Lecture 8 STA 103

STA 103: PRINCIPLES OF STATISTICS
TOPIC EIGHT
DESCRIPTIVE STATISTICS - MEASURES OF DISPERSION / VARIABILITY /
MEASURES OF SPREAD
Objectives
By the end of this topic the learner should be able to:

(a) Discuss the nature merits and demerits of the various measures of dispersion /
variability.
(b) Compute and interpret the various measures of dispersion
(c) (i) What do you understand by the term normal distribution
(ii) Discuss the properties of normal distribution
1.1 Introduction
These are the measures that describe the extent to which individual differences within the group
that occurs
The following are the different measures of variability
(i) Range
(ii) Mean deviation (MD)
(iii) Variance (S2)
(iv) Standard deviation
8.2 Range
This measures the distance between the highest and lowest score in any distribution
e.g 40, 42, 43, 39, 33
The highest value = 43
The lowest value = 33
Hence the Range = 43 – 43
= 10
1
8.3 Mean Deviation (Average Deviation)

This describes the absolute amount by which a distribution of scores deviates form its mean.
Mean deviation is computed as under.
MD = ∑ (Xi – X)
N
Illustration
Calculate the mean deviation for the following scores : 4, 5, 6, 10, 11, 12
Solution
Xi Xi – X Xi – X
4 4 – 8 = -4 4
5 5 – 8 = -3 3
6 6 – 8 = -2 2
10 10 – 8 = 2 2
11 11 – 8 = 3 3
12 12 – 8 = 4 4
n=6 ∑1 Xi – X = 18
Mean has been obtained by

∑ Xi = 4 + 5 + 6 + 10 + 11 + 12
6
= 48
6
MD = ∑1 Xi – X1 = 18 = 3
6 3
2
8.4 Variance
This is the average sum of squared deviations of the scores about their mean.
It is computed using the formular below
S2 = ∑ (Xi – X) 2
Illustration
Calculate variance for the following distribution : 4, 5, 6, 10, 11, 12
Xi Xi – X (Xi – X)2
4 -4 16
5 9
-3
6 4
10 -2 4
11 9
2
12 16
3
4
∑n = 6 ∑1 Xi – X = 58
S2 = ∑ (Xi – X) 2 = 58 = 9.67
6
8.5 Standard deviation

This is a measure of spread / variability that measures the extent to which the individual scores in
the distribution deviation from the mean.
It is calculated using the following formular
S.D = ∑ (Xi – X)
N
3
Illustration
Calculate the standard deviation in the following set of data
4, 5, 6, 10, 11, 12
Solution
Xi Xi – X (Xi – X)2
4 -4 16
5 9
-3
6 4
10 -2 4
11 9
2
12 16
3
4
N=6 ∑ (Xi – X)2 = 58
S.D = ∑ (Xi – X)2

N
= 58
6
= 3.11
Characteristics of standard deviation

(i) It is very sensitive to extreme scores e.g 6, 6, 68, 74, 76, in this case 6 , is an
outlier and will make SD artificially large.
(ii) It is very sensitive to change in the distribution. Such that any score that is shifted
further from the mean, the SD increases accordingly.
4
8.6 The Standard Z Scores

This is the score that stats the position of a given score in relation to the mean of the distribution
using standard deviation as the unit of measurement. It is calculated using the formular below.
Zi = Xi – X
Sx
Where
Zi = Z – Score
Xi= Individual raw score
Sx = The standard deviation from which Xi has come from
X = Mean
Illustration
The mean of mathematic marks in form 2 class was 60 while the standard deviation was 6.
Calculate the Z-score of the following marks of some student 32, 75, 54, 69
Solution
Z - Score of 32 = 32 - 60 = -4.67
6
Z - Score of 75 = 75 - 60 = 2.5
6
Z - Score of 54 = 54 - 60 = -1
6
Z - Score of 69 = 69 - 60 = 1.5
6
Characteristics of Z-scores
(i) Every distribution of Z- scores has a mean of zero

(ii) Every distribution of Z-scores has a standard deviation of one
(iii) The shape of Z- distribution is identical to the shape of original distribution i.e the Z
scores are a linear transformation of the raw scores
5
Advantages of Z- Scores over Raw scores

(i) It facilitates comparison from different tests, skills aptitudes.
(ii) The Z-scores that conform to the normal curve can be used to interpret scores i.e it is
possible to work out the probability (P) of any particular Z-score occurring
Criticisms of Z-Scores
Interpretation of Z – scores is not realistic because of the following reasons
(i) Negative scores – Any score below the mean e.g 32 and 54 in the above illustration
will have a negative value. The tendency to forget to writing the negative sign is very
high and this would change the value of the score significantly.
(ii) Zero average is very difficult for people to comprehend that O= average score. Any
squire value in the distribution equivalent to the mean score will have a Z-score of
zero.
(iii) Decimal tractions – The majority of Z-scores are small fractions. The values of Z-
scores irrespective of the size of n tend to lie between – 3 standard deviations and 3
standard deviation.
8.7 Normal distribution

Definition
Normal distribution is a theoretical distribution of scores defined by specific equation. The type
of curve is representative of the theoretical distribution.
It is represented as shown below.
0.5 0.5
O
X
6
Characteristics of Normal Curve

(i) It is bell shape symmetrical rising to a rounded Peak around the middle and tapering off
symmetrically at both tails.
(ii) The curve is asymptotic to the horizontal axis i.e the tails of the distribution tapper off
and come close the touching the abscissa but never do because they extend from negative
infinity to positive infinity.
(iii) A perpendicular line drawn from the apex of the curve to the base bisects the curve into
two equal parts.
(iv) The curve is based on Z-distribution such that the baseline (horizontal axis) is marked in
standard deviation units.
(v) The total area under the curve is equal to one unit or 100%
(vi) The area under the curve corresponds to the percentage of cases such that 100% of the
area under the curve and this corresponds to 100% of the scores or cases.
(vii) A certain percentage of the area (or cases) lies between the mean nada certain number of
standard deviation units above or below the mean.
Self-Test
(a) What do you understand by the term measures of variability

(b) Compute the mean deviation, variance and standard deviation from the following
grouped data given in the table below.
(c) Discuss the properties, merits and demerits of the following measures of dispersion
(i) Range
(ii) Variance
(iii) Standard deviation
7
Class Frequency
0–4 3
5–9 4
10 – 14 12
15 – 19 5
20 - 24 6
References
1. Chritensen, B.L. and Stoup, C.M. Introduction to statistics for Social Sciences. Belmont:
Brooks Cole 1991.
2. Cohen, J. West S.G & Aiken, L.S (2003) Applied Multiple Regression and Correlation
Analysis for Behavioural Science 3rd Edition.
3. Cooper, D.R & Schindler P.S (2011) Business Research Methods 9th Ed. Tata M.C Graw
Hill New Dechi.
4. Agarwal, B.L (2013) Basic Statistic 6th edition, New age international publishers.
5. Filed, A.P (2009) Discovering Statistics using SPSS 2nd Edition, London Sage
6. Freund, R.J, Wilson W.J. Mohr, D.L (2010). Statistical methods 3rd edition.

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STA 103: PRINCIPLES OF STATISTICS

By the end of this topic the learner should be able to:

8.3 Mean Deviation (Average Deviation)

Mean has been obtained by

8.5 Standard deviation

S.D = ∑ (Xi – X)2

Characteristics of standard deviation

8.6 The Standard Z Scores

(i) Every distribution of Z- scores has a mean of zero

Advantages of Z- Scores over Raw scores

8.7 Normal distribution

Characteristics of Normal Curve

(a) What do you understand by the term measures of variability

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