Question

The following is the frequency distribution for the speeds of a sample of automobiles traveling on an interstate highway.

Speed (MPH)	Frequency
50 - 54	2
55 - 59	4
60 - 64	5
65 - 69	10
70 - 74	9
75 - 79	5
	35

Find the median and the mode.

Answer 1

The formula for median of frequency distribution or grouped data is as follows

Speed (MPH)	Frequency	Cumulative Frequency
50 - 54	2	2
55 - 59	4	6
60 - 64	5	11
65 - 69	10	21
70 - 74	9	30
75 - 79	5	35

Median is the middle value, here there are 35 observations, so median will be average of 17th and 18th values which implies the median class is 65-69 since 17th and 18th values fall in that class (cumulative frequecy of that class is 21)

So from the formula

L = lower limit of median class = 65

n = 35 here

cf = 11 (cumulative frequency of the class 60-64)

f = 10 (frequency of class 65-69)

h = 4 which is the class size

so Median = 65 + ( [ 35/2 - 11 ] / 10 ) * 4

= 65 + ( [ 17.5 - 11 ] / 10 ) * 4

= 65 + ( 6.5 / 10 ) * 4

= 65 + 0.65 * 4

= 65 + 2.6

= 67.6

So Median = 67.6

The formula for mode is as follows

Modal class is the class with the highest frequency which is 65-69 here

L = 65 ( lower limit of class 65-69)

f₁ = 10 ( frequency of the class 65-69)

f₀ = 5 ( frequency of the class 60-64)

f₂ = 9 ( frequency of the class 70-74)

h = 4 which is the class size

So Mode = 65 + [  (10 - 5) / (2 * 10 - 5 - 9) ] * 4

= 65 + [ 5 / 6 ] * 4

= 65 + 3.33333

= 68.33333

So Mode = 68.33333