2

Measure of Variability

Measures of Dispersion Percentile

We discussed the commonly used measures of central tendency which are useful in providing descriptive information concerning a set of data. However, the information so obtained is neither exhaustive nor comprehensive, as the mean does not lead us to know whether the observations are close to each other or far apart. Median is the positional average and has nothing to do with the variability of the observations in the series. Mode is the largest occurring value independent of other values of the set.

This leads us to conclude that a central value or an average alone cannot describe the distribution adequately. Moreover, two or more sets may have the same mean but they may be quite different. To clear this point, let us consider the scores of two groups of students on the same test:

Here, both the groups have the same mean score of 60. From first inspection, we might say that the two sets of scores are equal in nature. In the first group, the range of scores is from 24 to 90, while in the second, the range is from 50 to 70. This difference in range shows that the students in the second group are more homogeneous in scoring than those in the first. So both the groups differ widely in the variability of scores.

Thus, while studying a distribution it is equally important to know whether the observations are clustered around or scattered away from the point of central tendency. Measures of dispersion help us in studying the extent to which observations are scattered about the average or the central value.

Measures of Dispersion

Measures of Dispersion are the following:

1. Range
2. Quartile deviation (Q)
3. Mean deviation (MD)
4. Variance and standard deviation (SD)
5. Combined variance and standard deviation
6. Coefficient of variation
7. When to use various measures of dispersion

The Range

The range of a set of observations is defined as the difference between the largest and the smallest value. The range is a measure of observations among themselves and does not give an idea about the spread of the observations around some central value. The range is defined by

 

 

where X_L is the largest of the observed values and X_S is the smallest of the observed values.

The Quartile Deviation

The quartile deviation is defined as the difference between Q₃ and Q₁ in a frequency distribution. It is computed by the formula:

where Q is the quartile deviation, Q₃ is the third quartile and Q₁ is the first quartile.

For computing Q, we must first get the values of Q₁ and Q₃, the first and third quartiles, respectively.

The Quartiles

In the previous chapter, we discussed about the median as a measure of central tendency. We know that the median is the value of the score which divides the distribution into two equal halves. Similarly, quartiles are the points which divide a distribution into quarters. Q₁ is the first quartile, Q₃ is the third quartile and Q₂ is the second quartile or median.

The quartile can be computed by the following formula:

where L is the lower limit of the class in which the quartile lies, Cf is the ‘less than’ cumulative frequency of the class preceding the quartile class, f is the frequency of the quartile class and i is the size of the class interval.

To find Q₁ and Q₃, we must first find quartile class.

The quartile class for Q₁ is the class in which N/4th case falls. Similarly for Q₃, the quartile class is the class in which 3N/4th case falls. The computation of Q₁ and Q₃ may be explained through the following example:

Calculation of Q₁: Since = 15th cases fall in the class interval 8–9, the quartile class for Q₁ is 8–9, with its class interval 2 and true limits ‘7.5–9.5’.

Therefore, L = 7.5, Cf = 14, f = 20, i = 2,

Substituting these values in the formula

Calculation of Q₃: Since = 45th cases fall in the class interval 10–11, the quartile class for Q₃ is 10–11 with its class interval 2 and true limits ‘9.5–11.15’.

Therefore, L = 9.5, Cf = 34, f = 15, i = 2.

Substituting these values in the formula

Merits and Demerits of Quartile Deviation

1. It is simple to understand.
2. It is not based on all the observations a of the data.
3. It cannot be treated algebraically.
4. It is affected by sample fluctuations.
5. It gives a rough estimate of the variability.
6. It is suitable when the data are concentrated around the median.

Mean Deviation

The average of the absolute deviations of every variate value from the mean is called the mean deviation (MD) or the average deviation (AD). On averaging deviations to find the mean deviation, no account is taken of signs, and all deviations whether plus or minus are treated as positive. Thus, if some of the deviations are +4, −9, +2, +5, − 1 and −3, we simply add 4, 9, 2, 5, 1 and 3, getting 24.

Computation of Mean Deviation from Ungrouped Data

The formula for computing mean deviation from ungrouped data is

where N is the number of observations and signifies the sum of absolute deviations (irrespective of positive or negative sign) taken from the mean of the series.

The computation of mean deviation will be clearer by taking an example.

Mean Deviation from Grouped Data

In the case of grouped data, the formula for calculating the mean deviation is as follows:

Computation of mean deviation will be clear from the following example:

Merits and Demerits of Mean Deviation

1. It is rigidly defined.
2. It is based on all observations.
3. It can be calculated by using any of the three central tendencies, i.e. mean, median and mode.
4. It is a clear average of the separated deviation and hence easy to understand.
5. It ignores the sign of deviation and considers magnitude only.
6. It cannot be calculated even if a single term is missing from the data.
7. It is not a very accurate measure of dispersion, particularly when calculated from median or mode.

Variance and Standard Deviation

The most widely used measures for showing the variability of a set of scores are variance and standard deviation (SD). The variance is defined as the average of the squares of deviations of the observations from the arithmetic mean. The standard deviation is defined as the positive square root of the arithmetic mean of the squares of deviations of the observations from the arithmetic mean. It may also be called as ‘root mean square deviation from mean’ and is generally denoted by the small Greek letter (sigma).

Computation of Variance and Standard Deviation for Ungrouped Data

There are two ways of computing variance and standard deviation (SD) for ungrouped data.

1. Direct method
2. Short-cut method

Direct Method.    The formulae for finding the variance and SD for a set of scores are as follows:

where X is the individual score, is the arithmetic mean and N is the total number of observations.

The steps involved in the computation procedure may be listed as follows.

1. Calculate the arithmetic mean of the given data.
2. Obtain deviation of each variate from as (X − ).
3. Square each deviation to get (X − )².
4. Obtain the sum of squared deviations as Σ (X − )².
5. Using the formula calculate variance Σ σ² as the arithmetic mean of the squared deviations.
6. The positive square root of variance is then calculated to get SD as .

The following example will show how these steps are followed.

The following steps are involved in the computation procedure of variance and SD:

       Calculate the arithmetic mean of the given data.

1. Square of each raw score to obtain x².
2. Sum of squares to get Σ x².
3. Use the formula

   to obtain variance.

4. The position square root of variance is then calculated to get SD
   

These steps will be clear from the following example

Short-Cut Method.    In most of the cases, the arithmetic mean of the given data happens to be a fractional value and then the process of taking deviations and squaring them becomes tedious and time-consuming in the computation of variance and SD. To facilitate computation in such situations, the deviations may be taken from an assumed mean. The short-cut formula for calculating SD then becomes

where d is the deviation of the variate from an assumed mean, say a, i.e. d = (x ∔ a), d² is the square of the deviations, Σ d is the sum of the deviations, Σ d² is the sum of the squared deviations and N is the total number of variates.

The following are the steps of this method for calculating variance and SD:

Take some assumed mean, say a (as in the case of calculating arithmetic mean).

1. Find deviation from a for each score to obtain d = x − a.
2. Sum all d to obtain Σ d.
3. Square each deviation d to obtain d².
4. Sum all d² to obtain Σ d².
5. Use formula to obtain variance.
6. The positive square root of variance is then calculated by to get SD.

The computation procedure is clarified in the following example.

From Examples (2.7) and (2.8), what have we found? We find that variance (11.5) and SD (3.39) are the same in both raw score and short-cut methods.

Computation of Variance and Standard Deviation for Grouped Data

For calculating variance and standard deviation (SD) for grouped data, there are two methods analogous to ungrouped data:

1. Direct method
2. Short-cut method

Direct Method.    This method uses actual arithmetic mean while considering deviations of given observations. The formulae for calculating variance and SD are as follows:

The method can be described by the following steps:

1. Find the mid-value of all the classes to obtain x.
2. Calculate the arithmetic mean of the given data by the formula .
3. Take deviation from mean of each variate to obtain X − .
4. Take square of each (X − ) and then multiply with the corresponding frequency to get f (X − )².
5. Sum all f (X − )² to get Σ f (X − )².
6. Use the above formulae to obtain variance and SD.

The computation procedure is clarified in the following example.

The method illustrated above is quite tedious. Let us consider the ‘raw score’ formulae for grouped data to compute variance and SD as already discussed for ungrouped data. The ‘raw score’ formulae are

The steps involved in the computation procedure may be listed as

1. Find the mid-value X of all the classes.
2. Multiply each X by the corresponding frequency to obtain fX and
3. Calculate mean by the formula
   
4. Take square of each X and multiply by the corresponding frequency to obtain fX².
5. Sum all f X² to obtain ΣfX².
6. Use the above formulae to obtain variance and standard deviation.

The following example will clear these steps.

Short-Cut Method.    In the short-cut method, our main aim is to reduce the computations. To simplify the procedure, we take deviations from some assumed mean instead of the actual mean. The calculations can be further simplified by dividing these deviations from class interval. The formulae for calculating variance and SD by this method are

where d is the deviation from the assumed mean, i.e. , N is the total number of observations and i is the class interval.

Computation by this method can be described by the following steps:

1. Take mid-value of each class to obtain x.
2. Take some assumed mean say a near the centre of the distribution.
3. Subtract a from x and divide by i to obtain d.
4. Multiply each d by the corresponding frequency to get fd.
5. Take square of each d and multiply by the corresponding frequency to obtain fd².
6. Sum all fd and fd² to get Σ fd and Σ fd².

Use the above formula to calculate variance and SD. The following example will illustrate how these steps are followed.

Combined Variance and Standard Deviation

If two frequency distributions have means ₁ and ₂ and standard deviations σ₁ and σ₂ respectively, then the combined variance, denoted by σ²₁₂, and SD, denoted by σ₁₂, of the two distributions is obtained by using

where N₁ is the total number of observations in the first frequency distribution, N₂ is the total number of observations in the second frequency distribution, σ₁ is the SD of the first frequency distribution, σ₂ is the SD of the second frequency distribution, D₁ = (₂ − ₁) is the difference between the combined mean and the mean of the first frequency distribution and D₂ = (₁₂ − ₁) is the difference between the combined mean and the mean of the second frequency distribution.

The formula can be extended to any number of observations. An example will illustrate the use of the formula.

Merits and Demerits of Standard Deviation

1. Standard deviation possesses most of the characteristics of an ideal measure of dispersion.
2. It is always definite, rigidly defined and based on all the observations.
3. It considers the square of deviation, so signs are not ignored. However, it is not easy to understand and its calculation is difficult when compared with the other measures of dispersion.
4. On squaring the deviations give more weightage to the extreme items and less to those which are near to the mean.

Graphical Representation of Data

The data are first tabulated in a frequency distribution. The next step is to analyse the data. It can be done by finding measures of central tendency and variability. But for quick and easy understanding, the data can be represented graphically. The following are the methods of graphical presentation of the data:

1. Histogram
2. Frequency polygon
3. Cumulative frequency curve or Ogive

The Histogram

The histogram is the most popular and widely used method of presenting a frequency distribution graphically.

Steps:

1. Draw two mutually perpendicular lines intersecting at 0.
2. Fix the horizontal line OX as X-axis and vertical line OY as Y-axis. 0 is the origin.
3. Mark the class intervals along the X-axis starting from the lowest and ending with the highest class interval of the frequency distribution. The class intervals in a histogram should be continuous. If they are discontinuous make them continuous.
4. Mark frequencies along Y-axis by selecting an appropriate scale.
5. Frequency of each class interval is drawn against both limits of the interval, thus making the rectangles with base as interval length and height equal to the respective frequencies.

Frequency Polygon

A frequency polygon is a many-sided figure representing the graph of a frequency distribution. In a frequency polygon, a mid-point of the class interval represents the entire interval. The frequency of the interval is drawn against the mid-point of the interval. The assumption is that all the scores are centred at the mid-point of the interval.

 

The following are the steps of construction:

1. Draw X- and Y-axis.
2. Mark class intervals along the X-axis.
3. Mark frequencies along the Y-axis. The scale of the axes should be chosen carefully so that the entire height of the Y-axis is approximately 3/4th of the length of the X-axis.
4. Draw points representing frequencies against mid-points of the respective intervals. For example, the mid-point of the first interval is 42 and frequency is 3. So draw a point on the graph representing the coordinate point (42, 3). Similarly draw points representing (47, 3), (52, 4) and (57, 6).
5. Join all the points in regular order by straight lines.
6. To complete the polygon, we have to join it on both ends with the X-axis. We consider an extra interval 35–39 at the lower end and one extra interval 95–99 at the upper end of the distribution on the X-axis. The frequency against each of these intervals is zero.

Smoothing the Frequency Polygon

Sometimes we find irregularities in the frequency distribution or data on a small sample. The frequency polygon of such distributions is jagged. To remove the irregularities and get a more clear perception of the data, the frequency polygon may be smoothed as shown in the figure. To smooth the polygon, running averages of frequencies are taken as new or adjusted or smoothed frequencies. To find the smoothed frequencies, we add on the given interval and the fs of two adjacent intervals and divide the sum by 3. For example, the smoothed frequency of interval 45–49 is (3 + 3 + 4)/3 = 3.33.

The process is illustrated as

Class Interval	Frequency (f)	Smoothed (f)
90–64	1	(0 + 1 + 4)/3 = 1.66
85–89	4	(1 + 4 + 2)/3 = 2.33
80–84	2	(4 – 2 – 8)/3 = 4.66
75–79	8	(2 + 8 + 9)/3 = 6.33
70–74	9	(8 + 9 + 14)3 = 10.33
65–69	14	(9 + 14 + 16)/3 = 9.66
60–64	6	(14 + 6 + 6) = 8.66
54–59	6	(6 + 6 + 4)/3 = 5.33
50–54	4	(6 + 4 + 3)/3 = 4.33
45–49	3	(4 + 3 + 3)/3 = 3.33
40–44	3	(3 + 3 + 0)/3 = 2

Cumulative Frequency Curve or Ogive

The cumulative frequency of an interval is found by adding to the frequency against it. It is the sum of all the frequencies against the interval below it. We start from the bottom for cumulating the frequency. Then we plot the cumulated frequency. Then we plot the cumulated frequencies instead of the respective frequencies against the intervals. It is to be noted that in a cumulative frequency curve, each cumulative frequency is plotted against the upper limit of the intervals. The process of finding cumulative frequencies and drawing the curve is illustrated below. The cumulative frequency curve starts at the lowest interval touching the X-axis, it rises gradually and becomes almost parallel to the X-axis after reaching the highest point.

Class Interval	Frequency (f )	Smoothed (f )
90–94	1	59 + 1 = 60
85–89	4	55 + 4 = 59
80–84	2	53 + 2 = 55
75–79	8	45 + 8 = 53
70–74	9	36 + 9 = 45
65–69	14	22 + 14 = 36
60–64	6	3 + 3 + 4 + 6 + 6 = 22
54–59	6	3 + 3 + 4 + 6 = 16
50–54	4	3 + 3 + 4 = 10
45–49	3	3 + 3 = 6
40–44	3	3
	N = 60

Figure 2.4 Cumulative frequency curve

Coefficient of Variation

All the measures of dispersion discussed so far have units. If two series differ in their units of measurement, their variability cannot be compared by any measure so far. Also, the size of measures of dispersion depends upon the size of the values. Hence, in situations where either of the two series has different units of measurements, or their means differ sufficiently in size, the coefficient of variation should be used as a measure of dispersion.

It is sometimes called the coefficient of relative variability. It is a unitless measure of dispersion and also takes into account the size of the means of the two series. It is the best measure to compare the variability of two series or sets of observations. A series with less coefficient of variation is considered more consistent.

Coefficient of variation of a series of variate values is the ratio of standard deviation to the mean multiplied by 100.

If σ is the standard deviation and is the mean of the set of values, the coefficient of variation is

An example will illustrate the use of the formula.

The Percentile

Percentile is nothing but a sort of measure used to indicate the relative position of a single item of individual in context with the group to which the item of individual belongs. In other words, it is used to tell the relative position of a given score among other scores. A percentile refers to a point in a distribution of scores or values below which a given percentage of the cases occur.

The percentile is named for the percentage of cases below it. Thus, 67 per cent of the observations are below the sixty-seventh percentile, which is written as P₆₇. The middle of a distribution or a point below which 50 per cent of cases lies in the fiftieth percentile P₅₀, which is the median, has been discussed in detail in the previous section. Similarly P₂₅ and P₇₅ are the first quartile Q₁ and third quartile Q₃, respectively, which have been previously discussed in the section.

Computation of Percentile

When we wish to compute the percentile, we will determine the score below which a given per cent of cases will fall. First the class in which the P^th percentile lies may be identified. This is the class in which PN/100th frequency falls. The formula for computing percentile is as follows:

where L is the lower limit of the P^th percentile class, N is the total number of cases, Cf is the less than cumulative frequency of class preceding to percentile class, f is the frequency of the percentile class and i is the class interval.

The computation procedure is clarified in the following example.

Percentile Ranks

The percentile rank of a given score in a distribution is the per cent of the total scores which fall below the given score. A percentile rank then indicates the position of a score in a distribution in percentile terms. If for example, a student had a score which was higher than 70 per cent of the scores in the distribution, but not higher than 71 per cent, his percentile rank would be 70.

Computation of Percentile Ranks

To compute the percentile rank for a score from the grouped data, we will need to determine the number of cases below the score in order to determine what per cent of the total cases would fall below that score. The formula for computing percentile rank is

where the symbols have the same meaning as in the case of percentiles.

Let us explain this in the following example.

Summary

1. There is a tendency for data to be clustered around or scattered away from the point of central tendency. This tendency is known as dispersion or variability. Measures of dispersion help us in studying the extent to which observations are scattered about the average or the central value.
2. There are five commonly used measures of dispersion - range, quartile deviation, mean deviation, standard deviation and variance. These have been discussed in detail in this chapter. However, in computation of further statistics from the measure of dispersion, we always prefer to compute standard deviation to all other measures of dispersion.
3. The computation of percentile and percentile ranks has also been discussed in this chapter. It helps in indicating the clear-cut relative position of an individual with respect to the same attribute in his own group.
4. The measures of variability are connected with spread of the scores and are a range, quartile deviation, mean deviation and standard deviation.
5. The data can also be represented graphically through histogram, frequency polygon and cumulative frequency curve or ogive.

Key Words

1. Measure of Dispersion: Measures the scatteredness from the central value.
2. Range: The difference between the largest and the smallest value in a set of observations.
3. Quartiles: Divides the whole distribution into four equal parts.
4. Quartile Deviation (Q): Half of the difference between the third quartile and the first quartile.
5. Mean Deviation (MD): The average of absolute deviations of every variate value from the mean.
6. Standard Deviation (SD): The positive square root of the arithmetic mean of the squares of deviation of the observations from the arithmetic mean.
7. Variance: The average of the squares of deviations of the observations from the arithmetic mean.
8. Coefficient of Variation (CV): The ratio of standard deviation to the mean multiplied by 100.
9. Percentile: A point in a distribution of scores or values below which a given percentage of the cases occur.
10. Percentile Rank: The number representing the percentage of the total number of cases lying below the given score.

References

1. Ferguson, G.A. (1980), Statistical Analysis in Psychology and Education. McGraw Hill Book Co.: New York, p. 20–21.

2. Garrett, H.E. (1986), Statistics in Psychology and Education. McGraw Hill: Tokyo, p. 15.

3. Guilford, J.P. (1978), Fundamental Statistics in Psychology and Education. McGraw Hill Book Co.: New York, p. 18–19.

4. Kurtz, A.K. and Mayo S.T. (1980), Statistical Methods in Education and Psychology. Narosa Publishing House: New Delhi, p. 10–11.

5. Mangal, S.K. (1987), Statistics in Psychology and Education. Tata McGraw Hill Publishing Company Ltd.: New Delhi, p. 24.

6. McCall, R.B. (1980), Fundamental Statistics for Psychology. Harcourt Brace Jovanavich Inc.: New York, p. 23.

7. Aggarwal, Y.P. (1988), Statistical Methods: Concepts Application and Computation. Sterling Publishers Pvt. Ltd.: New Delhi, p. 25–26.

Additional Readings

1. Grant, E.L. (1952), Statistical Quality Control IInd Ed. McGraw Hill Book Co. Inc.: New York.

2. Downie, N.M. and Heath, R.W. (1970), Basic Statistical Method. Harper & Row Publishers: New York.

3. Stanedecor, C.W. (1956), Statistical Methods. Eyova State College Press: Ames.

4. Pathak, R.P. (2007), Statistics in Educational Research. Kanishka Publishers & Distributors: New Delhi.

5. McNemer, Q. (1962), Psychological Statistics. John Wiley and Sons: New York.