Question

2. The following represents 50 test scores on an exam.

49, 48, 51, 50, 53, 52, 55, 54, 57, 56, 59, 58, 61, 60, 63, 62, 65, 64, 67, 66, 69, 68, 71, 70, 73, 72, 75, 74, 77, 76, 79, 78, 81, 80, 83, 82, 85, 84, 87, 86, 89, 88, 91, 90, 93, 92, 95, 94, 97, 96

1. Find the position of the median

2. Find the 67^th percentile

Answer 1

a) Given that,

49, 48, 51, 50, 53, 52, 55, 54, 57, 56, 59, 58, 61, 60, 63, 62, 65, 64, 67, 66, 69, 68, 71, 70, 73, 72, 75, 74, 77, 76, 79, 78, 81, 80, 83, 82, 85, 84, 87, 86, 89, 88, 91, 90, 93, 92, 95, 94, 97, 96

a) for calculating median

Here, n=50

The given data is even

Sorted given data in ascending order,

48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97

Median =(50+1)/2=25.5th item

Thus, the median is the average of the 25th and 26th item. 25th and 26th items are 72 and 73

Thus the exact position of median is 25th and 26th value.

Median is (72+73)/2 = 72.5

Median will be –

Thus, the Median = 72.5

b) 67th percentile =?

Arrange the given data in ascending order:

48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97

Compute the position of pth percentile,

i=(p/100)×n

=(67/100)×50

i=33.5

Since index ii is not an integer, round up:

i= 34i = 34.

The percentile is at the position i= 34i= 34: it is 81.

Answer: the 67^th percentile is 81.