Question

Use the display of data items to find the standard deviation.
A bar graph has a horizontal axis labeled Score from 6 to 12 in increments of 1 and a vertical axis labeled Frequency from 0 to 12 in increments of 2 contains vertical bars with labels and heights as follows, where the horizontal axis label is listed first and the height is listed second: 6, 5; 7, 0; 8, 0; 9, 3; 10, 0; 11, 0; 12, 5.
6
7
8
9
10
11
12
0
2
4
6
8
10
12
Frequency

Answer 1

First find the mean:
Total frequency N = 5 + 3 + 5 = 13.
Sum of scores = 6(5) + 9(3) + 12(5) = 30 + 27 + 60 = 117.
Mean μ = 117/13 = 9.

Compute squared deviations:
For 6: (6−9)^2 = 9, contribution 9·5 = 45.
For 9: (9−9)^2 = 0, contribution 0·3 = 0.
For 12: (12−9)^2 = 9, contribution 9·5 = 45.
Sum of squared deviations = 45 + 0 + 45 = 90.

Population variance = 90/13 ≈ 6.9231, so population standard deviation ≈ sqrt(6.9231) ≈ 2.63.

(If you need the sample standard deviation, divide by 12 instead: s = sqrt(90/12) ≈ 2.74.)