Question

A statistics professor gave a 5-point quiz to the 50 students in his class. Scores on the quiz could range from 0 to 5: The following frequency table resulted: (1.5 points)

Quiz Score	f	rf	cf	crf	c%
5	4	.08	50	1.00	100%
4	10	.20	46	.96	96%
3	14	.28	36	.72	72%
2	10	.20	22	.44	44%
1	8	.16	12	.24	24%
0	4	.08	4	.08	8%

1. Compute the values that define the following percentiles:

a. 25^th   2     b. 50^th   3     c. 55^th   3    d. 75^th   4e. 80^th   4     f. 99^th    5    

2. What is the interquartile range of the data in #1?

3. Compute the exact percentile ranks that correspond to the following scores:

a. 2        b. 3        c. 4        d. 1   

Answer 1

25th percentile is the first score where the cumulated percentage exceeds 25%.
Here it is 2 (c% of score 2 = 44%).
If the students are ranked by Quiz score, the last student with a score of 1 is ranked 12 (c% = 24 %) and the last student with a score of 2 is ranked 22 (c% = 44 %).

Any student between 24 % and 44 % will have a score of 2

25 th percentile = 2

50 th percentile = 3

55 th percentile = 3

75 th percentile = 4

80 th percentile = 4

99 th percentile = 5

Below table is for reference

b) IQR = 75 th percentile - 25 th percentile

IQR = 4 - 2 = 2

c) The formula to calculate the percent rank is

Position of the number = (n + 1 ) * Percent rank/100

n= 50

Position of number 2 = 13

Therefore,

13 = 51 * Percent rank of 2/100

13 * 100/ 51 = Percent rank of 2

Percent rank of 2 = 25.4

Percent rank of 3 = 23 * 100/51 = 45.1

Percent rank of 4 = 37 * 100/51 = 72.5

Percent rank of 1 = 5 * 100/51 = 9.8

Also, easier way of approximation is as below

The percentile rank of a quiz score is defined as the percentage of the scores in the data set which are at or below the particular score.

Percentile rank for 2 = 24

Percentile rank for 3 = 44

Percentile rank for 4 = 72

Percentile rank for 1 = 8