasting impression;

They make easier the summation of items and detection of errors and

omissions;

A tabular arrangement makes it unnecessary to repeat explanations,

phrases and headings;

All unnecessary details and repetitions are avoided.

2.1.3. Types of frequency distributions

18


19/81

There are two types of frequency distributions: simple frequency distribution orone-way table and grouped frequency distribution

A. Simple frequency distributions

A simple frequency distribution consists of a list of data values, each showing the

number of items having that value.

a) Quantitative

Variable X Frequenciesx1 ni

xn nnTotal N

Example: There are data from a classroom marks in probability exam in 2005.

16, 14,5, 8, 15, 15, 9, 12, 10, 9, 11, 11, 10, 17, 12, 10,14,5

Table 2.3. Frequency distribution of the marks obtained by 18 students

Marksxi 5 8 9 10 11 12 13 14 15 16 17 Total N

Tally marks II I II II III I I II II I I 18

frequencies ni 2 1 2 2 2 1 1 2 2 1 1 18Tally chart is used to record the occurrence of repeated values systematically

b) Qualitative

Example: The experience consists to know the number of students in Level I statistics in

2010 according to their sex. There are 45 students in Level I STA, then gender is coded

as G for girl and B for boy.

Table 2.4. Distribution of 45 students in Level I STA according to their sex clothes in

2009.

Tallymarks

frequencyni

B

19


20/81

GTotal

To convert a frequency distribution to relative frequency distribution, each the

frequencies is divided by the total number of observations.

When a relative frequency is multiplied by hundred it gives percentage. It is a

percentage distribution.

Cumulative frequency distribution is used when we require information on number of

observations whose characteristic is less than a given value. Data may be arranged in

such a way as to form a cumulative frequency distribution.

This is obtained by adding the numbers of observations in value cumulatively.

Cumulative distributions may be constructed for relative frequencies and percentages by

adding either the relative frequencies or the percentages in a cumulative way as has been

for absolute frequencies.

i

n

i

i-1

ii

i

n

i

i-1

1 1

% *100, total of f % equal 100

f % 100

nrelative frequency :f = ,

Ntotal of f equal 1

f 1

cumulative frequency

or

ii

p p

i i

i i

nf

N

cum of ni n f = =

=

=

=

=

Example 2.3 :

20


21/81

Marksxi 5 8 9 10

11

12

13

14

15

16

17

Total N

frequencies ni 2 1 2 2 2 1 1 2 2 1 1 18

Relative frequency fiCumulative Frequency

B .Grouped frequency distribution

When the number of distinct data values in a set of raw data is large more than 20, a

simple frequency distribution is not appropriate, since there will be too much

information, not easily assimilated.

In this case, a grouped frequency distribution is used. A grouped frequency distributionorganizes data items into groups or classes of values, each showing how many items have

values included within the group, known as the class frequency. The number of classes is

usually between 5 and 15

Definitions associated with frequency distribution classes

a) Class limits : are the lower and upper values of the classes;

b) The lower class limit represents the smallest data value that can be

included in the class;c) The upper class limit represents the largest data value that can be

included in the class;

d) Class boundaries: are the lower and upper values of a class that mark

common points between classes. These classes are used when there are

the closed intervals.

e) Class width (or length): is the difference between the lower and upper

class boundaries. If all class intervals of a frequency distribution haveequal widths, this common width is denoted by C in such case C is equal

to the difference between two successive lower class limits or two

successive upper class limits. Class width = Upper boundary lower

boundary;

21


22/81

f) Class mark or class midpoint: the class midpoint mX is obtained by

adding the lower and upper class limits and dividing by 2, or adding the

lower and upper boundaries and dividing by 2

lower boundary + upper boundary

2

lower limit +upper limit

2

m

m

X

or

X

=

=

Formulation of grouped frequency distributions

A tabulation of n data values into k classes called bins, based on values of data. The bin

limits are cutoff points that define each bin. Bins must have equal widths and their limits

cannot overlap.

1) calculate the range : highest value minus lowest value (W)

2) find number of classes (K) using following formula:

K number of classes is 2K N.

(rule of Sturge1)

42,5 Yule's rule.K N=

3) Calculate class interval or class widths ( lengths )

1h

K

=

or

1 Herbert Sturge proposed 21 logk N= +

22


23/81

4) The first classs boundary of the frequency distribution equal lowest value of series

-2

h

The last classs boundary of the frequency distribution equal the first class boundary +

Hk

The completed frequency distribution is:

Class

limits

Frequency Cumulative

Frequency

Relative

Frequency

Percentage

Total

EXCLUSIVE AND INCLUSIVE CLASS-INTERVAL

Class-interval of the type ( ): ( , )x a x b a b< < = are called exclusive (opened) since they

exclude the upper limit of the class. The following data are classified on this basis.

Income 50-100 100-150 150-200 200-250 250-300No.of

persoms

88 70 52 30 23

In this method, the upper limit of one class is the lower limit of the next class.

Class intervals of the type { } [ ]: ,x a x b a b< < = are called inclusive since they include

the upper limit of the class. The following data are classified on the basis.

Income 50-99 100-149 150-199 200-249 250-299No.of

persoms

60 38 22 16 7

However, to nsure continuity and to get correct class-limits, exclusive method of

classification should be adopted. To convert inclusive class-intervals into exclusive, we

have to make an adjustment.

23


24/81

Adjustment: find the difference between the lower-limit of the second class and upper

limit of the first class. Divide it by 2, subtract the value so obtained from all the lower

limits and add the value to all upper limits. In the above example, the adjustment factor is

100 99 0.52 =

The adjusted classes would then be as follows:

Income 49.5-99.5 99.5-149.5 149.5-199.5 199.5-249.5 249.5-299.5No.of

persoms

60 38 22 16 7

Example: the following data show the height in millimeters for 106 maize plants

after 2 weeks.

129 148 139 141 150 148 138 141 140 146 153 141 148 138

145 141 141 142 141 141 143 140 138 138 145 141 142 131

142 141 140 143 144 135 134 139 148 137 146 121 148 136

141 140 147 146 144 142 136 137 140 143 148 140 136 146

143 143 145 142 138 148 143 144 139 141 143 137 144 133

146 143 158 149 136 148 134 138 145 144 139 138 143 141145 141 139 140 140 142 133 139 149 139 142 145 132 146

140 140 140 132 145 145 142 149

Construct a grouped frequency distribution for the data.

Solution

The procedure for constructing a grouped frequency distribution for numerical data

follows:

1. Determine the classes intervals:

Find the highest value and lowest value: H = 158 and L = 121

Find the range: R = highest value lowest value = H L, so R = 158 121 = 37

Find the class width by dividing the range by the number of classes.

Width =R 37

3,7 4number of classes 10

= =

24


25/81

Find the lower limit of the first class of distributions by taking: the lower limit of

series -width

2=

4121 119

2 =

The upper class limit of the first class = the lower limit + width = 119 + 4 = 123Find the upper class limit, the high value of distributions by taking: the lower value

of distributions+ width* number of classes = 119 + 4 *10 = 159

The completed frequency distribution is:

Class

limits

Class mark

or

Mid point

Frequency Cumulative

Frequency

Relative

Frequency

Percentage

119 - 123

123 - 127

127 - 131

131 135

135 139

139 143

143 147

147 151

151 155

155 - 159

121

125

129

133

137

141

145

149

153

157

1

0

1

7

15

39

28

13

1

1

1

1

2

9

24

63

91

104

105

106

0.009

0.000

0.009

0.066

0.142

0.368

0.264

0.123

0.009

0.009

0.9

0.0

0.9

6.6

14.2

36.8

26.4

12.3

0.9

0.9

Total 106 1.000 100

Exercise

Example: construct a grouped frequency distribution of students of applied statistic Level

I in INES in 2010 according to: height, weight, age.

25


26/81

2.1.4. Two way Frequency Distribution ( Bivariate Frequency Distribution)

A two way Frequency Distribution is used when two variables are involved.

A two way frequency table has class intervals for one variable as columns and for the

other variables as rows. The boxes formed at the intersection of rows and columns thus

represent a joint class.

The column and row where are the total are named marginal distributions.

The others columns and rows are named conditional distribution.

The frequency of this joint class is the number of items that has the value of the first

variable in the class given by the column heading and the value of the second variable in

the class given by the row heading.

The method of constructing of the two way table consists of the following steps:

Determine the class intervals for each of the variables;

Place one of the variables at the top of the table and the other on the left hand

side;

Place each item in the approximate box;

Total the tallies in each box and in each row and column. The grand total of rows

and columns should check with the total number of items.

Example1: the following table shows the performance of students in two subjects:

statistics and Accountancy.

Roll number of students Marks in Statistics Marks in Accountancy1

2

3

4

5

6

7

8

9

10

15

1

1

3

16

2

18

5

4

17

13

1

2

7

8

9

12

9

17

16

26


27/81

11

12

13

14

15

16

17

18

19

20

21

2223

24

6

19

14

9

8

13

10

13

11

11

12

189

7

6

18

11

3

5

4

10

11

14

17

18

1515

3

Construct a two way frequency for data, take class interval of two variables (Statistics

and Accountancy) as 1 5; 6 11, etc. Use of 4 classes of width 5 for each variable.

The Two way Frequency table for marks in Statistics and Accountancy is shown

as:

Statistics

ACC

1 - 5 6 - 10 11 - 15 16 -20 Total

1 - 5 2 3 1 6

6 - 10 3 2 2 6

11 - 15 1 4 2 7

16 - 20 1 2 5 5

Total 6 6 7 5 24

Example 2: The age of 20 husbands and wives are given below. Form a two way

frequency table showing the relationship between the ages of husbands and wives with

the class-intervals 20-24, 25-29, etc.

27


28/81

S. No. Age ofhusband

Age of wife S. No Age of

husband

Age of wife

1 28 23 11 27 242 37 30 12 39 343 42 40 13 23 20

4 25 26 14 33 315 29 25 15 36 296 47 31 16 32 357 37 35 17 22 238 35 25 18 29 279 23 21 19 38 3410 41 38 20 48 47

Solution

Frequency Distribution of Age of Husbands and WivesAge of W

Age of H

20-24 25-29 30-34 35-39 40-44 45-49 Total

20-24 III 3

25-29 II III 5

30-34 I I 235-39 II III I 6

40-44 I I 2

45-49 I I 2

Total 5 5 4 3 2 1 20

Exercises

1. Prepare a two-way frequency table and marginal frequency tables for 25 values of

the two variables x and y given below. Take class interval of x as 10-20, 20-30,etc., and that of y as 100-200, 200-300, etc.

x y x y12 140 51 25024 256 27 550

28


29/81

33 360 42 36022 470 43 57044 470 52 29037 380 57 41626 280 44 380

36 315 48 45255 420 48 37048 390 52 31227 440 41 33057 390 69 59021 590

2. Prepare a bivariate frequency distribution for the following data:

Marks in Law Marks in

Statistics

Marks in Law Marks in

Statistics

10 20 13 2411 21 12 2310 22 11 2211 21 12 2311 23 10 2214 23 14 2212 22 12 2012 21 13 2413 24 10 2310 23 14 24

2.2 . GRAPHIC REPRESENTATION OF A FREQUENCY DISTRIBUTION

After the data have been organized into a frequency distribution, they can be presented in

graphical form. It is easier to comprehend the meaning of data presented graphically than

data presented numerically in tables or frequency distributions.

The three most commonly used graphs in research are:

1. The histogram ;

2. The frequency polygon;

3. The cumulative frequency graph or ogive.

29


30/81

1. Histogram

A histogram is a graphic presentation of a frequency distribution, in which the classes

are marked on the horizontal axis and the class frequencies on the vertical axis. The class

frequencies are represented by the heights of the rectangle. Each rectangle represents just

one class; the rectangle width corresponds to the class width and the rectangles are drawn

adjacent to each other.

Notice: in drawing histograms class intervals must be equal and exclusive.

Example: For the following frequency distribution of height of students drawn the

histogram.

Height 140-145 145-150 150-155 155-160 160-165 165-170 170-175No.of

Students

4 10 18 20 19 6 3

Solution

Histogram of distribution of height of

students

0

5

10

15

20

25

Height( Class)

Numbersofstuden

ts(

frequencies)

140-145

145-150

150-155

155-160

160-165

165-170

170-175

In the frequency distribution, if the class intervals are of unequal width, we have

first to calculate frequency density on a convenient scale.

30


31/81

i

d

ii

i

i

i

nd

a

f

a

=

=Some time we can multiply densities to the smallest class interval

Otherwise to multiply to a predetermined interval or choose the smallest in yourdistribution.

0

i 0

a

d a

ii

i

i

i

nd

a

f

a

=

= With a0 the smallest interval

Example: Average monthly earning of 1035 employees in construction industry

Monthly earning Numberof workers

Width0a

ii

i

nd

a=

60-70 25 10 2570-80 100 10 10080-90 150 10 150

90-100 200 10 200100-120 240 20 120120-140 160 20 80140-150 50 10 50150-180 90 30 30

180 and more 20 - -

Draw the histogram

Histogram of average monthly earninng of

1035 employees

0

50

100

150

200

250

Average monthly earning ( classes)

Numberof

workers(

frequencies)

60-70

70-80

80-90

90-100

100-120

120-140

140-150

150-180

31


32/81

If the frequency distribution has inclusive class intervals, they should be converted

into the exclusive type and only then, the histogram should be drawn.

Example: Draw histogram to present the following data.

Income No.of Employees Income No.of Employees100-149

150-199

200-249

250-299

21

32

52

105

300-349

350-399

400-449

450-499

62

43

18

9

Solution: here the grouped frequency distribution is not continuous because the classintervals are inclusive. We first convert it into a continuous distribution as follows:

Adjustment factor150 149

0.52

= . Subtract it from each lower limit and add to each

upper limit so as to have exclusive class intervals. Thus

Income No.of Employees

Income No.of Employees

99.5 -149.5

149.5-199.5

199.5-249.5

249.5-299.5

21

32

52

105

299.5-349.5

349.5-399.5

399.5-449.5

449.5-499.5

62

43

18

9

32


33/81

Frequency distribution of employees by

earned income (HISTOGRAM)

0

20

40

60

80

100

120

Income

Numberofemp

loyees 99.5-149.5

149.5-199.5

199.5-249.5

249.5-299.5

299.5-349.5

349.5-399.5

399.5-449.5

449.5-499.5

2. A frequency Polygon

A frequency polygon is a graph of class marks. Class marks are values of middle points

of class intervals. The polygon is drawn by placing the class marks on the horizontal axis,

and on the vertical axis are placed the frequency of observations.

If the class intervals are of equal width, the class frequencies are plotted against the class

mid values. If the class intervals are of unequal width, the graph is obtained by plotting

frequency density against class mid values.

Description of a frequency polygon:

1) Each class is represented by a single point. The height of the point represents the

class frequency; the position of the point must be directly above the

corresponding class mid point;

2) The points are joined by straight lines.

3) The extremities of the graph are joined with the mid- values of the class preceding

the first class and the class following the last class at zero frequency i.e on the x-

axis.

A curve of relative frequencies can also be drawn, and so can a curve of percentages.

These are called frequency curves.

Example: For the following frequency distribution, draw a frequency polygon.

33


34/81

Income 300-400 400-500 500-600 600- 700 700-800 800-900 900-1000workers 18 32 35 30 21 12 4

Solution

midpoint 350 450 550 650 750 850 950workers 18 32 35 30 21 12 4

Frequency polygon of distribution of Income

3. Cumulative Frequency Curve or the Ogive

A cumulative frequency distribution (traditionally called an ogive) is a graph that

represents the cumulative frequencies for the classes in a frequency distribution.

Cumulative frequency graph is used to visually how many values are below a certain

upper class boundary.

There are two types of ogives:

A) Less than ogive: Plot the points with the upper limits of the classes as abscissae and

the corresponding less than cumulative frequency as ordinates.

For less than distributions, the cumulation will proceed from the least to the greatest size,

and the series so obtained will be called less than cumulative frequency distribution.

B) For more than distributions, the cumulation will proceed from the greatest to the least,

and the series so obtained will be called more than cumulative frequency distribution.

To form cumulative frequency distributions, the points are joined with straight lines.

34


35/81

Example

Draw the two ogives for the following distribution showing the number of marks of 59

students.

Marks No. Of students Marks No. Of students

0-10

10-20

20-30

30-40

4

8

11

15

40-50

50-60

60-70

12

6

3

Solution

Construction of two Ogives

marks No.of students ( f) Less than cumulat f More than Cumul f0-1010-2020-3030-4040-5050-6060-70

4811151263

4122338505659

595547362193

Plotting the points ( 10, 4), ( 20,12), (30,23), ( 40, 38), ( 50,50 ), ( 60, 56), ( 70, 59) andjoining them by free hand, the smooth rising curve so obtained is less than ogive.

Plotting the points (0, 59), (10, 55), (20, 47), (30, 36), (40, 21), (50, 9), (60, 3) andjoining them by free-hand, the smooth falling curve so obtained is the more than ogive.

Less-than and more than cumulative frequency of marks distribution

35


36/81

EXERCISES

1. This table represents sex, age, height, weight of 24 students of Level I AST at INES in

2010.

Order Sex Age Height (en cm) Weight (en kg)

1 F 22 160 58

2 F 19 170 60

3 M 23 161 50

4 M 26 180 61

5 M 22 159 49

6 M 27 172 70

7 M 23 150 45

8 M 22 150 48

9 F 23 170 65

10 M 23 160 58

11 F 25 155 59

12 F 23 162 6013 F 24 171 80

14 F 24 170 62

15 F 24 165 64

16 F 23 173 61

17 F 22 160 57

18 F 18 163 52

19 F 19 143 48

20 F 25 167 67

21 F 23 168 59

22 F 22 172 63

23 F 24 162 5524 F 22 174 63

Draft a form of tabulation to show:

Sex and age, weight and height, age and weight, age and height. Present absolute , relative, %, cumulative frequency distributions. Draw Histogram, frequency polygon and ogive for age, height and weight.

2. Draw a histogram for the following frequency distribution of heights of students. Fromthe histogram, obtain the frequency polygon.

Height 140-150 150-160 160-165 165-170 170-180 180-190 No. of students

5 15 15 20 10 2

36


37/81

3. Daily wages of works of a factory has the following distribution. Draw the less than

cumulative frequency graph for the wages.

wages 30-39 40-49 50-59 60-69 70-79 80-89 90-99 100-109 Total

No of

works

9 25 34 25 19 13 7 2 134

4. Draw a histogram and frequency polygon for the following data:

Marks No. Of students Marks No. Of students

0-10

10-20

20-30

30-4040-50

5

13

12

118

50-60

60-70

70-80

80-9090-100

4

1

3

12

5. The following are the weights of 30 students. Draw up a frequency distribution with:

a) Class intervals 40-44, 45-49, 50-54,kgs.

b) Class intervals of width 6 kgs each.

Weights ( kgs) : 51, 47, 50, 54, 62, 52, 42, 49, 52, 49, 44, 50, 53, 58, 46, 50, 51, 53, 48,

50, 55, 52, 55, 58, 63, 54, 52,49,50,58.

2.3 . DIAGRAMATIC AND CHARTS REPRESENTATION

In section 2.2, graphs such as the histogram, frequency polygon, and ogive showed how

data can be represented when the variable displayed on the horizontal axis is quantitative,

such as heights and weights.

On the other hand, when the variable displayed on the horizontal axis is qualitative or

categorical several types of charts are used such that: pictograms, statistical maps orcartogram, spider chart, Gantt charts, bar chart, pareto charts, time series graphs, pie

graphs and so on.

37


38/81

This section is concerned with the presentation of non numeric or qualitative frequency

distributions data. The types of diagram described in this section include various types of

bar charts, pie charts, pareto charts and time series graphs.

1) Bar charts

a) Simple bar charts

It is a chart constructing of a set of non-joint bars. A separate bar for each class is drawn

to a height proportional to the frequency.

%

0.00

20.00

40.00

60.00

80.00

%

0 . 00 2 0. 0 0 4 0. 0 0 6 0. 0 0 8 0 . 00

tuerculoid

indeterminate

%

%

0.00

50.00

100.00

%

% 6 0. 64 2 7. 31 7 .2 3 4 .8 2

tuercul leprom indeter bordeli

%

60.64

27.31

7.23

4.82

%

The following bar charts is used for discrete variable

38


39/81

Example: This table shows the details of monthly expenditure of two families.

Draw a bar diagram to the data.

Family items of expenditure Family A Family B

Food

Clothing

House Rent

Education

Fuel and Lighting

Miscellaneous

Saving

140

80

100

30

40

40

70

240

160

120

80

40

80

80Total 500 800

Solution

Detail for monthly expenditure of family A

0

20406080

100120140160

Food

Clothin

g

HouseRent

Educ

ation

FuelandLig

hting

Misc

ellaneo

us

Savin

g

Expenditure

Revenue

Series1

39


40/81

Details for monthly expenditure of family

B

050

100150200250300

Food

HouseRent

Fueland

Lighting

Saving

Expenditure

Rev

enue

Series1

ii. Multiple bar charts

These charts are used as extension of simple bar charts, where another dimension of the

data is given.

0 . 0 0

1 0 . 0 0

2 0 . 0 0

3 0 . 0 0

4 0 . 0 0

tu e r c ul o i dl e pr o ma t o usi nde t e r mi na tebo r de l i ne

ma l e

f e ma l

male

female

0.00

20.00

40.00

60.00

80.00

female

male

Example: draw bar charts to show the details of monthly expenditure of two families.

Solution

40


41/81

Details for monthly expenditure of two

families A and B

050

100150200250300

Food

HouseRent

Fueland

Lighting

Saving

Expenditure

Rev

enue

Series1

Series2

2) Pie charts

A pie chart shows the totality of the data being represented using a single circle. The

circle is split into sectors, the size of each one being drawn in proportion to the class

frequency. Each sector can be shaded or colored differently if desired.

Procedures of drawing a pie graph are:

Step 1: Calculate the proportion of the total that each frequency represents, using the

formulaf

nwhere f = frequency of the class and n = total number of values.

Step 2: Find the number of degrees for each class, using the formula

Degrees = o360f

ng

or

Step 3: Find the percentage of values in each class by using the formula

% 100f

n= .

Step 4: Using a protractor and compass, graph each section and write its name and

corresponding degrees or percentage

41


42/81

Advantages and disadvantages of Pie charts

Advantages: easy to construct; easy to understand, a sense of continuity is given by line

diagram which is not present in a bar chart.

Disadvantages: might be confusing if too many diagrams with closely associated values

are compared together. Where several diagrams are displayed, there is no provision for

total figures.

Example1: construct a pie charts for the following data

Monthly expenditure

of family A

Family A

Rs % age Cumulative % age

Food

Clothing

House Rent

Education

Fuel and Lighting

Miscellaneous

Saving

140

80

100

30

40

40

70

28

16

20

6

8

8

14

28

44

64

70

78

86

100

Monthly expenditure of Family A

28%

16%

20%

6%

8%

8%

14% Food

Clothing

House Rent

Education

Fuel and Lighting

Miscellaneous

Saving

This graph shows that food is the most expenditure of family A.

42


43/81


44/81

For extra money

For something different to do

Other

18

12

8

1. A questionnaire about how people get news resulted in the following information

from 25 respondents. Construct a frequency distribution and a pie graph for the data

(N = newspaper, T = television, R = radio, M = magazine).

N N R T T R N T M R M M N R M T R M

N M T R R N N

2. A questionnaire on housing arrangements showed this information obtained from 25

respondents. Construct a frequency distribution and pie graph for the data (H =house, A = apartment, M = mobile home, C = condominium).

H C H M H A C A M C M C A M A C

C M C C H A H H M

4) Pareto chart

A pareto chart is used to represent a frequency distribution for a categorical or qualitative

variable, and the frequencies are displayed by the heights of vertical bars, which are

arranged in order from highest to lowest.

Procedures of drawing a pareto chart

1. Arrange the data from the largest to smallest according to frequency

2. Draw and label the x and y axes

3. Draw the bars corresponding to the frequencies.

Example: The following data are based on a survey from American Travel Survey onwhy people travel. Construct a pareto for the data and comment.

Purpose Number Personal business

Visit friends or relatives

146

330

44


45/81

Work related

Leisure

225

299Source: USA TODAY

0

50

100

150

200

250

300

350

1

Visit friends or relatives

Leisure

Work related

Personal business

This chart shows that the majority of American travel for visiting friends or relatives and

the minority travel for personal business.

5) Time series

When data are collected over a period of time, they can be represented by a time series

graph. A time series graph represents data that occur over a specific period of time.

Procedures of drawing a time series

Step 1: Draw and label the x and y axes

Step 2: Label the x axis for years and the y axis for the number of

Step 3: Plot each point according to the table

Step 4: Draw line segments connecting adjacent points.

Example 1: the number of bank failures in the United States during the years 1989 2000is shown. Draw a time series graph to represent the data and comment the results.

Year 198

9

199

0

199

1

199

2

199

3

199

4

199

5

199

6

199

7

199

8

199

9

2000

N. of 207 169 127 122 41 13 6 5 1 3 8 7

45


46/81

failures

0

50

100

150

200

250

1985 1990 1995 2000 2005

Series1

The graph shows the bank failures from 1989 trough 2000. The most bank failed was

between 1989 and 1992.

Example 2: The following table shows meat production for lamb for the years 1960

2000 (data are in millions of pounds), construct a time series for the data.

year 1960 1970 1980 1990 2000Lamb 769 551 318 358 234

46


47/81

0

100

200

300

400

500

600

700

800

900

1950 1960 1970 1980 1990 2000 2010

Year

Meatprod

uctionforLamb

Series1

The graph shows a decline in the quantity of meat production for lamb from 1960

through 2000.

47


48/81

Chapter 3: DATA DESCRIPTION: MEASURES OF CENTRAL TENDENCY,

MEASURES OF DISPERSION, MEASURES OF POSITION.

3.1 INTRODUCTIONThis chapter explains the basic ways to summarize data. These include measures of

central tendency, measures of variation or dispersion, and measures of position.

Central tendency refers to the location of a distribution. A measure of central tendency is

any of a number of ways of specifying this "central value". Several types of averages can

be defined, the most important being the mean, the median, the mode and midrange.

Means could be arithmetic, geometric, or harmonic mean.

The three most commonly measures of variation are the range, variance, and standard

deviation. The most common measures of position are percentiles, quartiles, and deciles

3.2 MEASURES OF CENTRAL TENDENCY

A. The arithmetic mean

1. Definition of the arithmetic mean

The arithmetic mean of a set of values is the simple arithmetic average of theobservations. This is defined as the sum of the values of all the observations divided by

the number of observations"

The arithmetic mean is normally abbreviated to just the "mean or average

The mean or average, of a population is represented by, the Greek letter ( mu); and for

a sample, by the Roman letterX (read X bar ).

That is arithmetic mean=the sum of all the values of observations in the sample

the number of values in the sample

48


49/81

2. The arithmetic mean for ungrouped data

The formula for calculating the arithmetic mean is:

j

11 2 3 N

n

j

j=11 2 3 n

xxx x x ... x

for the populationN N N

xxx x x ... x

X for the samplen n

N

j

n

=+ + + += = =

+ + + += = =

Where:

The symbol ( Geek capital letter "sigma")stands for summation: it meansthe total of";

x represents any particular value of an observation;

x is the sum of all values in the sample or population; N represents the total number of observations in the population;

n refers to the number of observations in the sample.

Assume that the data are obtained from samples unless otherwise specified.

Example 1: Find the arithmetic mean (the average) of the numbers 8, 3, 5, 12, and 10.

Solution: in this data set, 1 2 3 4 58, 3, 5, 12, 10 x x x x x= = = = = , n = 5

Then8 3 5 12 10 38

x 7.65 5

+ + + += = =

49


50/81

3. The mean of a simple( discrete ) frequency distribution

The mean for a simple frequency distribution is calculated using the following

formula:

k

j

j=11 1 2 2 3 3 x

1 3 3

1

xx xx + x x ... x

Mean, xn

j

k

k

kj

j

ff f f f f f

f f f f f f

=

+ += = = =

+ + +

Where

X represents values

f represents frequencies f is the total frequency or the total number of observations ( n) fx refers to the sum of each value x times its frequency f

Example : calculate the arithmetic Mean of the marks of 46 students given in the

following table.

Table 3.1 Frequency of marks of 46 students

Marks ( X) Frequency ( f ) fx9

10

11

12

13

14

15

16

17

18

1

2

3

6

10

11

7

3

2

1

9

20

33

72

130

154

105

48

34

18Total 46 623

50


51/81

The total of all these values ( fx ) = 623Total number of observations ( n) = 46

Therefore, the arithmetic mean of the marks of 46 students is,623

13.54

46

fxx

n

= = =

4. The mean of a grouped frequency distribution

For grouped data, and x are calculated byx x

and x =N n

f f=

Where f is the frequency, x the mid-point of the class interval and n the total number of

observation.

Procedure of finding the Mean of grouped frequency distribution

Characteristics of the Arithmetic Mean

1. Make a table as shown.

Class interval Frequency( f) Midpoint (x) of

class interval

f.x

2. Find the midpoints of each class

3. Multiply the frequency by the midpoint for each class

4. Find de sum of the frequency f of each class times the class midpoint X.

4. Divide the sum obtained by the sum of the frequencies.

51


52/81

Example 1: Calculate the arithmetic mean of the following data:

Table 3.2 shows profit per shop

Profit in N.of shops( f) Mid-point of

Class interval

f.x

0-10 12 5 6010-20 18 15 27020-30 27 25 67530-40 20 35 70040-50 17 45 76550-60 6 55 330Total 100 2800

The mean profit is:2800

28100

fx

n= =

Example 2: The following data relates to the number of successful sales made by the

salesmen in a particular quarter.

Number of sales: 0- 4 5 9 10 14 15- 19 20 24 25 29

Number of salesmen 1 14 23 21 15 6

Calculate the mean number of sales

Answer:

Number of sales

( class interval)

Number of

Salesmen (f)

Class midpoint ( x) ( fx)

0 to 4 1 2 25 to 9 14 7 9810 to 14 23 12 27615 to 19 21 17 35720 t0 24 15 22 33025 to 29 6 27 162Totals 80 1225

1225

80

122515.3

80

fx

f

fxx

f

=

=

= = =

The advantages of the mean

52


53/81

The mean is the most commonly used measure of central tendency

Every set of interval- or ratio- level data has a mean;

It is easily understood;

All the values are included in computing the mean;

A set of data has only one mean. The mean is unique;

It is used in performing many other statistical procedures and tests.

It is not necessary, to know the value of each individual observation in order to

calculate the arithmetic mean. Only the total of the observations and the number

of observations are required.

The disadvantages of the mean are:

The mean is affected by extremely high or low values, called outliers, and may

not be the appropriate average to use in these situations;

It is time consuming to compute for a large body of ungrouped data;

It cannot be calculated when the last class of grouped data is open ended ( i.e., it

includes the lower limit of the last class " and over ");

The sum of the deviations of each value from the mean will always zero:

Expressed symbolically:

( ) 0X X =As an example, the mean of 3, 8, and 4 is 5. Then:

( ) ( ) ( ) ( )3 5 8 5 4 5X X = + +

B. THE MEDIAN

The median is generally considered as an alternative average to the mean

53


54/81

The value of the variable which divides the distribution so that exactly half of the

distribution has the same or larger values and exactly half has the same or lower values is

called the median.

1. The median for ungrouped dataThe median of a set of data is the middle value that separates the higher half from the

lower half of the data set after they have been ordered from the smallest to the largest, or

the largest to the smallest.

Procedure for obtaining the median of a set of data:

order the given data from the smallest to the largest or the largest to the

smallest;

Select the middle point.

Example: Find the median of the following five observations

1 2 3 4 5x 10, x 15, x 6, x 12 and x 11= = = = =

Solution: We must:

1. order the given numbers from the smallest to the largest: 6, 10, 11, 12, 15

2. Select the middle point: the middle value is 11.Therefore the Median (MD) = 11

Note 1.Whena set of data contains an even number of items; there is no unique middle or

central value. The convention in this situation is to use the mean of the middle two items

to give a median

.

Example 1: Find the median of the following six observations:

1 2 3 4 5 6x 10, x 15, x 6, x 12 and x 11, x 17= = = = = =

Solution: As before arrange all the values of the observations in numerical order:

6, 10, 11, 12, 15, 17

54


55/81

Evidently there is no middle value. However two numbers lie in the middle: 11 and 12.

The two must be added together and divided by 2; thus obtaining their average:

11+12

MD = 11.52

=

Example 2: calculation of the median for the data given in table 3.1

Solution:

Arranging all the 24 values in ascending order of magnitude, we get the following data:

2.90 3.57 3.73

2.98 3.61 3.75

3.30 3.62 3.76

3.43 3.66 3.76

3.43 3.68 3.77

3.45 3.71 3. 84

3.55 3.72 3.88

The 12th value is 3.66 and 13th is 3.68; the median is the average of these two.

Median =3.66 3.68

3.67 %2

g+

=

Note 2. For a set with an odd number ( n) of items, the median can be precisely identified

as the value of the1

2

nth

+item. Thus in a size ordered set of the 15 items, the median

would be the15 1

8th item along.2

th the+

=

2. Median for a simple frequency distribution

Where there is a large number of discrete items in a data set, but the range of values is

limited, a simple frequency distribution will probably have been compiled.

The median for a simple frequency distribution is calculated by the following formula:

MD =1

2

f+

55


56/81

Where f is cumulative frequency, represented by F or N

Procedure for calculating the median

To calculate the median for a simple (discrete) frequency distribution, the followingprocedures should be followed

1. Calculate the value of1

2

f+ ;

2. Form a F ( cumulative frequency) column;

3. Find that F value which first exceeds1

2

f + ;

4. The median is that x value corresponding to the F value identified in 3.

Example: calculate the median for the following distribution of delivery times of orders

sent out from a firm.

Delivery time (days) 0 1 2 3 4 5 6 7 8 9 10 11

Number of orders 4 8 11 12 21 15 10 4 2 2 1 1

Answer

STEP 1 The median is the1

2

Nth

+

= 91 12th+

= 46th item

STEP 2 The F Column is shown in the following table:

Delivery time Number of orders

(Days) orders cum

( x ) ( f ) ( F)

0 4 4

1 8 12

2 11 23

3 12 35

4 21 56

5 15 71

56


57/81

6 10 81

7 4 85

8 2 87

9 2 89

10 1 90

11 1 91

STEP 3 The first F value to exceed 46 is F = 56

STEP4 The median is thus 4 (days)

3: Median for a grouped frequency distribution

There are two methods commonly employed for estimating the median for a grouped

frequency distribution.

a) using an interpolation formula;

b) by graphical interpolation

a) Estimating the median by formula

Given a grouped frequency distribution, the best that can be done is to identify the

class or group that contains the median item. From there, using cumulative

frequencies and the fact the median must lie exactly one half of the way along the

distribution.

The formula for calculating the median for a grouped distribution is:

Median =2 .

NF

L c

f

+

57


58/81

Where

lower bound(limit) of the median class ( the class contains the middle

item of distribution)

sum of frequecies of all classes lower than the median class

= median class widt

L

F

c

=

=

h(interval of median class)

= frequency of the median class

N = total number of obsrvations

f

Example1: calculation of Median for the Data of table 3.2

Protein intake/consumption

unit (g) /day ( class interval)

N.of families

Frequencies ( f)

Cumulative

frequency15-25 30 3025-35 40 7035-45 100 17045-55 110 28055-65 80 36065-75 30 39075-85 10 400Total 400

Median class is 45 -55 N=400

Median = ( ).

200 170 .10245 47.73

110

NF C

L gf

+ = + =

Procedure for estimating the median by formula

The procedure for estimating the median (by formula) for a grouped frequency

distribution is:

1. Form a cumulative frequency (F) Column;

2. Find the value ofN

( where N = ).2

f

58


59/81

3. Find that F value first exceeds, which identifies the median class M.

4. Calculate the median using the following interpolation formula:

2 .

NF

L c

f

+

Example: Estimate the median for the following data, which represents the ages of a set

of 130 representatives who took part in a statistical survey.

Age in years 20 and 25 and 30 and 35 and 40 and 45 and

Under 25 under 30 under 35 under 40 under 45 under 50

Number of 2 14 29 43 33 9

Representatives

Answer

1.

Age ( years) Number of representatives ( f) ( F )

20 and under 25 2 225 and under 30 14 1630 and under 35 29 4535 and under 40 43 8840 and under 45 33 12145 and under 50 9 130

2.130

652 2

N= =

3. The median class is the class that has the first F greater than 65. Here, it is 35 to 40.

4. The median can now be estimated using the interpolation formula.

59


60/81

35; 43; 5

2Thus, median = .

65-45= 35 + 5

43

= 37.33

Median = 37.33years

L F c

NF

cf

= = =

b) Estimating the median graphically

A percentage cumulative frequency curve (or ogive ) is drawn and the value of the

variable that corresponds to the 50% point is read off and gives the median estimate.

Procedure for estimating the median graphically

1. Form a cumulative ( percentage ) frequency distribution

2. Draw up cumulative frequency curve by plotting class upper bounds against

cumulative percentage frequency and join the points a smoth curve.

3. Read off 50% point to give median.

Properties of Median

1. The median is particularly useful where :

a) a set or distribution has extreme values present and

b) Values at the end of a set or distribution are not known. This means that

median is used for an open ended distributions.

2. The median can be determined for all levels of data except nominal

3. the median is unique; there is only one median for a set of data

The advantages of the median

The advantages of the median are:

it is not affected by extremely large or small values ;

60


61/81

it is easily understood ( i.e half the data are smaller than the

median and half are greater);

it can be calculated even when the last class is open ended and

when the data ere qualitative rather than quantitative;

The disadvantages of the median

It does not use much of the information available;

It requires that observations be arranged into any array, which is time

consuming for a large body of ungrouped data.

C. THE MODE

1. Definition

The mode is the value of the observation that appears most frequently, or equivalently

has the largest frequency. Especially, the mode is used in describing nominal and ordinal

levels of measurement

It is possible for data not to have any mode at all; like in a case where observations occur

with equal frequency.

Example: The mode of the set 2, 1, 3, 3, 1,1, 2, 4 is 1, since this value occurs most often.

For the data in table 3.1 is 3.76 this observation is most commonly occurring

The mode of the following simple discrete frequency distribution :

X 4 5 6 7 8 9 10

f 2 5 21 18 9 2 1

Is 6, since this value has the largest frequency

2. The mode for grouped data

For a grouped frequency distribution, the mode cannot be determined exactly and so must

be estimated. The technique used is one of interpolation. There are two methods that can

be used to estimate the mode:

Using an interpolation

61


62/81

Graphically, using a histogram.

Mode of a grouped frequency distribution by formula

An estimate of the mode for a grouped frequency distribution can be obtained using the

following procedure:

1. Determine the modal class ( that class which has the largest frequency)

2. Calculate D1= difference between the largest frequency and the frequency

immediately preceding it.

3. Calculate D2 = difference between the largest frequency and the frequency

immediately following it.

4. Use the following interpolation formula:

Interpolation formula for the mode

1

1 2

DMode = L+ .

DC

D

+

Where: L = lower bound of modal class

C = modal class width

And: D1, D2 are as described above in 2 and 3

Example 1: Estimate the mode of the following distribution of ages.

Age (years) 20-25 25-30 30-35 35-40 40-45 45-50

Number of employees 2 14 29 43 33 9

Answer:

Age (years) number of employees

20 and under 25 2

25 and under 30 14

30 and under 35 29

35 and under 40 43

40 and under 45 33

45 and under 50 9

62


63/81

D1= 43 29 = 14

D2= 43-33 = 10

The lower class bound of the modal class, L = 35

The class width of the modal class, C = 5 (from 35 to 40 )

1

1 2

Thus: mode= .C

14= 35+ .5

14+10

mode = 37.92 years

DL

D D

+ +

Graphical estimation of the modeThe graphical equivalent of the above interpolation formula is to construct three

histogram bars, representing the class with the highest frequency and the ones on either

side of it, and to draw two lines. The mode estimate is the x value corresponding to the

intersection of the lines.

Example 2: Estimation of the mode of a frequency distribution using the graphical

formula.

Using the data of ex 1:

Age (years) number of employees

30 and under 35 29

35 and under 40 43

40 and under 45 33

Draw the graph

63


64/81

The advantages of the mode

The mode has the advantage of not being affected by extremely

high or low values;

It is easily understood ( half the data are smaller than the median

and half are greater), not difficult to calculate and can be used

when the last class of a distribution is open ended;

The mode is used for al levels of data: nominal, ordinal, interval,

and ratio.

The disadvantages of the mode

The disadvantages of the mode are:

The mode does not use much of the information available;

For many sets of data, there is no mode because no value

appears more than once. For example, there is no mode for this

set of price data: RWF250 , RWF 400, RWF 650 and RWF

1250 ;

The mode is not always unique. Example: suppose the ages ofthe individuals in a scout Club is 14, 16, 17, 18, 18, 20, 20, 22,

24, 24, and 25. Both the ages 27 and 35 are modes.

In general, the mean is the most frequently used measure of central tendency and the

mode is the least used.

lowest value highest valueMR

2

+=

Example:

D. THE MIDRANGE

The midrange is defined as the sum of the lowest and highest values in the data set,

divided by 2. The symbol MR is used for the midrange.

Find the midrange of these numbers: 2, 3, 6, 8, 4, and 1

64


65/81

1 8 9MR 4.5

2 2

+= = =

Then, the midrange is 4.5

The Relationship between the Arithmetic Mean, the Median and the Mode

In a symmetrical frequency distribution the mode, median, and mean are located

at the center and are always equal illustrates this for a normal distribution .Fig

(a ) in this case one of these measures may be used.

Mean

Median

Mode

If the distribution of the variable is not symmetrical, we have a skew distribution:

the arithmetic mean is not so typical of the distribution. In a positively skewed

distribution, the mean is not at the centre. The mean is dragged to the right of

centre by a few extremely high values of the variable that have been observed.

The median is generally the next largest measure in a positively skewed

frequency distribution. The mode is the smallest of the three measures. If the

distribution is highly skewed, the mean would not be a good measure to use. The

median and mode would be more representative.

mode median mean

65


66/81

In a negatively skewed distribution the mean is reduced by a few extremely low

values of the variable and hence will be left of centre. The median is greater than

the arithmetic mean, and the modal value is the largest of the three measures.

Again, if the distribution is highly skewed, the mean should not be used to

represent the data.

In a moderately skew distribution the following relationship holds approximately:1. Mean - Mode= 3 (mean-Median);

2. Median mode = 2 ( mean median );

3. Median =2 mean + mode

3;

4. Mode = 3 median 2 mean ;

5. Mean =3 median - mode

2

THE GEOMETRIC MEAN G

The geometric mean is useful in finding the average of percentages, ratios, indexes, or

growth rates. It has a wide application in business and economics because we are often

interested in finding the percentage changes in sales, salaries, or economic figures, such

as the Gross Domestic Product, which compound or build on each other.

The geometric mean G of a set of N positive numbers 1 2 3, , ,... nx x x x , is calculated using

the formula: Geometric mean= 1 2 3...n nx x x x

Where n is the number of observation made of the variable x and 1 2 3, , ,..., n x x x x are the

values of these observations.

Example: the geometric mean of the numbers 3, 25 and 45 is:

G = 3 3 25 45 = 3 3375

66

Mean median mode


67/81

THE HARMONIC MEAN H

The harmonic mean is another specialized measure of location used only in particular

circumstances; namely when the data consists of a set of rates, such as prices, speeds or

productivity.

The harmonic mean H of a set of N numbers 1 2 3, , ,... nx x x x , is the reciprocal of the

arithmetic mean of the reciprocals of the numbers:

H =

1

111 1n

i i

n

xn x=

= Where n is the number of observations.

Example: the harmonic mean of the numbers 2, 4, and 8 is:

H =

3 33.43

1 1 1 72 4 8 8

= =+ +

The relation between the arithmetic, geometric, and harmonic means.

The geometric mean of a set of positive numbers 1 2 3, , ,... nx x x x is less than or equal to

their arithmetic mean but is greater than or equal to their harmonic mean. In symbols:

XH G

The equality signs hold only if all the numbers 1 2 3, , ,... nx x x x are identical.

Example: The set 2, 4, 8 has arithmetic mean 4.67, geometric mean 4, and harmonic

mean 3.43.

67


68/81

3.2 MEASURES OF DISPERSION

Dispersion refers to the variability or spread in the data. A small value for a measure of

dispersion indicates that the data are clustered closely, say, around the arithmetic mean. A

large measure of dispersion indicates that the mean is not reliable.

The most important measures of dispersion are:

1. Range is the difference between the largest and the smallest values in

a data. The range is the simplest of the three measures and is defined

now. The symbol Ris used for the range.

R= Largest value smallest value

1. Find the range of the following distribution.

35, 45, 30, 35, 40, 25

R = 45- 25 = 20

2. Mean Deviation (MD) is the arithmetic mean of the deviations of the

observations from the arithmetic mean ignoring the sign of these

deviations.

a) The formula for the mean deviation for ungrouped data is

MD = for populationsX

N

MD = for samplesX X

n

mean

Where:

X is the value of each observation;

X is the arithmetic mean of the values;

68


69/81

is th

Date post:	07-Apr-2018
Category:	Documents
Upload:	utcm77
View:	213 times
Download:	0 times

47721775 Ines Descriptive Statistics Level i Asta 2010

Documents