To find the 30th and 90th percentiles of a dataset, you first need to understand the concept of percentiles. Percentiles are values below which a certain percentage of the data falls. For example, the 30th percentile is the value below which 30% of the observations fall.
Here’s how to calculate the 30th and 90th percentiles for the dataset: 129, 113, 200, 100, 105, 132, 100, 176, 146, 152.
Step 1: Organize the Data
First, arrange the data in ascending order:
100, 100, 105, 113, 129, 132, 146, 152, 176, 200
Step 2: Calculate the Percentile Ranks
The formula to find the rank (position) of a percentile is:
P = (N + 1) * (percentile/100)
where P is the position, N is the number of data points, and percentile is the desired percentile.
For our dataset, N = 10.
Calculating the 30th Percentile
Using the formula:
Position = (10 + 1) * (30/100) = 11 * 0.3 = 3.3
The rank is 3.3, indicating that the 30th percentile falls between the 3rd and 4th values in the sorted list. We interpolate between the 3rd (105) and the 4th (113):
30th Percentile = 105 + (0.3 * (113 - 105)) = 105 + 2.4 = 107.4
Calculating the 90th Percentile
Now, for the 90th percentile:
Position = (10 + 1) * (90/100) = 11 * 0.9 = 9.9
The rank is 9.9, which indicates the 90th percentile falls between the 9th (176) and the 10th (200) values in the sorted list. We interpolate between the two:
90th Percentile = 176 + (0.9 * (200 - 176)) = 176 + 21.6 = 197.6
Final Results
So, the 30th percentile is approximately 107.4, and the 90th percentile is approximately 197.6.