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LECTURE NOTE
ON
DESCRIPTIVE STATISTICS
(STS 102)
BY
ADEOSUN SAKIRU ABIODUN
E-mail: [email protected]
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COURSE CONTENTS Statistical data: types, sources and methods of collection. Presentation of data: tables, charts and graphs. Error and Approximations. Frequency and cumulative distributions. Measures of location, partition, dispersion, Skewness and kurtosis. Rates, Proportion and index numbers.
READING LISTS 1. Adamu S.O and Johnson Tinuke L (1998): Statistics for Beginners; Book 1.
SAAL Publication. Ibadan. ISBN: 978-34411-3-2 2. Clark G.M and Cooke D (1993): A Basic course in statistics. Third edition.
London: Published by Arnold and Stoughton. 3. Olubosoye O.E, Olaomi J.O and Shittu O.I (2002): Statistics for
Engineering, Physical and Biological sciences. Ibadan: A Divine Touch Publications.
4. Tmt. V. Varalakshmi et al (2005): Statistics Higher Secondary - First year. Tamilnadu Textbook Corporation, College Road, Chennai- 600 006
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INTRODUCTION In the modern world of information and communication technology, the
importance of statistics is very well recognised by all the disciplines. Statistics has
originated as a science of statehood and found applications slowly and steadily in
Agriculture, Economics, Commerce, Biology, Medicine, Industry, planning, education
and so on. As of today, there is no other human walk of life, where statistics cannot
be applied.
Statistics is concerned with the scientific method of collecting, organizing,
summarizing, presenting and analyzing statistical information (data) as well as
drawing valid conclusion on the basis of such analysis. It could be simply defined as the science of data. Thus, statistics uses facts or numerical data, assembled, classified and tabulated so as to present significant information about a given
subject. Statistic is a science of understanding data and making decisions in the face
of randomness.
The study of statistics is therefore essential for sound reasoning, precise
judgment and objective decision in the face of up- to- date accurate and reliable
data. Thus many researchers, educationalists, business men and government
agencies at the national, state or local levels rely on data to answer operations and
programs. Statistics is usually divided into two categories, which is not mutually
elution namely: Descriptive statistics and inferential statistics.
DESCRIPTIVE STATISTICS
This is the act of summarizing and given a descriptive account of numerical
information in form of reports, charts and diagrams. The goal of descriptive
statistics is to gain information from collected data. It begins with collection of data
by either counting or measurement in an inquiry. It involves the summary of specific
aspect of the data, such as averages values and measure of dispersion (spread).
Suitable graphs, diagrams and charts are then used to gain understanding and clear
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interpretation of the phenomenon under investigation keeping firmly in mind
where the data comes from. Normally, a descriptive statistics should:
i. be single valued ii. be algebraically tractable
iii. consider every observed value.
INFERENTIAL STATISTICS
This is the act of making deductive statement about a population from the
quantities computed from its representative sample. It is a process of making
inference or generalizing about the population under certain conditions and
assumptions. Statistical inference involves the processes of estimation of
parameters and hypothesis testing. However, this concept is not in the context of
this course.
USES OF STATISTICS
Statistics can be used among others for:
1) Planning and decision making by individuals, states, business organizations,
research institution etc.
2) Forecasting and prediction for the future based on a good model provided
that its basic assumptions are not violated.
3) Project implementation and control. This is especially useful in on-going
projects such as network analysis, construction of roads and bridges and
implementation of government programs and policies.
4) The assessment of the reliability and validity of measurements and general
points significance tests including power and sample size determination.
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STATISTICAL DATA
Data can be described as a mass of unprocessed information obtained from
measurement of counting of a characteristics or phenomenon. They are raw facts
that have to be processed in numerical form they are called quantitative data. For
instance the collection of ages of students offering STS 102 in a particular session is
an example of this data. But when data are not presented in numerical form, they
are called qualitative data. E.g.: status, sex, religion, etc.
Definition: Statistical data are data obtained through objective measurement or
enumeration of characteristics using the state of the art equipment that is precise
and unbiased. Such data when subjected to statistical analysis produce results with
high precision.
SOURCES OF STATISTICAL DATA
1. Primary data: These are data generated by first hand or data obtained
directly from respondents by personal interview, questionnaire,
measurements or observation. Statistical data can be obtained from:
(i) Census complete enumeration of all the unit of the population (ii) Surveys the study of representative part of a population (iii) Experimentation observation from experiment carried out in
laboratories and research center.
(iv) Administrative process e.g. Record of births and deaths.
ADVANTAGES Comprises of actual data needed It is more reliable with clarity Comprises a more detail information
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DISADVANTAGES Cost of data collection is high Time consuming There may larger range of non response
2. Secondary data: These are data obtained from publication, newspapers, and
annual reports. They are usually summarized data used for purpose other
than the intended one. These could be obtain from the following:
(i) Publication e.g. extract from publications
(ii) Research/Media organization
(iii) Educational institutions
ADVANTAGES The outcome is timely The information gathered more quickly It is less expensive to gather.
DISADVANTAGES Most time information are suppressed when working with secondary data The information may not be reliable
METHODS OF COLLECTION OF DATA
There are various methods we can use to collect data. The method used depends
on the problem and type of data to be collected. Some of these methods include:
1. Direct observation
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2. Interviewing
3. Questionnaire
4. Abstraction from published statistics.
DIRECT OBSERVATION
Observational methods are used mostly in scientific enquiry where data are
observed directly from controlled experiment. It is used more in the natural
sciences through laboratory works than in social sciences. But this is very useful
studying small communities and institutions.
INTERVIEWING
In this method, the person collecting the data is called the interviewer goes to ask
the person (interviewee) direct questions. The interviewer has to go to the
interviewees personally to collect the information required verbally. This makes it
different from the next method called questionnaire method.
QUESTIONNAIRE
A set of questions or statement is assembled to get information on a variable (or a
set of variable). The entire package of questions or statement is called a
questionnaire. Human beings usually are required to respond to the questions or
statements on the questionnaire. Copies of the questionnaire can be administered
personally by its user or sent to people by post. Both interviewing and
questionnaire methods are used in the social sciences where human population is
mostly involved.
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ABSTRACTIONS FROM THE PUPLISHED STATISTICS
These are pieces of data (information) found in published materials such as figures
related to population or accident figures. This method of collecting data could be
useful as preliminary to other methods.
Other methods includes: Telephone method, Document/Report method, Mail
or Postal questionnaire, On-line interview method, etc.
PRESENTATION OF DATA
When raw data are collected, they are organized numerically by distributing
them into classes or categories in order to determine the number of individuals
belonging to each class. Most cases, it is necessary to present data in tables, charts
and diagrams in order to have a clear understanding of the data, and to illustrate the
relationship existing between the variables being examined.
FREQUENCY TABLE
This is a tabular arrangement of data into various classes together with their
corresponding frequencies.
Procedure for forming frequency distribution
Given a set of observation , for a single variable. 1. Determine the range (R) = L S where L = largest observation in the raw
data; and S = smallest observation in the raw data.
2. Determine the appropriate number of classes or groups (K). The choice of K is
arbitrary but as a general rule, it should be a number (integer) between 5 and
20 depending on the size of the data given. There are several suggested guide
lines aimed at helping one decided on how many class intervals to employ.
Two of such methods are:
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(a) K = 1 +3.322 (b) K = where = number of observations.
3. Determine the width of the class interval. It is determined as 4. Determine the numbers of observations falling into each class interval i.e. find
the class frequencies.
NOTE: With advent of computers, all these steps can be accomplishes easily.
SOME BASIC DEFINITIONS
Variable: This is a characteristic of a population which can take different values.
Basically, we have two types, namely: continuous variable and discrete variable.
A continuous variable is a variable which may take all values within a given range.
Its values are obtained by measurements e.g. height, volume, time, exam score etc.
A discrete variable is one whose value change by steps. Its value may be obtained
by counting. It normally takes integer values e.g. number of cars, number of chairs.
Class interval: This is a sub-division of the total range of values which a
(continuous) variable may take. It is a symbol defining a class E.g. 0-9, 10-19 etc.
there are three types of class interval, namely: Exclusive, inclusive and open-end
classes method.
Exclusive method:
When the class intervals are so fixed that the upper limit of one class is the lower
limit of the next class; it is known as the exclusive method of classification. E.g. Let
some expenditures of some families be as follows:
0 1000, 1000 2000, etc. It is clear that the exclusive method ensures continuity of data as much as the upper limit of one class is the lower limit of the next class. In
the above example, there are so families whose expenditure is between 0 and
999.99. A family whose expenditure is 1000 would be included in the class interval
1000-2000.
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Inclusive method:
In this method, the overlapping of the class intervals is avoided. Both the lower and
upper limits are included in the class interval. This type of classification may be used
for a grouped frequency distribution for discrete variable like members in a family,
number of workers in a factory etc., where the variable may take only integral
values. It cannot be used with fractional values like age, height, weight etc. In case of
continuous variables, the exclusive method should be used. The inclusive method
should be used in case of discrete variable.
Open end classes:
A class limit is missing either at the lower end of the first class interval or at the
upper end of the last class interval or both are not specified. The necessity of open
end classes arises in a number of practical situations, particularly relating to
economic and medical data when there are few very high values or few very low
values which are far apart from the majority of observations.
Class limit: it represents the end points of a class interval. {Lower class limit &
Upper class limit}. A class interval which has neither upper class limit nor lower
class limit indicated is called an open class interval e.g. less than 25, 25 and above
Class boundaries: The point of demarcation between a class interval and the next
class interval is called boundary. For example, the class boundary of 10-19 is 9.5 19.5
Cumulative frequency: This is the sum of a frequency of the particular class to the
frequencies of the class before it.
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Example 1: The following are the marks of 50 students in STS 102:
48 70 60 47 51 55 59 63 68 63 47 53 72 53 67 62 64 70 57 56
48 51 58 63 65 62 49 64 53 59 63 50 61 67 72 56 64 66 49 52 62
71 58 53 63 69 59 64 73 56.
(a) Construct a frequency table for the above data.
(b) Answer the following questions using the table obtained:
(i) how many students scored between 51 and 62?
(ii) how many students scored above 50?
(iii) what is the probability that a student selected at random from the class will
score less than 63?
Solution:
(a) Range (R) = No of classes Class size Frequency Table
Mark Tally frequency
7 51- 54 7
55 - 58 7
59 62 8 63 66 11 67 70 6 71 74 4
(b) (i) 22 (ii) 43 (iii) 0.58
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Example 2: The following data represent the ages (in years) of people living in a
housing estate in Abeokuta.
18 31 30 6 16 17 18 43 2 8 32 33 9 18 33 19 21 13 13 14
14 6 52 45 61 23 26 15 14 15 14 27 36 19 37 11 12 11
20 12 39 20 40 69 63 29 64 27 15 28.
Present the above data in a frequency table showing the following columns; class
interval, class boundary, class mark (mid-point), tally, frequency and cumulative
frequency in that order.
Solution:
Range (R) No of classes Class width Class interval Class boundary Class mark Tally Frequency Cum.freq
2 11 1.5 11.5 6.5 7 7 12 21 11.5 21.5 16.5 21 28 22 31 21.5 31.5 26.5 8 36 32 41 31.5 41.5 36.6 7 43 42 51 41.5 51.5 46.5 2 45 52 61 51.5 61.5 56.5 2 47 62 71 61.5 71.5 66.5 3 50
Observation from the Table
The data have been summarized and we now have a clearer picture of the
distribution of the ages of inhabitants of the Estate.
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Exercise 1
Below are the data of weights of 40students women randomly selected in Ogun
state. Prepare a table showing the following columns; class interval, frequency, class
boundary, class mark, and cumulative frequency.
96 84 75 80 64 105 87 62 105 101 108 106 110 64 105 117
103 76 93 75 110 88 97 69 94 117 99 114 88 60 98 77
96 96 91 73 82 81 91 84
Use your table to answer the following question
i. How many women weight between 71 and 90?
ii. How many women weight more than 80?
iii. What is the probability that a woman selected at random from Ogun state
would weight more than 90?
GRAPHICAL PRESENTATION OF DATA
It is not enough to represent data in a tabular form. The most attractive way
of representing data is through charts or graphs.
PICTOGRAM
Pictograms or pictographs are representations in form of pictures. They convey
broad meanings and relationships among data. Also they are the simplest way of
presenting information. Pictograms are popularly used in newspapers by journalists
and advertisers.
EXAMPLE 1: In a certain secondary school, there are 4 Eng teachers, 3 math
teachers, 2 Biology teachers, 2 government teachers and 1 Physics teacher. Draw a
pictogram to represent this information.
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SOLUTION: Let us use to represent a teacher
The different subject Teachers in a certain school
English Teachers
Maths teachers
Biology teachers
Government teachers
Physics teachers
PIE CHART
A pie chart is a circular graph in which numerical data are represented by sectors of
a circle. The angles of the sectors are proportional to the frequencies of the items
they represent
EXAMPLE 2: In ADAS international school, the lesson periods for each week are
given below.
English 7, Maths 10, Biology 3, Physics 4, Chemistry 3, others 9. Draw a pie chart to
illustrate this information.
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Solution: Total no of periods in a week Subject No of period Angle of sector
English 7
Maths 10
Biology 3
Physics 4
Chemistry 3
Others 9
Total 36
The pie chart showing the lesson period in ADAS international school
English
Mathematics
Biology
Physics
Chemistry
Other
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BAR CHART
A bar chart is a statistical graph in which bars (rectangular bars) are drawn such
that their lengths or heights are proportional to the quantities or item they
represent. Each bar is separated by equal gaps.
Example 3: The allotment of time in minutes per week for some of the university
courses in second semester is;
Courses Minutes
GNS 104 60
MTS 102 120
STS 102 180
ECO 102 120
BFN 108 120
PHS 192 140
Construct a bar chart to represent the above data.
Solution:
Bar chart showing the allotment of time in minutes
0
20
40
60
80
100
120
140
160
180
200
GNS 104 MTS 102 STS 102 ECO 102 BFN 108 PHS 192
Series1
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HISTOGRAM
A histogram is a graphical representation of a frequency distribution. It consists of a
number of rectangles. The area of each rectangular bar of a histogram is
proportional to the corresponding frequency. Unlike bar charts, the rectangles are
joined together for histogram.
Example 4: Draw a histogram to represent the data in the table below.
Height (cm) Frequency 6 9 15 5 2
Solution:
Histogram showing the heights
CUMMULATIVE FREQUENCY CURVE (Ogive)
This curve is the plotting of cumulative freq. against upper class bounding of
the class intervals. The points, when joined, result usually in a smooth curve which
is the form of an elongated S.
17
4.5
16
4.5
15
4.5
14
4.5
13
4.5
12
4.5
Fre
qu
en
cy
Height
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Example 5: Using the data below:
Weight (kg) Frequency Cumulative frequency
20 24 5 5 25 29 4 9 30 34 3 12 35 39 5 17 40 44 5 22 45 49 5 27 50 54 18 45 55 59 12 57 60 64 3 60
The cumulative frequency or Ogive will appears as follows:
Stem plots (Stem and leaf plots) A stem-and-leaf plot is used to visualize data. To set up a stem-and-leaf plot,
we follow some simple steps. First we have a set of data. Then we identify the lowest
and greatest number in the data set and then we draw a vertical line. On the left
hand side of the line we write the numbers that correspond to the tens and on the
right hand side of the line, we will write the numbers that corresponds with the
units. The digits should be arranged in order.
Example : Draw the stem plot for the following data
(a) 13, 24, 22, 15, 33, 32, 12, 31, 26, 28, 14, 19, 20, 22, 31, 15.
(b) 46 59 35 41 46 21 24 33 40 45 49 53 48 54 61 36 70 58 47 12
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Solution: (a) Stem Leaf
1 2 3 4 5 5 9
2 0 2 2 4 6 8
3 1 1 2 3
(b) Stem Leaf
1 2
2 1 4
3 3 5 6
4 0 1 5 6 6 7 8 9
5 3 4 8 9
6 1
7 1
Exercise 2:
1. The population by classes of Adas international school is as shown below
Class I II III IV V VI
No of pupils 30 25 27 26 22 26
Draw a pictogram, pie chart and a bar chart to represent the information.
2. The age distribution (in years) of a group of 100 individual is given below.
Ages(x) 2 4 5 9 10 14
15 19
50 24
25 29
30 34
35 39 No. of
individuals(f)
8 11 21 20 17 10 10 3
Draw a histogram and a cumulative frequency curve for the data.
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ERRORS AND APPROXIMATION
Rounding of statistical data is the method of approximating a number such
that the last digits affected are changed to zero and the number become clearer.
Since in reality most measurement is not exact, hence offence is not much
committed when data are rounded up.
Methods of rounding are rounding to specific units. E.g. 4,234,120 to the
nearest million is 4,000,000. Also rounding to specific significant figures e.g. to 5
significant figures will give 4,235,100. Again when an amount, say N750.3 is used to
denote the average salary per person in a month, the figure in this wise is rounded
to one place of decimal and the actual figure is between the ranges N750.25 to less
than N 750.35. This implies rounded figures have errors.
Error: For any given figure the correct figure lies in a certain range. Half of this
range is the Error.
Suppose correct figure; rounded figure e = error; Absolute error: . Relative error Percentage Error . i.e. percentage Error
Example: The length of a pole is measured as 10meters to the nearest meter. What
is the range of its actual length? Calculate the percentage error.
Solution The actual length of the pole will be between 9.5 and 10.5
Error Percentage Error
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MEASURES OF LOCATION
These are measures of the centre of a distribution. They are single values that
give a description of the data. They are also referred to as measure of central
tendency. Some of them are arithmetic mean, geometric mean, harmonic mean,
mode, and median.
THE ARITHMETIC MEAN (A.M)
The arithmetic mean (average) of set of observation is the sum of the
observation divided by the number of observation. Given a set of a numbers , the arithmetic mean denoted by is defined by Example 1: The ages of ten students in STS 102 are determine the mean age.
Solution: .
If the numbers times respectively, the ( for short.)
Example 2: Find the mean for the table below
Scores (x) 2 5 6 8
Frequency (f) 1 3 4 2
Solution
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.
Calculation of mean from grouped data
If the items of a frequency distribution are classified in intervals, we make the
assumption that every item in an interval has the mid-values of the interval and we
use this midpoint for . Example 3: The table below shows the distribution of the waiting items for some
customers in a certain petrol station in Abeokuta.
Fin
d the average waiting time of the customers.
Solution:
Waiting (in min) No of customers Class mark mid-value(X) 1.5 1.9 3 1.7 5.1 2.0 2.4 10 2.2 22 2.5 2.9 18 2.7 48.6 3.0 3.4 10 3.2 32 3.5 3.9 7 3.7 25.9 4.0 4.4 2 4.2 8.4
= 2.84
Waiting time(in
mins)
1.5 1.9 2.0 2.4 2.5 2.9 3.0 3.4 3.5 3.9 4.0 4.4 No. of customers 3 10 18 10 7 2
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Use of Assume mean
Sometimes, large values of the variable are involve in calculation of mean, in
order to make our computation easier, we may assume one of the values as the
mean. This if A= assumed mean, and d= deviation of from A, i.e. Therefore, since . If a constant factor C is used then
.
Example 4: The exact pension allowance paid (in Nigeria) to 25 workers of a
company is given in the table below.
Pension in N 25 30 35 40 45
No of person 7 5 6 4 3
Calculate the mean using an assumed mean 35.
Solution
Pension in N frequency 25 7 25 35 = -10 - 70 30 5 30 35 = -5 - 25 35 A 6 35 35 = 0 0 40 4 40 35 = 5 20 45 3 45 35 = 10 30 25 - 45
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Example 5: Consider the data in example 3, using a suitable assume mean, compute
the mean.
Solution:
Waiting time 1.5 1.9 3 1.7 -1 -3 2.0 2.4 10 2.2 -0.5 -5 2.5 2.9 18 2.7 A 0 0 3.0 3.4 10 3.2 0.5 5 3.5 3.9 7 3.7 1 7 4.0 4.4 2 4.2 1.5 3 50 7
NOTE: It is always easier to select the class mark with the longest frequency as the
assumed mean.
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ADVANTAGE OF MEAN
The mean is an average that considers all the observations in the data set. It is single
and easy to compute and it is the most widely used average.
DISAVANTAGE OF MEAN
Its value is greatly affected by the extremely too large or too small observation.
THE HARMONIC MEAN (H.M)
The H.M of a set of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers. It is used when dealing with the rates of the
type per (such as kilometers per hour, Naira per liter). The formula is expressed thus:
H.M If has frequency , then H.M Example: Find the harmonic mean of . Solution:
H.M . Note:
(i) Calculation takes into account every value
(ii) Extreme values have least effect
(iii) The formula breaks down when o is one of the observations.
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THE GEOMETRIC MEAN(G.M)
The G.M is an analytical method of finding the average rate of growth or
decline in the values of an item over a particular period of time. The geometric mean
of a set of number is the root of the product of the number. Thus G.M If is the frequency of , then G.M
Example: The rate of inflation in fire successive year in a country was . What was the average rate of inflation per year? Solution:
G.M Average rate of inflation is Note: (1) Calculate takes into account every value.
(2) It cannot be computed when 0 is on of the observation.
Relation between Arithmetic mean, Geometric and Harmonic
In general, the geometric mean for a set of data is always less than or equal to
the corresponding arithmetic mean but greater than or equal to the harmonic mean.
That is, H.M G.M A.M The equality signs hold only if all the observations are identical.
THE MEDIAN
This is the value of the variable that divides a distribution into two equal
parts when the values are arranged in order of magnitude. If there are (odd)
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observation, the median is the center of observation in the ordered list. The location of the median is th item. But if is even, the median is the average of the two middle observations in the ordered list.
i.e.
Example 1: The values of a random variable are given as Find the median.
Solution: In an array: . is odd, therefore The median, Example 2: The value 0f a random variable are given as and . Find the median. Solution: is odd. Median,
Calculation of Median from a grouped data
The formula for calculating the median from grouped data is defined as
Where: Lower class boundary of the median class
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Total frequency Cumulative frequency before the median class Frequency of the median class. Class size or width.
Example3: The table below shows the height of 70 men randomly selected at Sango
Ota.
Height 118-126 127-135 136-144 145-153 154-162 163-171 172-180
No of rods 8 10 14 18 9 7 4
Compute the median.
Solution
Height Frequency Cumulative frequency
118 126 8 8 127 135 10 18 136 144 14 32 145 153 18 50 154 162 9 59 163 171 7 66 172 180 4 70 70 . The sum of first three classes frequency is 32 which therefore means that the median lies in the fourth class and this is the median class. Then
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.
ADVANTAGE OF THE MEDIAN
(i) Its value is not affected by extreme values; thus it is a resistant measure of
central tendency.
(ii) It is a good measure of location in a skewed distribution
DISAVANTAGE OF THE MEDIAN
1) It does not take into consideration all the value of the variable.
THE MODE
The mode is the value of the data which occurs most frequently. A set of data
may have no, one, two or more modes. A distribution is said to be uni-model,
bimodal and multimodal if it has one, two and more than two modes respectively.
E .g: The mode of scores 2, 5, 2, 6, 7 is 2.
Calculation of mode from grouped data
From a grouped frequency distribution, the mode can be obtained from the
formula.
Mode, Where: lower class boundary of the modal class Difference between the frequency of the modal class and the class before it.
Difference between the frequency of the modal class and the class after it.
Class size.
Example: For the table below, find the mode.
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Class 11 20 21 30 31 40 41 - 50 51 60 61 70 frequency 6 20 12 10 9 9
Calculate the modal age.
Solution: Mode,
ADVANTAGE OF THE MODE
1) It is easy to calculate.
DISADVANTAGE OF THE MODE
(i) It is not a unique measure of location.
(ii) It presents a misleading picture of the distribution.
(iii) It does not take into account all the available data.
Exercise 3
1. Find the mean, median and mode of the following observations: 5,
6,10,15,22,16,6,10,6.
2. The six numbers 4, 9,8,7,4 and Y, have mean of 7. Find the value of Y.
3. From the data below
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Class 21 23 24 26 27 29 30 32 33 35 36 38 37 41 Frequency 2 5 8 9 7 3 1
Calculate the (i)Mean (ii)Mode (iii) Median
MEASURES OF PARTITION
From the previous section, weve seen that the median is an average that divides a distribution into two equal parts. So also these are other quantity that
divides a set of data (in an array) into different equal parts. Such data must have
been arranged in order of magnitude. Some of the partition values are: the quartile,
deciles and percentiles.
THE QUARTILES
Quartiles divide a set of data in an array into four equal parts.
For ungrouped data, the distribution is first arranged in ascending order of
magnitude.
Then
First Quartiles: Second Quartile: Third Quartile:
For a grouped data
Where The quality in reference
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Lower class boundary of the class counting the quartile Total frequency Cumulative frequency before the class The frequency of the class Class size of the class.
DECILES
The values of the variable that divide the frequency of the distribution into
ten equal parts are known as deciles and are denoted by the fifth deciles is the median.
For ungrouped data, the distribution is first arranged in ascending order of
magnitude. Then
For a grouped data Where
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PERCENTILE
The values of the variable that divide the frequency of the distribution into
hundred equal parts are known as percentiles and are generally denoted by The fiftieth percentile is the median.
For ungrouped data, the distribution is first arranged in ascending order of
magnitude. Then
For a grouped data,
Where
Example: For the table below, find by calculation (using appropriate expression)
(i) Lower quartile, (ii) Upper Quartile, (iii) 6th Deciles,
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(iv) 45th percentile of the following distribution
Mark 20 29 30 39 40 49 50 59 60 69 70 79 80 89 90 99 Frequency 8 10 14 26 20 16 4 2
Solution
Marks frequency cumulative frequency
20 29 8 8 30 39 10 18 40 49 14 32 50 59 26 32 60 69 20 58 70 79 16 78 80 89 4 98 90 99 2 100 100
(i) Lower quartile, w (ii) Upper Quartile,
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(iii)
(iv)
MEASURES OF DISPERSION
Dispersion or variation is degree of scatter or variation of individual value of a
variable about the central value such as the median or the mean. These include
range, mean deviation, semi-interquartile range, variance, standard deviation and
coefficient of variation.
THE RANGE
This is the simplest method of measuring dispersions. It is the difference
between the largest and the smallest value in a set of data. It is commonly used in
statistical quality control. However, the range may fail to discriminate if the
distributions are of different types.
Coefficient of Range
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SEMI INTERQUARTILE RANGE This is the half of the difference between the first (lower) and third quartiles
(upper). It is good measure of spread for midrange and the quartiles.
THE MEAN/ABSOLUTE DEVIATION
Mean deviation is the mean absolute deviation from the centre. A measure of
the center could be the arithmetic mean or median.
Given a set of the mean deviation from the arithmetic mean is defined by:
In a grouped data
Example1: Below is the average of 6 heads of household randomly selected from a
country. 47, 45, 56, 60, 41, 54 .Find the
(i) Range
(ii) Mean
(iii) Mean deviation from the mean
(iv) Mean deviation from the median.
Solution:
(i) Range
(ii) Mean ( )
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(iii) Mean Deviation
(iv) In array: 41, 45, 47, 54, 56, 60
Median
Example2: The table below shown the frequency distribution of the scores of 42
students in MTS 101
Scores 0 9 10 19 20 29 30 39 40 49 50 59 60 69 No of student 2 5 8 12 9 5 1
Find the mean deviation from the mean for the data.
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Solution:
Classes midpoint 0 9 4.5 2 9 -29.52 29.52 59.04 10 19 14.5 5 72.5 -19.52 19.52 97.60 20 29 24.5 8 196 -9.52 9.52 76.16 30 39 34.5 12 414 0.48 0.48 5.76 40 49 44.5 9 400.5 10.48 10.48 94.32 50 59 54.5 5 272.5 20.48 20.48 102.4 60 69 64.5 1 64.5 30.48 30.48 30.48 42 1429 465.76
THE STANDARD DEVIATION
The standard deviation, usually denoted by the Greek alphabet (small signal for population is defined as the positive square root of the arithmetic mean of the squares of the deviation of the given observation from their arithmetic mean. Thus, given as a set of observations, then the standard deviation is given by:
(Alternatively, . )
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For a grouped data
The standard deviation is computed using the formula
If A= assume mean and is deviation from the assumed mean, then Note: We use when using sample instead of the population to obtain the standard deviation.
MERIT
(i) It is well defined and uses all observations in the distribution.
(ii) It has wider application in other statistical technique like skewness,
correlation, and quality control e.t.c
DEMERIT
(i) It cannot be used for computing the dispersion of two or more
distributions given in different unit.
THE VARIANCE
The variance of a set of observations is defined as the square of the standard
deviation and is thus given by
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COEFFICIENT OF VARIATION/DISPERSION
This is a dimension less quantity that measures the relative variation between two
servers observed in different units. The coefficients of variation are obtained by
dividing the standard deviation by the mean and multiply it by 100. Symbolically
The distribution with smaller C.V is said to be better.
EXAMPLE3: Given the data 5, 6, 9, 10, 12. Compute the variance, standard deviation
and coefficient of variation
SOLUTION Hence C.V
EXAMPLE4: Given the following data. Compute the
(i) Mean
(ii) Standard deviation
(iii) Coefficient variation.
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Ages(in years) 50 54 55 59 60 64 65 69 70 74 75 79 80 84 Frequency 1 2 10 12 18 25 9
SOLUTION
Classes 50 54 52 1 52 -20.06 402.40 55 59 57 2 114 -15.06 453.61 60 64 62 10 620 -10.06 1012.04 65 69 67 12 804 -5.06 307.24 70 74 72 18 1296 -0.06 0.07 75 79 77 25 1925 4.94 610.09 80 84 82 9 738 9.94 889.23 77 5549 3674.68
C.V
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Exercise 4
The data below represents the scores by 150 applicants in an achievement text for
the post of Botanist in a large company:
Scores 10 19 20 29 30 39 40 49 50 59 60 69 70 79 80 89 90 99 Frequency 1 6 9 31 42 32 17 10 2
Estimate
(i) The mean score
(ii) The median score
(iii) The modal score
(iv) Standard deviation
(v) Semi interquartile range (vi) D4
(vii) P26
(viii) coefficient of variation
SKEWNESS AND KURTOSIS
Skewness means lack of symmetry. We study skewness to have an idea about the shape of the curve which we can draw with the help of the given data. If in
a distribution mean in median mode, then that distribution is known as symmetrical distribution. If in a distribution, mean median mode, then it is not a symmetrical distribution and it is called a skewed distribution and such a
distribution could either be positively skewed or negative skewed.
(a) Symmetrical distribution:
mean median mode
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Mean, median and mode coincide and the spread of the frequencies is the
same on both sides of the centre point of the curve.
(b) Positively Skewed distribution:
mode median mean
In a positive skewed distribution, the value of the mean is the maximum and
that of the mode is the least, and the median lies in between the two. The
frequencies are spread out over a grater range of values on the right hand side
than they are on the left hand side.
(c) Negatively Skewed distribution:
mean median mode
In a negatively skewed distribution, the value of the mode is the maximum
and that of the mean is the least. The median lies in between the two. The
frequencies are spread out over a greater range of values on the left hand side
than they are on the right hand side.
Measures of Skewness
The important measures of skewness are:
(i) Karl-Pearsons coefficient of skewness (ii) Bowleys coefficient of skewness
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(iii) Measures of skewness based on moments.
Karl-Pearsons coefficient of skewness This is given by:
Karl-Pearsons Coefficient Skewness . In case of mode is ill-defined, the coefficient can be determined by the formula:
Coefficient of Skewness .
Bowleys coefficient of skewness This is given by: Bowleys Coefficient of Skewness .
Measure of skewness based on moment
First, we note the following moments:
1st moment about mean: ; . 2nd moment about mean: ; . 3rd moment about mean: ; . 4th moment about mean: ; . Then the measure of skewness based on moments denoted by is given by: .
KURTOSIS
The expression Kurtosis is used to describe the peakedness of a normal curve. The three measures central tendency, dispersion and skewness, describe the characteristics of frequency distributions. But these studies will not give us a
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clear picture of the characteristics of a distribution. Measures of kurtosis tell us the
extent to which a distribution is more picked or more flat topped than the normal
curve, which is symmetrical end bell shaped, is designated as Mesokurtic. If a curve is relatively more narrow and peaked at the top, it is designated as
Leptokurtic. If the frequency curve is more flat than normal curve, it is designated
as Platykurtic.
Measures of Kurtosis
The measure of kurtosis of a frequency distribution based on moment is denoted by and is given by: . If , the distribution is said to be normal and the curve is Mesokurtic. If , the distribution is said to be more peaked and the curve is Leptokurtic. If , the distribution is said to be flat peaked and the curve is Platykurtic.
Example 1: Calculate Karl Pearsons coefficient of skewness for the following data: 25, 15, 23, 40, 27, 25, 23, 25, 20.
P
M
L
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Solution: Computation of mean and standard deviation using assume mean method:
Size Deviation from A 25 25 0 0
15 -10 100
23 -2 4
40 15 225
27 2 4
25 0 0
23 -2 4
25 0 0
20 -5 25
Mean Mode , as this size of item repeats three times. Karl Pearsons coefficient of skewness
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Example 2: Find Karl Pearsons coefficient of skewness for the given distribution: Class 0 5 6 11 12 17 18 23 24 29 30 35 36 41 42 47 F 2 5 7 13 21 16 8 3
Solution: Mode lies in 24 29 group which contains the maximum frequency. Mode Computation of mean and standard deviation:
Class Mid Point ( ) 0 5 2.5 2 5 -23.28 541.96 1083.92 6 11 8.5 5 42.5 -17.28 298.59 1492.95 12 17 14.5 7 101.5 -11.28 127.24 890.68 18 23 20.5 13 266.5 -5.28 27.88 362.44 24 29 26.5 21 556.5 0.72 0.52 10.92 30 35 32.5 16 520 6.72 45.16 722.56 36 41 38.5 8 308 12.72 161.79 1294.32 42 47 44.5 3 133.5 18.72 350.44 1051.32
Mean
Standard deviation
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. Therefore,
Karl Pearsons coefficient of skewness
Example 3: Find the Bowleys coefficient of skewness for the following series: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22.
Solution: The given data in order: 2, 4, 6, 10, 12, 14, 16, 18, 20, and 22.
size of th item size of th item size of 3rd item
size of th item size of th item size of 9th item Median size of th item size of th item size of 6th item Bowley coefficient of skewness
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Since skewness , the given series is a symmetrical data.
Example 4: Calculate (measure of skewness based on moment) and (measure of kurtosis based on moment) for the following data.
X 0 1 2 3 4 5 6 7 8
F 5 10 15 20 25 20 15 10 5
Solution:
First Moment: where Second Moment: Third Moment: Fourth Moment: . That is, symmetrical curve. . The value of , hence the curve is Platykurtic.
Example 5: For the data below, determine the and . Mark 30 33 34 37 38 41 42 45 46 49 50 53 Frequency 2 4 26 47 15 6
Solution:
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Mark Mid Point ( ) 30 33 31.5 2 63 -11.48 131.79 263.58 34 37 35.5 4 142 -7.48 55.95 223.80 38 41 39.5 26 1027 -3.48 12.11 314.87 42 45 43.5 47 2044.5 0.52 0.27 12.71 46 49 47.5 15 712.5 4.52 20.43 306.46 50 53 51.5 6 309 8.52 72.59 435.54 100 4298 1556.96
-1512.95 -3025.91 17368.71 34737.42
-18.51 -1674.04 3130.45 12521.79
-42.14 -1095.64 146.66 3813.16
0.14 6.58 0.07 3.44
92.35 1385.25 417.40 6261.02
618.47 3710.82 5269.37 31616.22
-692.94 88952.90
. Since , the curve is Leptokurtic.
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Exercise 5
Calculate the and for the data below and interpret your result. Class 10 14 15 19 20 24 25 29 30 34 35 39 40 44 45 49 f 1 4 8 19 35 20 7 5
Rates, Proportion and Index Numbers
Proportion
Proportion is the ratio of a number of items with certain characteristics (X)
and the total number of items exposed to such characteristics (N). It is defined as
. The above expresses the chance of occurrences of such characteristics (i.e.
Probability of event X).
Example: If the voting age population (people 18 years and above) in Ifo ward I
consists of 550 males and 600 females. What is the proportion of males?
Solution: , Total population . Proportion of males, .
Rates
When proportion refers to the number of events or cases occurring during
certain period of time, it becomes a rate and is usually expressed as so many per
1000. Thus we refer to birth rate as the number of birth per 1000 population in a
year. So also we have death rate, migration rate, marriage rate etc.
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INDEX NUMBERS
There are various types of index numbers, but in brief, we shall discuss three
kinds, namely: (a) Price Index (b) Quality Index (c) Value Index.
(a) Price Index
For measuring the value of money, in general, price index is used. It is an
index number which compares the prices for a group of commodities at a certain
time as at a place with prices of a base period. There are two price index numbers
such as whole sale price index numbers and retail price index numbers. The
wholesale price index reveals the changes into general price level of a country, but
the retail price reveals the changes in the retail price of commodities such as
consumption of goods, bank deposit etc.
(b) Quantity Index
This is the changes in the volume of goods produced or consumed. They are
useful and helpful to study the output in an economy.
(c) Value Index
Value index numbers compare the total value of a certain period with total
value in the base period. Here a total vale is equal to the price of commodity
multiplied by the quantity consumed.
NOTATION: For any index number, two time periods are needed for comparison.
These are called the Base period and the Current year. The period of
the year which is used as a basis year and the other is the current year.
The various notations used are as given below:
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Price of current year Price of base year Quantity of current year Quantity of base year Definition (The weight): Different weights are used in different part of the country
for a particular commodity. For instance, congo western part of Nigeria, mudu northern part etc.
Problems in the construction of Index numbers
No index number is an all purpose index number. Hence, there are many
problems involved in the construction of index numbers, which are to be tackled by
an economist or statistician. They are:
1. Purpose of the index numbers
2. Selection of base period
3. Selection of items
4. Selection of source of data
5. Collection of data
6. Selection of average
7. System of weighting
METHOD OF CONSTRUCTION OF INDEX NUMBERS
Index numbers may be constructed by various methods as shown below:
INDEX NUMBERS
Un weighted Weighted
Simple
average of
price
relative
Simple
aggregate
index
numbers
Weighted
aggregate
index
number
Weighted
average of
price
relative
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Simple Aggregate Index Number
This is the simplest method of construction of index numbers. The prices of
the different commodities of the current year are added and the sum is divided by
the sum of the prices of those commodities by 100. Symbolically,
S.A.P.I, Where total prices for the current year total prices for the base year And we noted that Price Relative Index .
Example 1: Calculate index numbers from the following data by simple aggregate
method taking prices of 2007 as base.
Commodity Price per unit (in N)
2007 2013
A 80 95
B 50 60
C 90 100
D 65 97
Solution:
Commodity Price per unit (in N)
2007 2013
A 80 95
B 50 60
C 90 100
D 65 97
Total 285 352
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S.A.P.I,
Simple Average Price Relative Index
In this method, first calculate the price relative for the various commodities
and then average of these relative is obtained by using arithmetic mean and
geometric mean. When arithmetic mean is used for average of price relative, the
formula for computing index is
S.A. of P. R. by arithmetic mean where n is the number of items. When geometric mean is used, the following formula is adopted:
Example 2: From the following data, construct an index for 2012 taking 2011 as
base by the average of price relative using (a) arithmetic mean (b) geometric mean.
Commodity Price in 2011 Price in 2012
Beans 50 70
Elubo 40 60
Rice 80 100
Garri 20 30
Solution:
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Commodity Price in 2011 ( ) Price in 2012( ) Beans 50 70 140 2.1461
Elubo 40 60 150 2.1761
Rice 80 100 125 2.0969
Garri 20 30 150 2.1761
Total 565 8.5952
(a) Simple average of price relative index by arithmetic mean
(b) Price relative index number using geometric mean
.
Weighted Aggregate Index Number
In order to attribute appropriate importance to each of the items used in an
aggregate index number, some reasonable weights must be used. There are various
methods of assigning weights and consequently index numbers have been devised
of which some of the most important ones are:
1. Laspeyres method
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2. Paasches method 3. Fishers )deal method 4. Bowleys method 5. Marshall Edgeworth method 6. Kellys method
. Laspeyres method This is a weighted aggregate price index, where the weights are determined
by quantity in the based period and is given by:
Laspeyres price index .
. Paasches method It is a weighted aggregate price index in which the weight are determined by
the quantities in the current year. The formula for constructing the index is given
by:
Paasches price index . 3. Fishers Ideal method This is the geometric mean of the Laspeyre and Paasche indices. Symbolically,
Fishers )deal index method .
4. Bowleys method This is the arithmetic mean of Laspeyres and Paasches method. Symbolically, Bowleys price index number .
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5. Marshall Edgeworth method In this method, the current year as well as base year prices and quantities are
considered. The formula in using to get this is given as follows:
Marshall Edgeworth price index .
6. Kellys method Kelly has suggested the following formula for constructing the index
number:
Kellys price index number where . Here the average of the quantities of two years is used as weights.
Example 3: Construct price index number from the following data by applying (a) Laspeyres method b Paasches method and c Fishers ideal method. Commodity 2000 2012
Price Quantity Price Quantity
A 2 8 4 5
B 5 12 6 10
C 4 15 5 12
D 2 18 4 20
Solution:
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Commodity A 2 8 4 5 16 10 32 20
B 5 12 6 10 60 50 72 60
C 4 15 5 12 60 48 75 60
D 2 18 4 20 36 40 72 80
172 148 251 220
L.A.P.I.N .
P.R.I.N .
F.I.I.N
Interpretation: The results can be interpreted as follows.
If N100 were used in the base year to buy the given commodities, we have to use
N145.93k in the current year to buy the same amount of the commodities as per the Laspeyres formula. Other values give similar meaning.
Example 4: Calculate the index number from the following data by applying
(a) Bowleys method b Marshall Edgeworth price index.
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Commodity Base year Current year
Quantity Price Quantity Price
A 10 3 8 4
B 20 15 15 20
C 2 25 3 30
Solution:
Commodity A 10 3 8 4 30 24 40 32
B 20 15 15 20 300 225 400 300
C 2 25 3 30 50 75 60 90
380 324 500 422
(a) B.P.I.N .
(b) M.E.P.I.N .
Example 5: Calculate a suitable price index from the following data.
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Commodity Quantity Price
1999 2000
X 20 2 4
Y 15 5 6
Z 8 3 2
Solution: (ere the quantities are given in common and so we can use Kellys index price number.
K.P.I.N .
Remark:
1. For Quantity or Volume index number, interchange and in the formula for Laspeyre, Paasche and Fishers ideal. 2. For Consumer Price index number:
(a) Aggregate Expenditure method:
Consumer Price Index Number (based on Laspeyre) (b) Family Budget method or Method of Weighted Relative
Consumer Price Index Number where and value weight i.e. .
Exercise 6
1. Five feed components are to be used in the construction of an animal
feedstuff index number. From the figures given in the following table,
calculate
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(i) simple average of price relative index by arithmetic mean and by
geometric mean.
(ii) Laspeyres price index number (iii) Paasche price index number (iv) Fishers ideal index number (v) Bowleys price index number (vi) Marshall Edgeworth price index number (vii) Kellys price index number
Component 1914 2014
Price per ton Consumption(tons) Price per ton Consumption
A 40 3600 41 2750
B 39 2750 53 1500
C 38 2050 35 2350
D 37 500 30 750
E 36 1475 24 2850
2. From the following data, compute quantity indices by i Laspeyres method ii Paasches method iii Fishers method.
2008 2010
Commodity Price Total value Price Total value
X 10 100 12 180
Y 12 240 15 450
Z 15 225 17 340
Hint: Quantity
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3. Construct the consumer price index number for 2012 on the basis of 2002
from the following data using Aggregate expenditure method.
Price in
Commodity Quantity consumed 2002 2012
A 100 8 12
B 25 6 7
C 10 5 8
D 20 15 18
4. Calculate consumer price index by using Family Budget method for year 2013
with 2012 as base year from the following data.
Price in
Items Weights 2012 (in N) 2013 (in N)
Food 35 150 140
Rent 20 75 90
Clothing 10 25 30
Fuel 15 50 60
Miscellaneous 20 60 80