Answer:
20th percentile: 22.8
25th percentile: 24
65th percentile: 26.6
75th percentile: 29
Explanation:
To compute the percentiles for the given data set, you can follow these steps:
First, arrange the data in ascending order:
16, 21, 24, 24, 26, 29, 31, 35
Calculate the position of each percentile:
20th percentile: 20% of 8 (number of data points) = 0.20 * 8 = 1.6
25th percentile: 25% of 8 (number of data points) = 0.25 * 8
= 2
65th percentile: 65% of 8 (number of data points) = 0.65 * 8
= 5.2
75th percentile: 75% of 8 (number of data points) = 0.75 * 8
= 6
Since the positions are not whole numbers, we need to interpolate to find the values at these percentiles.
For the 20th percentile, interpolate between the 1st and 2nd data points:
21 + (0.6 * (24 - 21))
= 21 + 0.6 * 3 = 21 + 1.8
= 22.8
For the 25th percentile, it falls exactly on the 2nd data point, which is 24.
Further explanation: The position of the 25th percentile is 2 (since 25% of 8 data points is 2). This means we're looking for the value that divides the lower 25% of the data from the upper 75%. In the sorted data set:
16, 21, [24], 24, 26, 29, 31, 35
You can see that the 25th percentile falls right in the middle of the two 24s, so it's logical to take 24 as the 25th percentile.
For the 65th percentile, interpolate between the 5th and 6th data points:
26 + (0.2 * (29 - 26))
= 26 + 0.2 * 3
= 26 + 0.6
= 26.6
For the 75th percentile, it falls exactly on the 6th data point, which is 29.
Further explanation: the position of the 75th percentile is 6 (since 75% of 8 data points is 6). This means we're looking for the value that divides the lower 75% of the data from the upper 25%. Again, in the sorted data set:
16, 21, 24, 24, 26, [29], 31, 35
The 75th percentile falls right on the 29, so we take 29 as the 75th percentile.
So, the percentiles are as follows:
20th percentile: 22.8
25th percentile: 24
65th percentile: 26.6
75th percentile: 29