Step Deviation Method for Grouped Data

First, we need to find the mean of the data set by using the formula:

Mean = ∑(Midpoint x Frequency) / ∑Frequency

Where ∑ denotes the sum of the values, and Midpoint is the middle value of each class interval.

Using the given data, we can find the mean as follows:

Mean = ((20+19)/2 x 1) + ((19+18)/2 x 2) + ((18+17)/2 x 4) + ((17+16)/2 x 8) + ((16+15)/2 x 11) + ((15+14)/2 x 10) + ((14+13)/2 x 7) + ((13+12)/2 x 4) + ((12+11)/2 x 2) + ((11+10)/2 x 1) / (1+2+4+8+11+10+7+4+2+1)

Mean = 15.75

Next, we need to find the step deviation for each class interval by subtracting the mean from the midpoint of each class interval and then dividing by the class width. The formula is:

Step Deviation = (Midpoint - Mean) / Class Width

Using the mean we found earlier, we can find the step deviation for each class interval as follows:

Class Interval | Midpoint | Frequency | Midpoint - Mean | Step Deviation

20-19 | 19.5 | 1 | 3.75 | 0.375

19-18 | 18.5 | 2 | 2.75 | 0.275

18-17 | 17.5 | 4 | 1.75 | 0.175

17-16 | 16.5 | 8 | 0.75 | 0.075

16-15 | 15.5 | 11 | -0.25 | -0.025

15-14 | 14.5 | 10 | -1.25 | -0.125

14-13 | 13.5 | 7 | -2.25 | -0.225

13-12 | 12.5 | 4 | -3.25 | -0.325

12-11 | 11.5 | 2 | -4.25 | -0.425

11-10 | 10.5 | 1 | -5.25 | -0.525

Finally, we can find the mean deviation by using the formula:

Mean Deviation = ∑(Step Deviation x Frequency) / ∑Frequency

Using the step deviation we found earlier, we can find the mean deviation as follows:

Mean Deviation = (0.375 x 1) + (0.275 x 2) + (0.175 x 4) + (0.075 x 8) + (-0.025 x 11) + (-0.125 x 10) + (-0.225 x 7) + (-0.325 x 4) + (-0.425 x 2) + (-0.525 x 1) / (1+2+4+8+11+10+7+4+2+1)

Mean Deviation = 1.84

Therefore, the mean

Arithmetic Mean = 15.5