Question

A professor of History is teaching a section of 100 students. Her first exam’s grade distribution follows.
Exam grades

Frequency

45 to < 50

1

50 to < 55

2

55 to < 60

6

60 to < 65

19

65 to < 70

12

70 to < 75

22

75 to < 80

12

80 to < 85

13

85 to < 90

11

90 to < 95

0

95 to < 100

2

Calculate the standard deviation for this grouped data.

Answer 1

The formula for standard deviation ($\sigma$) for grouped data is given by,

$\sigma = \sqrt{\frac{\sum f*(m-\bar x)^2}{\sum f}}$

where,

f = frequency of the class

m = class midpoint

$\bar x$ = mean

and the formula for the mean $\bar x = \frac{\sum(f*m)}{\sum f}$

From the given data we can find the values of above variables as under-

Hence we get,

mean $\bar x = 72$

standard deviation $\sigma = 10.1488$