A random sample of bonuses (in millions of dollars) paid by large companies to their executives is shown below.
Class boundary
Frequency
0.5----3.5
11
3.5----6.5
12
6.5----9.5
4
9.5----12.5
2
12.5---15.5
1
Find
P25
To find P25 (the 25th percentile), we need to find the value that separates the lowest 25% of the data from the rest of the data.
First, we calculate the total number of observations:
11 + 12 + 4 + 2 + 1 = 30
Next, we calculate the position of the 25th percentile in the data set:
25/100 * 30 = 7.5
Since the position is not an integer, we need to take the average of the 7th and 8th values in the data set.
Since 7 is in the 0.5-3.5 class, and the boundaries for this class are 0.5 and 3.5, we can calculate the width of this class:
3.5 - 0.5 = 3
To find the value at the 25th percentile, we can use the formula:
P25 = lower boundary of the class + [(position - cumulative frequency of the class before) / frequency of the class] * width of the class
P25 = 0.5 + [(7.5 - 0) / 11] * 3
P25 = 0.5 + (7.5/11) * 3
P25 = 0.5 + 0.6818181818181818 * 3
P25 = 0.5 + 2.0454545454545454
P25 ≈ 2.545
Therefore, the 25th percentile of the data set is approximately 2.545 million dollars.