**Type II error - An error caused by failing to reject a null hypothesis that is not true**

** CENTRAL TENDENCY OF DATA**

**5. The curve**

When the data are plotted on a graph paper, the curve will not be bell-shaped, or when cut along a vertical line through the centre, the two halves will not be identical.

Conversely stated, in the absence of skewness in the distribution:

(i) Values of mean, median and mode will coincide.

(ii) Sum of the positive deviations from the median or mode will be equal to the sum of negative deviations.

(iii) The two quartiles, deciles one and nine, and percentile ten and ninety will be equidistant from the median.

(iv) Frequencies on the either side of the mode will be equal.

(v) Data when plotted on a graph paper will take a bell-shaped form.

**Measures of Skewness**

To find out the direction and the extent of symmetry in a series statistical measures of skewness are calculated, these measures can be absolute or relative. Absolute measures of skewness tell us the extent of asymmetry and whether it is positive or negative. The absolute skewness can be known by taking the deference between mean and mode.

Symbolically,

**Absolute SK = X - Mo**

If the value of mean is greater than the mode (M > Mo) skewness will be positive. In case the value of mean is less than the mode (M < Mo) skewness will be negative. The greater is the amount of skewness, the more the mean and mode differ because of the influence of extreme items. The reason why the difference between mean and mode is taken for the measure of skewness is that in a symmetrical distribution, both the values along with median coincide, but in an asymmetrical distribution, there will be a

difference between the mean and mode.

Thus the difference between the mean and the mode, whether positive or negative, indicates that the distribution is asymmetrical. However such absolute measure of skewness is unsatisfactory, because:

(1) It cannot be used for comparison of skewness in tow distributions if they are in different units, because the difference between the mean and the mode will be in terms of the units of distribution.

(2) The difference between the mean and mode may be more in one series and less in another, yet the frequency curves of the two distributions may be similarly skewed. For comparison, the absolute measures of skewness are changed to relative measures, which are called Coefficient of Skewness.

There are four measures of relative skewness. They are:

1. The Karl Pearson's Coefficient of Skewness 2. The Bowley's Coefficient of Skewness.

3. The Kelly's Coefficient of Skewness

4. Measure of skewness based onmoments. Measures of Skewness
**1. The Karl Pearson's Coefficient of Skewness**

Karl Pearson has given a formula, for relative measure of Skewness. It is known as Karl Pearson's Coefficient of Skewness or Pearsonian Coefficient of Skewness. The formula is based on the difference between the mean and mode divided by the standard deviation.

The coefficient is represented by J

If in a particular frequency distribution, the mode is ill-defined, the coefficient of Skewness can be determined by the following changed formula.

This is based on the relationship between different averages in a moderately asymmetrical distribution. In such a distribution:

The Pearsonian coefficient of skewness has the interesting characteristic that it will be positive when the mean is larger than the mode or median, and it will be negative when the arithmetic mean is smaller than the mode or median. In a symmetrical distribution, the value of Pearsonian coefficient of skewness will be zero.

There is no theoretical limit to this measure, however, in practical the value given by this formula is rarely very high and usually lies between ±∞. The direction of the skewness is given by the algebraic sign of the measure; if it is plus then the skewness is positive, if it is minus, the skewness is negative. The degree of skewness is obtained by the numerical figure such as 0.9, 0.4, etc.

Thus this formula gives both the direction as well as the degree of skewness. There is another relative measure of skewness also based on the position of averages. In this, the difference between two averages is divided by the mean deviation. The formula is:

These formulas are not very much used in practice, because of demerits of mean deviation.

**DISPERSION**

**Measures of Dispersion**

An average nay give a good idea of the type of data, but it alone can't reveal all the characteristics of data. It cannot tell us in what manner all the values of the variable are scattered / dispersed about the average.

**Meaning of Dispersion**

The Variation or Scattering or Deviation of the different values of a variable from their average is known as Dispersion. Dispersion indicates the extent to which the values vary among themselves. Prof. W.I. King defines the term, 'Dispersion' as it is used to indicate

the facts that within a given group, the items differ from another in size or in other words, there is lack of uniformity in their sizes. The extent of variability in a given set of data is measured by comparing the individual values of the variable with the average all the values and then calculating the average of all the individual differences.

**Objectives of Measuring Variations**

1. To serve as a basis for control of the variability itself.

2. To gauge the reliability of an average

3. To serve as a basis for control of the variability itself
**Types of Measures of Dispersion**

There are two types of measures of dispersion. The first, which may be referred to as Distance Measures, describes the spread of data in terms of distance between the values of selected observations. The second are those which are in terms of an average

deviation from some measure of central tendency.

**Absolute and Relative Measures of Dispersion**

Measures of absolute dispersion are in the same units as the data whose scatter they measure. For example, the dispersion of salaries about an average is measured in rupees and the variation of time requires for workers to do a job is measured in minutes or hours. Measures of absolute dispersion cannot be used to compare the scatter in one distribution with that in another distribution when the averages of the distributions differ in size or the units of measure differ in kind. Measures of relative dispersion show some measure of scatter as a percentage or coefficient of the absolute measure of

dispersion. They are generally used to compare the scatter in one distribution with the scatter in other. Relative measure of dispersion is called coefficient of dispersion.

**Methods of Measuring Dispersion**

There are two meanings of dispersion, as explained above. On the basis of these two meanings, there are two mathematical methods of finding dispersion, i.e. methods of limits and methods of moment. Dispersion can also be studied graphically. Thus, the following are the methods of measuring dispersion:

**I. Numerical Methods**