Skewness:

Definition: Skewness is a measure of the asymmetry of a probability distribution. It indicates the extent to which a distribution differs from a symmetrical, bell-shaped curve.

There are two methods to calculate skewness: Karl Pearson's method and Prof. Bowley's method.

Karl Pearson's Method:

1. Calculate the mean (μ) and standard deviation (σ) of the data set.
2. Calculate the deviation of each value from the mean (x - μ).
3. Calculate the cube of each deviation.
4. Multiply each deviation cubed by its frequency.
5. Sum up all the frequency multiplied deviation cubes.
6. Divide the sum by (n * σ^3), where n is the total number of observations.
7. The result is the skewness coefficient.

Let's calculate the skewness using Karl Pearson's method:

Mean (μ) = ((6 * 10) + (11 * 20) + (12 * 30) + (14 * 40) + (20 * 50) + (18 * 60) + (13 * 70) + (9 * 80) + (6 * 90) + (3 * 100)) / (10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 100) = 48.3

Standard Deviation (σ) = sqrt((((10 - 48.3)^2 * 6) + ((20 - 48.3)^2 * 11) + ... + ((100 - 48.3)^2 * 3)) / (10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 100)) = 24.21

Deviations from the mean:
10 - 48.3 = -38.3
20 - 48.3 = -28.3
30 - 48.3 = -18.3
40 - 48.3 = -8.3
50 - 48.3 = 1.7
60 - 48.3 = 11.7
70 - 48.3 = 21.7
80 - 48.3 = 31.7
90 - 48.3 = 41.7
100 - 48.3 = 51.7

Cubed deviations:
(-38.3)^3 = -57178.647
(-28.3)^3 = -22991.267
(-18.3)^3 = -5992.647
(-8.3)^3 = -571.787
(1.7)^3 = 4.913
(11.7)^3 = 1941.963
(21.7)^3 = 10185.263
(31.7)^3 = 31000.747
(41.7)^3 = 68902.377
(51.7)^3 = 138334.717

Frequency multiplied deviation cubes:
-57178.647 * 6 = -343071.882
-22991.267 * 11 = -252902.937
-5992.647 * 12 = -71911.764
-571.787