Measures Of Central Tendency

& Dispersion.

ByTwino Ivan

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Session Content• Explain Measures of Central Tendency

as used in quantitative methods.• State the commonly used measures of

central tendency• Compute some measures of central

tendency using standard formulae.• Discuss their application in

management.Ivan Twino 2

Measures of Central tendency• Measures of Central tendency or location

describe the centre of the entire data set.• They indicate the location to which items

in the data set “tend to concentrate”.• They indicate where the centre of the

distribution of data is located on the scale that is being used.

• The most commonly used measures of central tendency are; mean (Arithmetic mean), median & mode.

• Measures of central tendency are measures of the location of the middle or the center of a distribution. Ivan Twino 3

Measures of Central tendency……

• Therefore, any statistical measure indicating the centre of a set of the data arranged in an increasing or decreasing order of magnitude is called the measure of central tendency (measure of central location).

• Measures of central tendency are appropriate for quick decision making among other uses in management.

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NOTE:• To clearly get the whole story of Measures of

Central tendency and dispersion as descriptive statistics, the investigator needs to organize and present the raw data collected into a frequency distribution table from which some descriptive statistic can then be computed or derived to simplify data analysis, interpretation and decision making /conclusion in relation to the problem being researched on. Refer to the case study in the next slide.

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QM Case Study The ministry of works is planning to construct a

dual carriage road from Kampala to Entebbe International Airport. In their pilot survey, it was found out that many people are to be compensated if the project is implemented. To get the levels of compensation estimates, the ministry then collected information from a sample of 50 households on amounts of their property values (in”000” $) for compensation as in the table 1 below:

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Table 1:Group/organize this data and construct the

frequency distribution table70 41 34 55 45 66 73 77 80 30

50 45 72 50 27 70 55 70 85 70

30 50 60 53 40 45 35 55 20 81

25 51 35 62 60 30 45 35 50 89

53 23 28 65 68 50 65 34 35 76

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Class Work• Using table 1 above and starting with the

class of 20-29 and constant class width, construct a cumulative frequency distribution table and:

• (a)Determine the mean, median and mode of the given raw data and comment on nature of distribution of compensation.

• (b) Determine the range, quartile deviation, the standard deviation and variance of the given raw data.

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Format of cumulative frequency distribution table for Organizing Data

Class interval

Class boundary

Class Mid-Mark

(x)

Tally Bars

Frequency

(f)

Cumulative frequency

(cf)

fX X2 fX2

- - - - - - -

- - - - - - -

- - - - - - -

- - - - - - -

Total ∑f = ∑fx =

∑fX2

=

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MEAN (Arithmetic Mean)• This refers to the total or sum of observations

divided by the number of observations. • It is also defined as the ratio of the sum of

observations to the number of observations.• It is the mostly widely used measure of

central tendency.• It indicates where the center of the

distribution is located on the scale. • The mean is a single value which is

representative of all items in the population or sample.

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Mean for discrete data(Ungrouped data)

If a set of data x1, x2… xn, not necessary all distinct, represents a finite sample of size n, It mean is given by: mean = Σ Xi

n

=X1 +X2 +….+Xn

nx

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Mean for discrete data(Ungrouped data)

• Given 10,22,31,9,24,27,29,9,23,12:• Mean = ∑x n• Mean=196/10= 19.6

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Arithmetic Mean Grouped data

Mean(average) for Grouped datais given by the formula:

= Σ fX Σf

Alternatively(see next slide),13

x

Mean For Grouped Data

.

n

ii

n

iii

f

xfMean

1

1x

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mean • Simple or arithmetic average of a range of values

or quantities, computed by dividing the total of all values by the number of values. For example, the mean of 1, 2, 3, 4, and 5 is (15 divided by 5) = 3. It is the most common and best general purpose measure of the mid-point (around which all other values cluster) of a set of values, but is prone to distortion by the presence of extreme values and may require use of a measure of distortion (such as mean deviation or standard deviation). Also called arithmetic mean. See also median and mode.

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Application/uses of mean• For decision making purposes: Used as a decision

making tool, like in estimating per capita incomes, average wage payment, factor productivity, average expenditures or profit/cost per unit output, etc

• For performance assessment/analysis; and generalization / inference on certain business indicators.

• For comparison purposes (for comparative studies of different distributions or communities /countries).

• For describing the distribution in a concise manner(around which value is data distributed).

• For monitoring & evaluation purposes given the baseline information( use mean before and after intervention)

• Used for research purposes and problem identification. • For computing other various basic characteristics of a mass

of data e.g. Variances and standard deviations. • Auditing, planning and quality control, ETC

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THE MEDIAN• It is another measure of the central tendency,

which corresponds to the half of the total frequency of the distribution of organized data.

• The median is the middle value of a set of observation arranged in an increasing or decreasing order of magnitude when the number of orderly arranged discrete observations is odd.

• Or the Arithmetic mean of the two middle values when the number of observation is even in case of discrete data.

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Median for discrete data(ungrouped data)

• Middle value of distribution.• But when N is even, the median will be the

value at position (N+1)/2• Given 10,22,31,9,24,27,29,9,23,12.• Arranged in ascending order:• 9,9,10,12,22,23,24,27,29,31• (N+1)/2 position=5.5.• Then median=(22 +23)/2=22.5

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Median for Grouped data.

Cf

cfN

LMedian

)

2(

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Cont’d.Where :• L = Lower class boundary of the median class• f = frequency of the median class• cf = Cumulative frequency of the previous class• C = class width of the median class or group• N/2= Middle item.• (N/2) – cf shows the position of median in the

median class.

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Median-Grouped data

Media = Lm + N/2 – Cfbm cm

fm

Where• Lm is the lower class boundary of the median

class • N is the total number of observations • CFbm is the cumulative frequency of the class

before the median class • Cm is the class interval of the median class

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THE MODE• Mode is any value in the distribution/data that occurs

most frequently (Most frequently occurring number or item in the data set).

• The mode of a set of observations is that value which occurs most often or one with “greater frequency” than others .

• The mode can be estimated from the class with the highest frequency.

• It is used as a tool for answering some research questions in relation to “Common Responses or occurrences or observations”.

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Estimation of Mode.

CfffLMode

ba

a

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Cont’d.

• Where:• L = Lower class boundary of the modal

group• = frequency of post-modal class

(after the modal group)• = Frequency of the pre-modal (class)

(before the modal group)• C = Class width

af

af

bf

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Estimation of Mode… Alternatively; mode can also be given as:

CLMode

21

1

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Cont’dWhere :• L = Lower class boundary of the modal class or

group• fm = frequency of the modal class

1 = fm-fb = Difference between frequency of the modal class & that of the pre-modal(before)class

2 = fm-fa =Difference between frequency of the modal class & that of the post-modal(after)class.

• C = Class width of the modal classIvan Twino 26

Common shapes for explaining nature of distribution ( using frequency Vs observations)

• Normal distribution(symmetric-bell shaped): In this case observations are evenly distributed, and distribution can be divided into almost two equal parts. For such distributions, the mean, mode and median are assumed to be the same/ very close to each other.

• Positive skewness(right skewness): most observations (mode) cluster around relatively low values resulting into a longer tail to the right i.e. long right tail compared to a short left tail.In this case, values for: mean>median> mode.

• Negative skewness(Left skewness):most observations (mode) cluster around relatively high values resulting into a longer tail to the left compared to a short right tail. In this case values for: mean<median<mode.

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Pearson’s coefficient of skewness

Sk =3(mean-median)/(standard deviation)OR Coefficient of sleekness=(Mean-mode)/SD It explains the nature of distribution of

observations or data. If it is equal or approximated to zero, it implies normal distribution, otherwise negatively or positively skewed depending on the sign(+ or -) of value got whose range is within -1 to +1.

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Estimated relationship between Mean, Median and Mode

Mode= 3(Median)-2(Mean)

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Common shapes for explaining nature of distribution ( using frequency Vs observations)

• Normal distribution(symmetric-bell shaped): In this case observations are evenly distributed, and distribution can be divided into almost two equal parts. For such distributions, the mean, mode and median are assumed to be the same/ very close to each other.

• Positive skewness(right skewness): most observations (mode) cluster around relatively low values resulting into a longer tail to the right i.e. long right tail compared to a short left tail.In this case, values for: mean>median> mode.

• Negative skewness(Left skewness):most observations (mode) cluster around relatively high values resulting into a longer tail to the left compared to a short right tail. In this case, values for: mode>median> mean.

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Measures of Central TendencyThe Shape of Distributions

• With perfectly bell shaped distributions, the mean, median, and mode are identical.

• With positively skewed data, the mode is lowest, followed by the median and mean.

• With negatively skewed data, the mean is lowest, followed by the median and mode.

Median vs. Mean Values (cont.)• If the distribution of the data is skewed, the mean is pulled

(relative to the median) in the direction of the long thin tail. – For example, income is distributed in a highly skewed fashion, with a

long thin tail in the direction of higher income. Thus, mean income is typically considerably higher than median income, with low incomes earned by most/majority like classroom teacher in MOES or police in IA

skewness

.

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Session II

Measures of Dispersion

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Measures of Dispersion• They are descriptive statistics/ measures that

explain or describe the nature of variation (deviation) or degree of scatterdness of data around some measures of central tendency(mainly the mean) that exists in the distribution of data.

• The mean, median and mode only estimate the location(value) where data tend to concentrate (the centre of data distribution), but do not describe how data is spread or distributed about them.

• Hence need for use of measures of dispersion for this purpose.

• The degree to which numerical data tend to spread about an average value (mean) is called the variation or dispersion of data.

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Measures of Dispersion

• They show how the observations/data are spread about(out) from the average or mean in the entire data set in the sample or population.

• For instance how wage payments and household incomes are spread around the average wage rate and average income respectively ( depict the degree/nature of distribution or inequality or variability within the set of data in considaration)

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Measures of Dispersion…• They enable the investigator know the extent to

which the values in the distribution are dispersed around the measure of central tendency for management, monitoring, evaluation, policy and decision making purposes.

• Such measures help in comparison of distributions of certain different phenomena or two sets of data for two communities whose SD values are given (Note: one with lower SD is often preferred to the other with higher disparities and risks).

• The larger the measures of dispersion, the greater the degree of deviations of values about the mean.

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Examples of Measures of Dispersion • Some important descriptive statistics for measuring

the dispersion or variability of the set of data are: RangeVariancestandard deviationCoefficient of variation (Which is standard

deviation as a percentage of the mean is also a good measure of comparing variability between two sets of data)

C.V ={Standard deviation/mean}*100Mean Absolute deviationThe Quartile deviationETC. NOTE: Use table 1 to compute the above measures.

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RANGE• It is the difference between the highest value

and the lowest value in a given set of distribution of observations or data (ungrouped data).

• The range R of the set of data is the difference between the largest and the smallest numbers in the set. R = H - L

• Where R = Range• H = Highest value of observation• L = Lowest value of observation.• However, range is the poor measure of variation

particularly when the sample is large.

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Range For Grouped Data• This is defined as the difference between the

largest and smallest values among all observations for ungrouped data.

• For Grouped data the range is the difference between the lower class limit of the lower extreme class and the upper class limit of the upper extreme class.

• R is difference between the lower class limit of the first class interval and the upper class limit of the last class interval of the frequency distribution table for grouped data.

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Variance• Variance is the arithmetic mean of the squares of

the deviations(X-µ)2 of individual values of observations from their arithmetic mean(µ).

• Its denoted by S2 for a sample distribution and σ2

for a population.• Variance is the average of sum of square of

deviations of values from the mean(µ) and is given as Variance σ2 = ∑(X-µ)2/n.

• Or Variance σ2 = ∑f(X-µ)2/∑f .

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Variance

.

Variance σ2 = 2

fxxf

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STANDARD DEVIATION• Standard Deviation (σ) is the positive square root of

variance.• Also known as Root-mean square deviation and is

described as the “square root of the average of the sum of the square of the deviations”. The square root of the mean squared deviation from mean of distribution.

• It is the Square root of the mean of the squared deviations.

• And this gives how far above or below the mean the observations are in the data distribution.

• It indicates the spread/ variability of the data around the mean.

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Standard Deviation (for grouped data)

Standard deviation σ is expressed as:

22

ffx

ffx

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Standard Deviation (for grouped data)Alternatively,

2

f

fSD

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Importance of SD• It is an important measure of degree of dispersion or

variation of other items in the distribution from the mean.• It makes distribution observations very easily understood.

The larger the SD the greater will be the degree of deviation /dispersion from the mean. Hence, applicable in the analysis of degree and nature of distribution of income or wages of people around the average figure or per capita income in an economy.

• NOTE :If there is no dispersion, it implies that all values in the distribution are the same (SD = 0)-equality

• It used for comparison purposes given two sets of data for two communities(whose SD values are got).

• For quality control-quality assurance and Auditing against set standards

• Applied in risk analysis and decision making.• . Ivan Twino 46

• For decision making purposes• For comparison purposes of different

distributions or situations for communities /countries before and after interventions.

• For describing the nature distribution in a concise manner

• For monitoring & evaluation purposes given the baseline information( use mean before and after intervention)

• Used for research purposes and problem identification.

• Auditing and quality control purposes,• For planning and policy purposes

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Quartiles• For n observations, arranged in order of size, the lower

quartile,Q1,is the value at the position of 25% of the way through the distribution of ordered data.

• The upper quartile, Q3, is the value at the position of 75% of the way through the distribution.

• The quartiles together with the median divide the distribution into 4 equal parts.

• For instance, given the ungrouped data: 3,3,5,6,8,9,12,14,19,20,24,30

Q1 (lower quartile) =5

Q2(median) =9

Q3(upper quartile)=19

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The Quartile for grouped data(apply the formula for median)

• Where;Q3 = Lq3 + 3N/4 –Cfbq3 Cq3 f

and Q1 = Lq1 + 1N/4 –Cfbq1 Cq1

f

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The Inter-quartile range and Quartile deviation

• Inter-quartile range is the difference between the upper quartile and the lower quartile i.e. (Q3 - Q1 ).

• Quartile deviation is equal to the Inter-quartile range (Q3 - Q1 ) divided by two.

• It is half of Inter-quartile range • Simply the average of the difference

between the upper quartile and lower quartile i.e. Q3 – Q1

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Example

• The table 2 below shows the scores of 30 applicants for secretarial positions in a large project obtained in aptitude test marked out of 50 designed to measure clerical skills. Complete the table and compute the descriptive statistics:

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Illustration(Table 2)Class interval Midpoint (x) Frequency(f) fx20 – 24 22 2 4425 – 29 27 3 8130 – 34 32 6 19235 – 39 37 5 18540 – 44 42 10 42045 - 49 47 4 188Total ∑f=30 ∑fx=1110

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Group Work

• Using table 2:• (a) Determine the mean, median and the

mode scores and state managerial implication of each where possible.

• (b) Estimate the values of range , variance, the standard deviation, quartile deviation and coefficient of variation, the degree of skewness and state their managerial implication of each.

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Conclusion Since today’s decisions are driven by data,

therefore measures of central tendency and dispersion are generally vital for statistical thinking for managerial business decisions.

Such statistical models help Managers and professionals to solve problems in diversity of context, add substance to decisions, reduce guess work and decisions based entirely on personal opinions or beliefs.

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Importance of Quantitative Methods in Management

Important for decision making based on relevant and reliable quantitative data/findings(data/evidence based decisions-statistical thinking for managerial decisions) ;but not only on personal opinions /attitudes or beliefs.

Statistics helps in predicting future trends of events (Forecasting the socio- economic, business, political and organizational performance trends/outcome of events) in an economy/country, organizations /markets.

They are important in determining the rates of change in various socio-economic variables over time

In quality control, planning, budgeting for scarce resources and control of activities subject to resource constraints

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Cont…• Help to measure effects/impacts of change in one

variable(IV) on the other(DV) & rates of changes.• Applied for comparison purposes that are vital for

policy formulation , policy/ project evaluation and monitoring ,mgt purpose and comparative studies of different distributions or communities /countries e.g. Per capita income, SOL, inflation & exchange rates,etc

• The Regression/ Correlation techniques help the managers or planners and firms to know the relationship between two or more variables (e.g. sales and advertising, rewards and org performance) from which mgt decisions and policies are based.

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• Quantitative approaches are faster in providing solutions to managerial problems of what, when, how and for whom to produce using scarce resources.

• Linear programming and optimization models are used by firms and organizations to estimate optimal resource combinations for cost minimize and profits or returns maximize, hence prioritizing activities subject to certain resource constraints .

• For describing the distribution of data in a concise manner ( e.g. average- is the value around which data is distributed).

• For monitoring & evaluation purposes given the baseline information (for performance assessment/analysis)

• Used problem identification or identification of knowledge/information gaps for research purposes

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Cont’d..• Probability Theory (as part of Management Science) is important

risk analysis for decision-making or management purposes.• Statistics helps a lot in Policy making. In order to form certain

policies, Statistics provide baseline information with adequate numerical data relevant to that phenomena e.g. Policy on Education depends on data related to Educational Issues, Policy on health is also dependent on medical data etc.

• H R -management: Study of Statistical data regarding wage rates, employment trends, COL indices, employees grievances, Labor turnover rates, records of performance appraisal etc and proper analysis of such data assist employers and the personnel department in formulating the personnel policies & strategies for manpower planning and organizational development.

• Index numbers help government ,individuals and firms in analyzing changes in certain variables ( prices, exchange rates , C OL ,poverty levels ,etc) overtime for appropriate interventions

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