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After reading this term paper you will learn about the various measures of dispersion used in social research.

**Measures of Dispersion
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**Term Paper Contents: **

- Term Paper on Range
- Term Paper on Semi-lnter-Quartile Range or Quartile Deviation
- Term Paper on Mean Deviation
- Term Paper on Standard Deviation
- Term Paper on the Coefficient of Variation

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In social research, we often wish to know the extent of homogeneity and heterogeneity among respondents with respect to a given characteristic. Any set of social data has values which may characterize heterogeneity. The set of social data is typically characterized by the heterogeneity of values.

In fact, the extent to which they are heterogeneous or vary among themselves, is of basic importance in statistics. Measures of central tendency describe one important characteristic of a set of data typically but they do not tell us anything about this other basic characteristic.

Consequently, we need ways of measuring heterogeneity — the extent to which data are dispersed. The measures which provide this description are called measures of dispersion or variability. The following three distributions shown in Fig. 18.4 will illustrate the importance of measuring the dispersion of statistical data.

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**Distribution of Mean Values for Samples of Different Sizes**:

It can be seen that the arithmetic mean of all the three curves in the above figure is the same, but the distribution of values as depicted by curve A shows less variability (dispersion) than that depicted by curve B, while curve B has less variability as compared to that shown by curve C.

If we consider only the measure of central tendency of distributions, we will miss an important difference among the three curves. To get a better understanding of the pattern of the data, we must also get the measure of its dispersion or variability, we now turn to consider various measures of dispersion.

**Term Paper # 1. Range: **

The range is defined as the difference between the highest and lowest values: Mathematically,

R (Range) = M_{n} – M_{L}

where M_{n} and M_{l} stand for the highest and the lowest value. Thus, for the data set: 10, 22, 20, 14 and 14 the range would be the difference between 22 and 10, i.e., 12. In case of grouped data, we take the range as the difference between the midpoints of the extreme classes. Thus, if the midpoint of the lowest interval is 150 and that of the highest is 850, the range will be 700.

The only advantage of range, which measure of dispersion is seldom used, is that it can be easily calculated and readily understood. Despite this advantage, it is generally not a very useful measure of dispersion; its main drawback being that it does not tell us anything about the dispersion of values which are intermediate between the two extremes.

**Term Paper # 2. ****Semi-lnter-Quartile Range or Quartile Deviation****: **

Another measure of dispersion is the semi-inter-quartile range, commonly known as Quartile Deviation. Quartiles are the points which divide the array or series of values into four equal parts each of which contains 25 per cent of the items in the distribution. The quartiles are then the highest values in each of these four parts. Inter-quartile range is the difference between the values of first and third quartiles.

Thus, where and Q_{1 }and Q_{3} stand for first and third quartiles, the semi-inter-quartile range or quartile deviation is given by formula _{=} Q_{3} –Q_{1}/ 2

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**Calculation of Quartile Deviation****:**

Quartile deviation is an absolute measure of dispersion. If quartile deviation is to be used for comparing the dispersions of series it is necessary to convert the absolute measure to a coefficient of quartile deviation.

**Term Paper # 3. ****Mean Deviation**:

Range and quartile deviation suffer from serious drawbacks, i.e., they are calculated by taking into consideration only two values of a series. Thus, these two measures of dispersion are not based on all observations of the series. As a result, the composition of the series is entirely ignored. To avoid this defect, dispersion may be calculated taking into consideration all the observations of the series in relation to a central value.

The method of calculating dispersion is called the method of averaging deviations (mean deviation). As the name clearly suggests, it is the arithmetic average of the deviations of various items from a measure of central tendency.

As we well know, the sum of deviations from a central value would always be zero. This suggests that in order to obtain a mean deviation (about the mean or any one of the central values), we must somehow or the other get rid of any negative signs. This is done by ignoring signs and taking the absolute value of the differences.

In our hypothetical example, the mean of the number 12, 14, 15, 16 and 18 is 15. This implies that difference of 15 from each of these numbers, ignoring the signs all along and then adding the results, we will get the total deviation.

**Dividing it by 5, we get: **

= 1.6 (where |d | stands for sum of absolute deviations).

We may therefore say that on an average the scores differ from the mean by 1.6.

**Calculation of mean Deviation in Ungrouped date (Individual Observations):**

**Calculation of mean deviation in Continuous Series:**

**Coefficient of Mean Deviation**:

To compare the mean deviation of series the coefficient of mean deviation or relative mean deviation is calculated. This is obtained by dividing the mean deviation by that measure of central tendency from which deviations were calculated. Thus,

Coefficient of Mean. Deviation /X

Applying this formula to the previous example, we have,

Coefficient of Mean Deviation = 148/400= 0.37

**Term Paper # 4. ****Standard Deviation**:

The most useful and frequently used measure of dispersion is the standard deviation or root-mean square deviation about the mean. The standard deviation is defined as the square root of the arithmetic mean of the square of the deviations about the mean. Symbolically,

σ =√Σd^{2 }/N

where σ (Greek letter Sigma) stands for the standard deviation, Σd^{2} for the sum of the square of the deviations measured from mean and N for the number of items.

**Calculation of Standard Deviation in Series of Individual Observations:**

**Short-Cut Method:**

**Calculation of Standard Deviation in Discrete Series**:

In a discrete series the deviations from an assumed mean are first computed and multiplied by the respective frequencies of items. The deviations are squared and multiplied by the respective frequencies of the items. These products are totaled and divided by the total of the frequencies. The standard deviation is calculated by the following formula:

**The following illustration would explain the formula: **

**Calculation of Standard Deviation in a Continuous Series**:

In a continuous series the class intervals are represented by their mid-points. However, usually the class-intervals are of equal size and thus, the deviations from the assumed average is expressed in class interval units. Alternatively, step deviations are arrived at by dividing the deviations by the magnitude of the class interval.

**Thus, the formula for computing standard deviation is written as under: **

where i stands for the common factor or the magnitude of the class-interval.

**The following example would illustrate this formula: **

**Term Paper # 5. ****Coefficient of Variation****: **

The standard deviation represents measure of absolute dispersion. It is also necessary to measure the relative dispersion of two or more distributions. When the standard deviation is related to its mean, it measures relative dispersion. Karl Pearson has worked out a simple measure of relative dispersion which is generally known as the coefficient of variation.

**The coefficient of variation for the problem in Table 18.47 is: **