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UNIT 14 MEASURES OF DISPERSIONStructure14.1 introduction

ObjectivesMeaning o f DispersionImportance. of the M easures of DispersionConcept of RangeCon cept of Quartile Deviation14.6.1 Calculation of Quartile Deviation14.6.2 Interpreta6ion of Quartile DeviationConcept of Percentiles14.7.1 Calculation of Percentiles14.7.2 lnterpretation of Percentiles14.7.3 Limitations of PercentilesConcept of M ean Deviation14.8.1 Calculation of Mean Deviation14.8.2 Interpretationof Mean DeviationConcept of Standard Deviation14.9.1 Calculation of Standard Deviation14.9.2 Interpretation of Standard Deviation

14.10 Use of S tandard Deviation, Quartile Deviation and Percentiles in C lassroom Situation

14.12 Unit-end Exercies14.13 Points for Discussion14.14 An sw ers to Check Your Progress14.15 Suggested Readings

14.1 INTRODUCTIONIn Unit 12 and 13 of this Block, you have studied about the 'Tabulation and GraphicalRepresentation of Data' and the 'Mea sures of Central Tendency '. Measures of central tendencycan desc ribe only on e of the important characteristics of a given distribution i.e. the measuresof location or the value of the variate aroun d which the distribution may c entre. Anotherimpo rtant characteristic of the distribution is its variability. It is equally nec essary to knowabout the variability of data, which may be concentrated or scattered around the measures ofc e h t r a ~ endency.In the present unit, you are.going to study about the meaning and the importance of themeasu res of variability or dispersion, calculation and interpretation of these measu res and theuse of these measures in the actual classroom situation. Ya u wouM then be in a better positionto teach these to your students. ' -- -14.2 OBJECTIVES --- -After goin g through this unit, you will be able to:

und~erstand he concept of dispersion;diffkrentiate between the measures of central tendency and the rrieasures of dispersion;40


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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 byyou;define standard deviation'd n your own words;

dcalculate 'o'rom ungrouped,as well as from grmpd data;interpret the index of 'a',alculated by you;use appropriate measure of dispersion in cla ssr am situation for the improvement ofteaching-learningprocess.

14.3 II~EAIWNGOFDISPERSIONThe measures of central tendency give one p i n t re~esentation f t k dia i$ut ion bere tkdecrsion making requires more than this information. For exampEe, let us consider th eperformance of the two groups of children i n a mathematics test. The sores obtained by tkchildren are as follows :GroupA: 8, 12,11,12,10,&~9,11,12,10,8,10,9,0,112,8,f8,9,10,TII.GroupB : 15,2 , 8, I2,4, 17,20,6,2, E8, 16,4 3, 9,6, I@ 15, 17,9, 11.By computing means, yo u will notice thaf the averageperformanceof the children of bothrkgroups is the same, which is 10.0. However, if y m go rhaogh the scores & a i d by thechildren s f the two groups, youwill find hat in Group A, no child has less than 8 ar m e hm12 marks, while in G r ~ u p , there are many chiidren getting less titan 8 sr m e km I2 -marks. Actually, thescoresvary frmO to20 in Group B. The twogroups arem cmprsbkin terms of homog&eity. Group A is hmoger~ews,while G r q B iswtek a ( ~ o gm o u ~ ~This information regarding the variability d data is not revealed by the measure of sentraItendency. The property which denaes the extent to which shevalues are dispersed a thecentral 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 themeasure of central tendency is a point on the scaleofm a w e m e n t , the:measwe of dispersioni s also a distancedong the scaleofmeaswemnt. Them e a s m s of dispersion incm m useare Range,Quartile Deviaaion, Mean Deviation and S t d a dDeviation.

Measures of Central Tendency can be used to &scribe a d a mpre differentdata, &H thewdcrno1give information about its variability. The varkbi9lty of dasa reveals much more thanwhat the measure of central tmdency alone can do . Tfre masure of ddiqmsisn provides aquantitative index of ,fkegree of v;rrislion arrtang a parriedar set of values cw sexes. Forcompirri~ghe relative&gra ofvariabilitywith r e p ~ do some trait ammg the indivirdwis int w o or more groups, the measure of diqersion of the trait in a& group is needed. Thequantitative ~ndicesequired far camprison purposes stPauld 6e available in dams of thesame unit. The measure of dispersion,especidfyek deviation, is required for m y 41


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N Measures of Dispersionbeing that for the median we consider - ases, while t'or the Q1 and 4 3 we have to take- 2N 3N-and - ases respectively.4 4

Let us consider the following example :Example 1 :The scores obtained by 36 students of a class in mathematics are shown in thetable. ~ i n dhe quartile deviation of the scores.

Table 14.1Scores f cf

In column 1 , we have taken Scores, in column 2, we have taken the frequency, and in colu~nn3. cumulative frequencies starting from the bottom have been written.

N 36Here N = 36, so for Q1 we have to take -=-=9 cases and for 4 3 we have to take4 4---= 36 27 cases: By looking into column 3, cf = 9 will be included in C155-59, whoseN4 4real limits are 54.5 - 59.5. So Ql would lie in the interval 54.5- 59.5. The value of Ql is tobe computed as fo ll ~ws

For calculating 43, cf = 27 will be included in CI 65-69, whose real limits are 64.5 - 69.5. SoQ3 would lie in the interval 64.5 - 69.5 and its value is to he computed as follows':

Q3-Q1 65.50-55.93- 9.57-hus, Q=-- --2 2 2


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Siatistierl Techniques of Analysis 14.6.2 Interpretation of Quartile DeviationLet 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 extrem e values,so it is mo re representative and reliable than range. Wh erever median is preferred as a measureof cenrral tendency, quartile deviation is preferred as mea sure of dispersion. How ever, likemedian, quartile deviation is not amenable to algebraic treatment, as it does not take intoconsideration all the values of the distribution.While interpreting the value of quartile deviation it is better to have the values of M edian, Q1and Q3, longwith Q. f the value of Q is more, then the dispersion will be mo re, but again thevalue depends on the scale of measurement. Two values of Q are to be compared only if thescale used is the same. Q measured for scores out of 20 cannot be compared directly with Qfor scbres out of 50. f median and Q are known, we can say that 50% of the cases lie between'Median - Q' and 'Median + Q'. These a re the middle 50% of the cases. Here , we come toknow about the range of only the middle 50% of the cases. How the lower 25% .of the casesand the upper 25% of the cases are distributed, is not known through this mea sure. Som etimes ,the extreme cases o r values are not known, in which case the only alternative available to us isto compute median an d quartile deviation as the mea sures of central tendency and dispersion.Thro ugh median and qu artiles we can infer about the symme try or ske wnes s of the distribution.Let ud therefo re, get some ide a of symm etrical and skew ed distributions.Sym aetr ical and Skewed Distributions :A distribution i s said to be symme trical when thefrequencies are symmetrically distributed around the measure of central tendency. In otherwords, we can say that the distribution is symmetrical if the values at equal distance on thetwo sides of the measure of central tendency have equal frequencies.Example 2 :Find whether the given distribution is symm etrical or not.-

Score 0 1 2 3 4 5 6 7 8 9 10Frequency 1 2 2 4 5 8 5 4 2 2 1

Here Ihe meas ure of central tendency, mean as well as median, is 5. f we start comparing thefreque ncies of the values on the two sid es of 5,we find that the values 4 and 6,3 nd 7,2 nd8, 1 an d 9, 0 an d 10 have the same number of frequencies. S o the distribution is perfectly~ y m m e t r i c a l .En a symm etrical distribution, mean and median a re equal and median lies at an equal distancefrom the two quartiles i.e. Q3 - Median =Median - Q 1 .If a distribution is not sym metric , then the departure from the symm etry refers to its skewne ssSkewness indicates that the curve is turned more towards o ne side than the other. So the curvewill have a longer tail on one side. The sk ewne ss is said to be positive if the longer tail is onthe right side and it is said to be negative if the longer tail is on the left side. Th e followingfigurks, show the appearance of a positively skewed curve and a negatively skewed curve.

Fig. 14.1(a) Pclsitively skewed (h) Negatively skewed


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For a positively skewed curve,,.r (Q3 -Median) is greater than (Median - Q1)

' For a negatively skewed curve,(Q3 - Median) is less than (Median - Q1)

.. ------ --.1.'.i.Gt-o:tr ~ ~ ~ v g r c ~ s s I'I ; 1 , ~ ~ J : ~ ! I ~ ~ ~ ~j:!;t;\:i


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Statistical Techniques ofAnalysis 20NFor comput ing P?,,we have to take -or 7.2 cases.l o oTh e cf against 50 - 54 is 7 and against 55 - 59 is 14. So 7.2 cases would lie upto a pointbetween 54.5 an d 59.5. Now I

7 .3 -7: P2,) = 54.5 + -=--x 57I= 54.5 + -7

= 54.5 + 0.14 .= 54.64

In Unit 12, you have learnt to draw a cumulative frequency curve and ogive, w herein alongw ithcumu lative:freq uency, he cu~ nula tive ercentage has been used on Y-axis. With the help ofthe Ogive, you can directly find the values of different percentiles without com puting them.

14.7.2 Interpretation of PercentilesPercenti les dre more frequently used in testing and interpreting test scores. For any standardisedtests, percentile norms are reported with the test, so that the obtained test results may beinterpreted droperl y. If the percentile rank of an individu al is 60, we come to know that 60%of the studebts have scored less ihan that individual. If only the score of an individual isgiven, it is difficult to judge the performance. It can be judged only w ith reference to a particulargroup. However, with the help of the cumulative percentage curve, we can find the percentilerank of that ihdividual and judge the performance on that basis,14.7.3 Limitations of PercentilesThe mastery of an individual is not judged by the use of percenliles, as the same person in apoor gro up will sh ow better rank and in an excellent group will show compa ratively poorerrank. Also, as in casc of simple ranks the difference in percentile ranks at different intervalsare not equal. !Asan example, P,,,,, P,,, is not comparab le to P,,, - P ,,. The posit ion of a s tudenton tota l achievement cannot be calculated from percentiles given in several tests.14.8 CONCEPT OF MEAN DEVIATIONTh e distance of a score from a central point is called a deviation. Th e simplest way to takeinto consideraqion the variation of all the values in a distribution i s to find the mean of all thedev iati ons of tlbese values from a selected point of central tendency. Usually, the deviation istaken from t1re:mean of the distribution. Th e avcrag e of the dev iati ons of all values fr om thearithmetic meah is know n as mean deviation o r average deviation.The me asure o t central tendency is such a point on the scale of measurem ent on both side s ofwhich there are a number of values. So, the deviations from this point, will be in oppositedirections, both positivc and negative. If the score is denotcd by X, and the mean by M , thenX-M denotes the deviation of scores from the mean. Th e deviation, where mean is greaterthan the scorc, will be negative. By definition of the mean, as measure of central tendency,the algebraic suln of all thcse deviations will come out to be zero, as the deviat ions on both thesides are equal .! To avoid this problem, th e absolu te values of these deviations, i.e. IX -MIirrespecti ve o f tlleil- sign is taken into consideration. Thus,

CIX - MIMean Deviation = N .Where X is the scorc, M is [he mean and N is the total frequency.14.8.1 Calculation of Mean ~eviationIn Unit 13, you v en t through some examples for ca lcula tion of means. Let us consider the4h


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Statistical Techniques of Analysis Step 3 :, Square the deviations and write the value of (X-M)2 against each score in column3. Find the sum of the squared deviations.

Step 4 : Calculate the mean of the squared deviations and then find out the positive squareroot for getting the value of standard deviation i.e. '0'.

The computation of standard deviation is shown below :Table 14.5

Scores X-M (X-M)'12 0 0

The requiredl standard deviation is 2.9.Calculation bf Standard Deviation for Grouped DataExample 7 : Find the standard deviation for the following distribution :

Here also, thelfirst step is to find the mean M, for which we have to take the mid points of theZfl The second step is toI, denoted bp X' and find the product fX'. Mean is given by .-

(ind the deviations of the mid points of class intervals X' from the mean i.e. X'-M denoted byX"\ ~he hird step is to square the deviations and find the product of the squared deviationsand the corresponding frequency.

Now, NTo solve the a k v e problem, CI is written in column 1, frequencies are written in column, 2mid points of TI i.e. X' are written in column 3, the product fx ' is written in column 4, thedeviation x' of lX' from mean is wr~tt en n column 5 , the squared deviation xtZ s written incolumn 6, and the product f ~ ' ~s written in column 7, as shown below :


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Table 14.6Mid Point

C I f x' fx ' x i X 12 jx'2

Total 30 333 674.70

c f X' 333Mean =-=-=I 1.1N 30

So, the deviations of the mid points are to be taken from 11.1.

= 1 / m = 4 . 7 4Thus, the required standard deviation is 4.74.Here, you must h ave observ ed that the deviations from the actual mean results in decim als andthe values of x'' and f X" are difficult t0,calculate. In order to avoid this problem we follow ashort-cut method for calculating standard deviation. In this method , instead of taking thedeviations from actual mea n, we take deviations from assumed mean. As a result , the form ulafor finding the standard d eviation also changes to :

i 4=-x N X f d - X fdN

Here , i is the length of each cla ss interval,N s the total number of cases and is the deviationof mid-point from the Assumed M ean, divided by the length of CI. This method is also know nas Step Deviat ion Method.Step Deviation Method for the Calculation of Standard DeviationIn this method. in column 1, we write the CI; in column 2, we write the frequency; in colum n

X' -AM3, we write the values of d , where d = . ; n column 4, we wri te the product fd, and in1column 5, we w rite the value of fd 2, as shown below

Measures of Dispersion


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1

Statistical Techniques of Analysis

Total 30 11 79

He re Assumed Mea n is the mid-point of the C I 9- 11 i.e. 10, so the deviations have been takenfrom 10 and divided by 3, the length of CI.

Now, O = ~ J ~

Here also o 4.7414,9.2 Interpretation of Standard DeviationLe t us first cons ider the use and limitations of Standard D eviation as a mea sure of dispersion.Standard Deviation is the most widely used and important measure of dispersion. It occupiesa centra l position in statistics. Like mean deviation, it is based on all the value s of thedistribution. Here, the signs of deviations are not disregarded, instead they ar e eliminated bysquaring eac h of the deviations. It is the master measure of va riability as it is amena ble toalgebraic treatmen t and is used in correlational work and in further statistical analyses. Theonly limitation is that, the extrem e values affect it to a large extent. For nterpreting the valueof the measure of dispersion we must understand that greater the value of 'o the more scatteredare the scores from the mean.A s in the case of mean deviation, the interpre tation of standard deviation requires the valuesof M nd N for consideration.In exaqples 6 and 7 , the required values o r ? , mean and N are as follows :

-Example 6Exam ple 7 11.1 4.74 30

I

Here, the dispersion is more in example 7 a s compared to example 6. It means the values aremore soatiered in example 7, as compared to the values of example 6.


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- -=-mw"T- I Me;lsums ofDispersion

14.10 USE OF STANDARD DEVIATION, QUARTILEDEVIATION AND PERCENTILES IN CLASSROOMSITUATIONIn classroom situations, w e are required to use statistics frequently. However, in such a situation,either the teacher avoids comp uting the required statistic or computes the simples t measure ofcentral tendency and dispersion without bothering for the suitability of these measures for thegiven distribution. A teacher should use the statistic when required and it should be appropriatefor the situation .While reporting the progress of the child, the teacher simply reports the subject-wise m a k s inan examination. No o ther information is provided w ith reference to the performance of thechildren of the class. To avoid this situation , teacher can prepare the individual profiles of thechildren in terms of their percentile ranks. These p rofiles can be reported to their parents; theywill be able to understan d the position of their child in the class.Example 8 :The scores of a student of class IX in the annual examination are reported asfollows : - - -Subject : English Hindi Maths Science Soc. St Total

Max. Marks 100 100 100 100 100 500Marks obtained 55 6 0 7 0 6 3 5 8 3 0 6

From the abov e data, it is difficult to have an objective assessment. We can simply see thepercentage of achievem ent in each of the subjects and in the total. However, if the teacherprovides subject-w ise PRs alongw ith marks obtain ed, it will be more meaningful. For this,individual's PRs reported are as follows :

Subject : English Hindi Maths Science So c. St Total!PRs P o Ps4 P~~ Pss P53 Pss

Total number of studen ts in class are 44.Here, one understands the relative position of the individual in the group, for each of thesubject. Further, individu als can be com pared with earlier and later performance, if reportedin this way.Th e measu re of dispe rsion helps the teacher to know ab out the variability in the class and planthe individualised instruction to suit the deprived and the intelligent children.


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Statistical Techniques Analysis For comparison of individual scores, the best way is to convert the obtained scores into thestandard scores and for comparing tw o different distributions in terms of their variability, thecoefficient of the variation may be used. The standa red score and the coefficient of variationis being discussed in the following sec tion.Standard ScoresIf the mean and S D of a set of values is known, then the given scor e may be expressed asdeviation from the mean a s a multiple of SD. Such a score is called standard s core or Z score.

X - Mz=-a

Stan datd scores are am enable to all algebraic treatmen t, as these scores are represented in theform of Ratio Scales of measurement. Any two scores are comparable when converted intostandard score s. If the raw score s of different subjects are converted into standard scor es, wecan calculate the composite scores also.The Coefficient of VariationWhen two series or distributions are kxpressed in the same unit and have approximately sameaverage value then the a s of the two series/distributions may be comp ared directly. But if theunits are different or the average value of the two distributions differ to a large extentthen direct comparison may not give the true picture.Example 9 :The heights of students of class VIII A have been measured in inches, whilethose of class VIII B have been measu red in cms. The standard deviations fo r the two groupsare as follows :

Class 0VIII A 5VIII B 12

These two a s cannot be compared d irectly, unless these are converted t o the same unit or inany other form not involving unit.For this we need a nother measure of variability that is independ ent of unit and takes averagesinto consideration. Such a measure is the co-efficient of variation o r relative standard deviationand is expressed as a percentage.

A s CV is the ratio of two quan tities, a and M, having the sa me unit, it is independ ent of unit.It als o provides a va lue relative to mean.Exceptional variation : n educational situations CV less than 5% or more than 35% wouldraise doubts regarding accuracy of computation of mea ns and SD s, or the appropriate measu resmay not have been com puted. For such cases, special analysis and interpretation is needed.

14.11 LET US SUM UPThe property which denotes the extent to which the values are spread about the centralvalue is called dispersion.The measure of dispersion is a distance along the scale of measurement.Commonly used measures of dispersion are Range, Quartile Deviation, Mean Deviationadd stan da rd Deviation. The simplest measu re is Range and the most trust worthy measureis Standard Deviation.Standard D eviation is required for further statistical comp utations.Before computing the measure of dispersion it is desirable to see the suitability of the

54 measure to a given situation.


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Measures of Dispersion


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StatisticalTechniques ofAnalysis 14.lb SUGGIESTED READINGSBlombners, P. and Lindquist , E.F. (1959), Elementary Statistical Methods, HoughtonMifflih Company, Boston.Garrett, H.E.(1956), Elementary Statistics. Longmans, Green & Co., New York.Guilford, J.P. (1965), Fundamental Statistics in Psychology and Education, McGraw HillBook Company, New York.Hannagan, T.J. (1982), Mastering Statistics, The Macmillan Press Ltd. , Surrey.Lindgren, B.W. (1975), Basic Ideas of Statistics, Macmillan Publishing Co. Inc.,New York.Tate, M.W. (1955), Statistjcs in Education, The Macmillan Company, New York.Walkeq, H.M. and Lev, J. (1965), Elementary Statistical Methods, Oxford& IBH PublishingCo., Chlcutta.Wine, R.L. (1976), Beginning Statistics, Winthrop Publishers Inc., Massachusetts.

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