Measure of skewness
- 3. SKEWNESS • Skewness means lack of symmetry. • In skewed distribution, the mean and the median are pulled away from the mode. • Mean, median and mode are not equal. • A skewed distribution is an asymmetrical distribution. • It has a long tail on one side and short tail on the other side. • Eg:- Income, Savings, etc
- 4. TEST OF SKEWNESS To test whether a distribution is skewed or not, the following are to be noticed. A distribution is skewed if 1. mean, median and mode are not equal. 2. sum of positive deviations from median or not equal to the sum of negative deviation from median
- 5. 3. (a) Q1 and Q3 are not equidistant from median. (b) D1 and D9 are not equidistant from median. (c) P10 and P90 are not equidistant from median. 4. Frequencies on either side of modes are not equal. 5. The frequency curve has longer tail on the left side or on the right side.
- 7. • Skewness maybe either positive and negative. • Skewness is said to be positive when the mean is greater than the median and median is greater than mode. • Skewness is sais to be negative when the mean is less than the median and the median is less than mode. • For a positively skewed curve, there is longer tail to the right and for a negatively skewed curve, there is longer tail to left.
- 8. MEASURE OF SKEWNESS • Idea about the direction and extent of asymmetry in a series. • Compare two or more series and say which series has more skewness. • Absolute or relative . • Relative measures of skewness are also known as coefficients of skewness.
- 9. 1.) First measure of skewness: For skewed distribution the values of mean, median and mode are not equal. The distance between the mean and the mode can be used to measure the skewness. Coefficient of skewness, 𝐽 = 𝑀𝑒𝑎𝑛 − 𝑀𝑜𝑑𝑒 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 J will be between -3 and 3. Mode is indeterminate, then Mean-Mode can be taken as 3(Mean-Median) Also known as Karl Pearson’s coefficient of skewness
- 10. 2.) Second measure of skewness:- Q1 and Q3 are not equidistant from median. Difference between M-Q1 and Q3-M gives the measure of skewness. Coefficient of skewness, 𝑄3−𝑀 −(𝑀−𝑄1) 𝑄3−𝑄1 = (𝑄3+𝑄1−2𝑀) 𝑄3−𝑄1 This formula is known as Bowley’s coefficient of skewness.
- 11. 3.) Third measure of skewness:- The difference between D9-Median and Median-D1 give the measure of skewness. Absolute measure. Also known as Kelley’s coefficient of skewness (𝐷9 − 𝐷1 − 2𝑀𝑒𝑑𝑖𝑎𝑛) 𝐷9 − 𝐷1 4.) Fourth measure of skewness:- Based on third moment. The value of µ3 gives the absolute measure and the coefficient of the skewness = µ3 √µ2³
- 12. USES Concentrations is in higher and lower values. to study the distribution is normal or not.