14.9.2 Interpretation of Standard Deviation. 14.10 Use of Standard Deviation,
Quartile Deviation and Percentiles in Classroom Situation. 14.12 Unit-end
Exercies.
UNIT 14 MEASURES OF DISPERSION Structure 14.1
introduction Objectives Meaning of Dispersion Importance. of the Measures of Dispersion Concept of Range Concept of Quartile Deviation 14.6.1 Calculation of Quartile Deviation 14.6.2 Interpreta6ion of Quartile Deviation
Concept of Percentiles 14.7.1 Calculation of Percentiles 14.7.2 lnterpretation of Percentiles 14.7.3 Limitations of Percentiles
Concept of Mean Deviation 14.8.1 Calculation of Mean Deviation 14.8.2 Interpretation of Mean Deviation
Concept of Standard Deviation 14.9.1 Calculation of Standard Deviation 14.9.2 Interpretation of Standard Deviation
14.10 Use of Standard Deviation, Quartile Deviation and Percentiles in Classroom Situation
14.12 Unit-end Exercies 14.13 Points for Discussion 14.14 Answers to Check Your Progress 14.15 Suggested Readings
14.1 INTRODUCTION In Unit 12 and 13 of this Block, you have studied about the 'Tabulation and Graphical Representation of Data' and the 'Measures of Central Tendency'. Measures of central tendency can describe only one of the important characteristics of a given distribution i.e. the measures of location or the value of the variate around which the distribution may centre. Another important characteristic of the distribution is its variability. It is equally necessary to know about the variability of data, which may be concentrated or scattered around the measures of cehtra~tendency. In the present unit, you are.going to study about the meaning and the importance of the measures of variability or dispersion, calculation and interpretation of these measures and the use of these measures in the actual classroom situation. Yau wouM then be in a better position to teach these to your students. '
14.2 OBJECTIVES
--
-
---
After going through this unit, you will be able to: und~erstandthe concept of dispersion; 40
diffkrentiate between the measures of central tendency and the rrieasures of dispersion;
-
state the importance of the measures of dispersion; define quanile deviation 'Q' in your own words; calculate 'Q' for given data; interpret the value of 'Q' obtained by you; +
define and calculate the specified percentiles; interpret the percentile obtained by you; define mean deviation in your own words; calculate mean deviation for ungrouped, as we11 as gmuped data; interpret the mean deviation obtained by you; define standard deviation 'din your own words; d
calculate 'o'from ungrouped, as well as from grmpd data; interpret the index of 'a',calculated by you; use appropriate measure of dispersion in classram situation for the improvement of teaching-learning process.
14.3 II~EAIWNGOF DISPERSION The measures of central tendency give one p i n t re~esentationof t k diai$ution bere t k decrsion making requires more than this information. For exampEe, let us consider the performance of the two groups of children in a mathematics test. The s o r e s obtained by t k children are as follows : GroupA:
8, 12,11,12,10,&~9,11,12,10,8,10,9, ~0,112,8,f8,9,10,TII.
Group B :
15,2, 8, I2,4, 17,20,6,2, E8, 16, 4 3, 9,6, I@ 15, 17,9, 11.
By computing means, you will notice thaf the average performance of the children of both r k groups is the same, which is 10.0. However, if y m go rhaogh the scores & a i d by the children s f the two groups, you will find that in Group A, no child has less than 8 ar m e ihm 12 marks, while in G r ~ u pB, there are many chiidren getting less titan 8 sr m e tkm I2 marks. Actually, the scores vary f r m O to 20 in Group B. The two groups are mc m p r s b k in terms of homog&eity. Group A is hmoger~ews,while G r q B is wte k a ( ~ o g m o u ~ ~ This information regarding the variability d data is not revealed by the measure of sentraI tendency. The property which denaes the extent to which she values are dispersed athe central value is called dispersion. Dispersion is also known as spread, scatter or variability.
-
Measures of dispersion, provide a quantitative m m r e ofthe v;rriabiIi:y of vdues. Just as the measure of central tendency is a point on the scale of mawement, the:measwe of dispersion i s also a distance dong the scale of measwemnt. The measms of dispersion in c m m use are Range, Quartile Deviaaion, Mean Deviation and S t d a d Deviation.
Measures of Central Tendency can be used to &scribe a d campre different data, &H thew dcr no1 give information about its variability. The varkbi9lty of dasa reveals much more than what the measure of central tmdency alone can do. Tfre masure of ddiqmsisn provides a quantitative index of ,fk degree of v;rrislion arrtang a parriedar set of values cw sexes. For compirri~gthe relative &gra ofvariability with r e p ~ to d some trait ammg the indivirdwis in t w o or more groups, the measure of diqersion of the trait in a & group is needed. The quantitative ~ndicesrequired far camprison purposes stPauld 6e available in dams of the same unit. The measure of dispersion, especidfy e k deviation, is required for m y
41
being that for the median we consider
N -and 4
N cases, while t'or the Q1 and 4 3 we have to take
-2
3N
- cases respectively.
4
Let us consider the following example :
Example 1 : The scores obtained by 36 students of a class in mathematics are shown in the table. ~ i n dthe quartile deviation of the scores. Table 14.1 Scores
f
cf
In column 1 , we have taken Scores, in column 2, we have taken the frequency, and in colu~nn 3. cumulative frequencies starting from the bottom have been written.
N 36 Here N = 36, so for Q1 we have to take -=-=9 cases and for 4 3 we have to take 4 4 3N ---=36 27 cases: By looking into column 3, cf = 9 will be included in C155-59, whose 4 4 real limits are 54.5 - 59.5. So Ql would lie in the interval 54.5 - 59.5. The value of Ql is to be computed as f o l l ~ w s:
For calculating 4 3 , cf = 27 will be included in CI 65-69, whose real limits are 64.5 - 69.5. So Q3 would lie in the interval 64.5 - 69.5 and its value is to he computed as follows':
Thus, Q=-- Q3 - Q1 - 65.50 - 55.93 --9.57 2 2 2
Measures of Dispersion
Siatistierl Techniques of Analysis
14.6.2 Interpretation of Quartile Deviation Let us first discuss the use and limitations of quartile deviation as a measure of dispersion. Quartile deviation is easy to calculate and interpret. It is independent of the extreme values, so it is more representative and reliable than range. Wherever median is preferred as a measure of cenrral tendency, quartile deviation is preferred as measure of dispersion. However, like median, quartile deviation is not amenable to algebraic treatment, as it does not take into consideration all the values of the distribution. While interpreting the value of quartile deviation it is better to have the values of Median, Q1 and Q3,alongwith Q. If the value of Q is more, then the dispersion will be more, but again the value depends on the scale of measurement. Two values of Q are to be compared only if the scale used is the same. Q measured for scores out of 20 cannot be compared directly with Q for scbres out of 50. If median and Q are known, we can say that 50% of the cases lie between 'Median - Q' and 'Median + Q'. These are the middle 50% of the cases. Here, we come to know about the range of only the middle 50% of the cases. How the lower 25% .of the cases and the upper 25% of the cases are distributed, is not known through this measure. Sometimes, the extreme cases or values are not known, in which case the only alternative available to us is to compute median and quartile deviation as the measures of central tendency and dispersion. Through median and quartiles we can infer about the symmetry or skewness of the distribution. Let ud therefore, get some idea of symmetrical and skewed distributions.
Symaetrical and Skewed Distributions :A distribution is said to be symmetrical when the frequencies are symmetrically distributed around the measure of central tendency. In other words, we can say that the distribution is symmetrical if the values at equal distance on the two sides of the measure of central tendency have equal frequencies. Example 2 : Find whether the given distribution is symmetrical or not. -
Score
0
1
2
3
4
5
6
7
8
9
10
Frequency
1
2
2
4
5
8
5
4
2
2
1
Here Ihe measure of central tendency, mean as well as median, is 5. If we start comparing the frequencies of the values on the two sides of 5,we find that the values 4 and 6,3and 7,2and 8, 1 and 9, 0 and 10 have the same number of frequencies. So the distribution is perfectly ~ymmetrical. En a symmetrical distribution, mean and median are equal and median lies at an equal distance from the two quartiles i.e. Q3 - Median = Median - Q 1. If a distribution is not symmetric, then the departure from the symmetry refers to its skewness. Skewness indicates that the curve is turned more towards one side than the other. So the curve will have a longer tail on one side. The skewness is said to be positive if the longer tail is on the right side and it is said to be negative if the longer tail is on the left side. The following figurks, show the appearance of a positively skewed curve and a negatively skewed curve.
Fig. 14.1
(a) Pclsitively skewed
(h) Negatively skewed
Measures of Dispersion
For a positively skewed curve, (Q3 -Median) is greater than (Median - Q1)
,.r
'
For a negatively skewed curve, (Q3 - Median) is less than (Median - Q1) ~~~vgrc~ss
I
:
%
;
I
I
1 ,
~
~
J
i
I I L !.I ~, $ .
:j:!;t;\:i
