UGANDA MANAGEMENT INSTITUTE

DPAM 2014/2015

Philip Odida: 14/DPAM/000/KLA/EVE/0040

Module: Quantitative Methods in Decision Making

Module lecture: Kimbugwe Hassan, Consultant,

Uganda Management Institute

Individual Coursework:

7th November, 2014

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The following data represents the monthly income (Shs’ 000) of 80 randomly selected

villagers after a poverty reduction exercise had been carried out in Goma sub-county. Before

the exercise, those in charge of the poverty reduction program had carried out a baseline

survey and established that, on average, the monthly income was Shs 26,500 per month.

Table 1. Sample data collected from Goma sub-county.

22 23 10 17 39 54 33 47

16 33 24 10 47 31 42 22

40 23 38 29 25 37 22 12

15 14 24 22 52 13 48 27

27 29 24 45 28 45 21 44

49 23 16 24 23 23 29 13

22 34 31 29 22 43 15 32

17 21 21 24 35 37 19 36

25 20 21 39 41 44 39 23

14 24 21 38 23 28 53 37

i. Simple random sampling and purposive sampling techniques will be employed for

selection of the sample size. These sampling methods will be as follows: To begin

with, four groups different parish will be randomly sampled for the study from Goma

Sub County. Then purposive sampling will be to select respondents. As for the simple

random sampling, the lottery technique will be used. Simple random sampling method

is appropriate because every item in the population has a calculable or known chance

of being included in the sample. This will end up giving reliable data for decision

makers.

The hypothetical illustration below shows the use of Slovin’s formula in determining

the minimum sample size, assuming that the population is made up of 80,000

respondents.

The Slovin’s formula was used to determine the minimum sample size.

n =

1

N

1+Na2

Where:

N=Target population

n=Sample size

α=0.05 that is the level of significance

n = 80

1 + 80 (0.05)²

n = 80

1 + 80 (0.0025)

n = 80

1 + 0.5075

n = 53

The sample size was selected from the study categories. Table 1 shows the distribution of

population and sample size.

ii. Frequency distribution table for randomly selected villagers in Goma sub-county

.

Class Tally F

10 – 19 IIII IIII IIII 14

20 – 29 IIII IIII IIII IIII IIII IIII IIII 35

30 – 39 IIII IIII IIII I 16

40 – 49 IIII IIII II 12

50 - 59 III 3

80

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iii. Computation of measures of central tendencies and measures of dispersions

Class Tally F X fx fx2 cf

10 – 19 IIII IIII IIII 14 14.5 203 2,943.5 14

20 – 29 IIII IIII IIII IIII IIII IIII IIII 35 24.5 857.5 21,008.72 49

30 – 39 IIII IIII IIII I 16 34.5 552 19,044 65

40 – 49 IIII IIII II 12 44.5 534 23,763 77

50 - 59 III 3 54.5 163.5 8,910.75 80

80 2,310 75,669.97

- The Mean of the sample (ㄡ) is derived from the following equation:

ㄡ = ΣfxΣf

ㄡ = 2,310

80

ㄡ = 28.875

- The Median of the sample (Md) is derived from the following equation:

Formula for median is given as

L + ¿)x C

The median class is that which lies 1/2th the population, 802 therefore

median class is class 20 - 29

L = 19.5

N2 = 80

2 = 40

Cfb = 14

fm = 35

C = 10

Substituting:

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Median = L + ( N

2−cfb)

fmx c

= 19.5 +( 80

2– 14)

35 x 10

= 19.5 + (40– 14)35

x 10

Md = 26.93

- The Mode of the frequency distribution is derived from the following equation:

L +Δ1

Δ1+Δ2 x C

Modal Class is class 20-29

L = 19.5

Δ1 = 35-14 = 21

Δ2 = 35-16 = 19

C = 10

Modal monthly income = 19.5 + 21

21+19x 10

= 19.5 + 2140 x 10

Mo = 24.75

Generally, in a frequency distribution, the values cluster around a central value. This

property of concentration of the values around a central value is called Central Tendency. The

central value around which there is concentration is called measure of central tendency

(measure of location, average).

The mean, median and mode of the frequency distribution is 28.91, 26.93, 24.75

respectively. The median of the class calculated in the above exercise (26.93), is closest to the

result of the baseline survey that established that on average, the monthly income of the 80

randomly selected villagers in Goma sub-county was 26,500 shillings.

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Desired qualities of an ideal measure of central tendency.

It should be easy to understand. Its computation procedure should be simple

It should be rigidly defined

It should be based on all the values

It should not be affected too much by abnormal extreme values

It should be capable of further algebraic treatment so that it could be used in

further analysis of the data

It should be stable. That is, the measure should be such that sampling variation

in the value of the measure should be least.

Basing on these qualities, mean stands up to be the best measure of central tendency

because it is an Algebraic sum of the deviations of a set of values from their arithmetic mean

is zero. That is, Sum of the squared deviations of a set of values is a minimum when

deviations are taken around the arithmetic mean.

iv. The data obtained in the random sample above pertain to the measures of central

tendency, which are useful for public administration and management in the sense that

they serve as a basis for analysis in a manner that can contribute to scientifically

informed decision making. In management terms, the data serves essentially as a basis

for planning, decision making with a view to reasonable resource allocation.

v. The standard deviation (σ) of the frequency distribution above is derived from the

following equation:

σ = √∑fx 2 - ㄡ 2

∑f

Where ∑fx2 = 75,669.97; ∑f = 80 and ㄡ= 28.875

σ = √75,669.97 - (28.875)2

80

Therefore, standard deviation is 10.588

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- The Coefficient of variation of the incomes in the frequency distribution is derived

from the following equation:

COV = SD

(r∨ㄡ)X 100 where r = ㄡ

COV = 10.588

(28.875) X 100

COV = 36.69%

Standard Deviation is the abbreviation and (read, sigma) is the symbol. Mean square

deviation of the value from their arithmetic mean is variance.

Standard Deviation is the positive square root of variance. Karl Pearson introduced the

concept of standard deviation in 1893. Standard Deviation is also called mean square

deviation.

It is a mathematical deficiency of mean deviation to ignore negative sign. Standard

deviation possesses most of the desirable properties of a good measure of dispersion.

It is the most widely used absolute measure of dispersion. The corresponding relative

measure is coefficient of variation.

Karl Pearson gave the definition of coefficient of variation, in the perspective that like

all other relative measures of dispersion, it is a pure number. All relative measures of

dispersion are free from units of measurement such as kg., metre, litre, etc. The

variations in two or more series (groups or sets of data) are compared based on a

relative measure of dispersion. For example, a residence in Goma Sub County may

have different income at various periods. His income is quoted in Ugx. The variations

in their incomes can be compared by using any relative measure of dispersion.

Coefficient of variation is the more widely used relative measure of dispersion and the

best measure of central tendency. It is a percentage. While comparing two or more

groups, the group, which has less coefficient of variation, is less variable or more

consistent or more stable or uniform or more homogeneous.

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i) Hypothesis testing

HO: µ = 26,500

H1: µ ≠ 26,500

N = 80

µ = 26,500

ㄡ = 28.875

SD = 10.588

ð = 0.05

Z = 1.96

Therefore;

Z =ㄡ - µ SD/√N

Z =28.875 – 26.5

10.588/√80

Z = 2.006

Comment: since the calculated value of Z as 2.006 is greater than 1.96 and falls

outside the area of acceptance bound by + or – 1.96 we cannot accept the null

hypothesis

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Brief Report to decision makers in Goma Sub County

This exercise was devoted to different measures used to summarize the data as well as

the measures of variation. Different measures discussed are Mean, Median and Mode.

Definition, method of computation, Interpretation and uses form the structure of the

explanation for each measure.

Generally, in a frequency distribution, the values cluster around a central value. This

property of concentration of the values around a central value is called Central

Tendency. The central value around which there is concentration is called measure of

central tendency (measure of location, average).

Generally, a simple comparison of frequency distribution is made by comparing their

measures of central tendency. For a frequency distribution, important measures of

central tendency are defined. They are: 1. Arithmetic Mean 2. Median and 3. Mode

Basing on thèse qualités, mean stands up to be the best measure of central tendency

because it is an Algebraic sum of the deviations of a set of values from their arithmetic

mean is zero. That is, Sum of the squared deviations of a set of values is a minimum

when deviations are taken around the arithmetic mean. Further, the following can be

concluded:

i. Residents from Goma Sub County have an average monthly income of

Ugx. 28,875

ii. The income distribution is uneven having a variation of Ugx 10,588

iii. It can be therefore concluded that there is no significance difference

between the average monthly incomes from the baseline survey of Ugx.

26,500 and that computed on Ugx. 28,875

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Reference

i. Xu, Gang. “Estimating Sample Size for a Descriptive Study in Quantitative

Research .” Quirk’s Marketing Research Review, June 1999.

ii. William Kaberuka. (2003). Statistical techniques, a basic course for social

scientists

iii. Chandan Sigh (2006). Business Statistics. Second revised edition

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