To find the 30th and 90th percentiles of the given data set, we first need to organize the data in ascending order:
Sorted Data: 100, 100, 105, 113, 129, 132, 146, 152, 176, 200
Next, we can use the formula for finding the percentile:
P = (n + 1) * (percentile / 100)
where n is the number of data points.
In our case, n = 10.
1. **Calculating the 30th percentile:**
P = (10 + 1) * (30 / 100) = 11 * 0.3 = 3.3
This means the 30th percentile falls between the 3rd and 4th values in the sorted data.
Values: 105 (3rd) and 113 (4th)
To calculate the 30th percentile, we can use linear interpolation:
P30 = 105 + 0.3 * (113 – 105) = 105 + 0.3 * 8 = 105 + 2.4 = 107.4
2. **Calculating the 90th percentile:**
P = (10 + 1) * (90 / 100) = 11 * 0.9 = 9.9
This means the 90th percentile falls between the 9th and 10th values in the sorted data.
Values: 176 (9th) and 200 (10th)
Using linear interpolation again:
P90 = 176 + 0.9 * (200 – 176) = 176 + 0.9 * 24 = 176 + 21.6 = 197.6
In conclusion:
- 30th Percentile: 107.4
- 90th Percentile: 197.6