EXAMPLE 2.9.6

The frequency table below represents the distribution of scores on a ten-point quiz. Compute the mean, median, and mode for this distribution.

Quiz Scores

Quiz Scores

VALUEFREQUENCY
Value 5Frequency 6
Value 6Frequency 8
Value 7Frequency 14
Value 8Frequency 22
Value 9Frequency 28
Value 10Frequency 36

 

EXAMPLE 2.9.6 SOLUTION

 

MODE

The mode = 10, since that is the value with the highest frequency.

 

The mean and median both require that we find n, the population size.

n = sum of frequencies = 6 + 8 + 14 + 22 + 28 + 36 = 114

 

 

MEAN

mean = [(5 times 6) plus (6 times 8) plus (7 times 14) plus (8 times 22) plus (9 times 28) plus (10 times 36)] divided by =964 divided by 114

= 8.46 (this has been rounded)

 

MEDIAN

To find the median, we use the fact that the position of the middle value is

(n + 1) divided by 2 = 115 divided by two
= 57.5

Since 57.5 is between 57 and 58, the median is determined by the average of the 57th and 58th numbers on the list. We will count frequencies to determine the 57th and 58th numbers. The first six numbers are all fives. The next 8 numberes are sixes, so after reading through all the fibes and sixes we have gone through 14 numbers.
The next 14 numbers are sevens, so after going through all the fives, sixes and sevens we have gone through 28 numbers. Next, we have 22 eights, so after we have read all the fives, sixes, sevens and eights from the list , we have gone through 50 numbers. The next 28 numbers are all nines, so now we have found the 57th and 58th numbers. In fact, the fifty-first through seventy-eighth numbers are all nines, so in particular, the fifty-seventh and fifty-eighth numbers are both nines.

The median is 9.