Answers:
Sample Variance = 31.03
Sample Standard Deviation = 5.57
See the table below.
===================================================
Explanation:
M refers to the midpoint of each interval. The midpoint is [tex]M = \frac{a+b}{2}[/tex] where a,b are the left and right endpoints.
For example, [tex]M = \frac{0+4}{2} = \frac{4}{2} = 2[/tex] in the first row.
Multiply the frequency with the midpoint to get the third column. Summing this column of values leads to 4+49+180+153+154 = 540 shown at the bottom of that column.
Divide this sum over the sum of the frequencies (2+7+15+9+7 = 40) and we arrive at [tex]\overline{x} = \frac{540}{40} = 13.5[/tex] which is the sample mean xbar.
--------------------------------------------------------------------------
After we determine xbar, we subtract it from each value of M. Then we square the result to get [tex](M - \overline{x})^2[/tex]. Multiply that with the column f to get the new column [tex]f(M-\overline{x})^2[/tex].
This column is added to arrive at 1210 shown in the table below. Divide this over the sum of the frequencies minus 1. So we divide by n-1 = 40-1 = 39 to get roughly 31.03
This is the sample variance.
The square root of this is sqrt(31.03) = 5.57 and this is the approximate sample standard deviation.
