To find the percentiles of a dataset, we can use the formula for the rank (position) of the percentile:
\[ P = \frac{n \cdot k}{100} \]
Where:
- \(P\) is the rank (position) of the percentile,
- \(n\) is the total number of observations in the dataset,
- \(k\) is the desired percentile.
After calculating \(P\), if it is not a whole number, we round it up and round down to find the nearest ranks, then interpolate if necessary.
Here, the dataset contains 60 values.
Given Dataset:
\[ 2, 4, 4, 5, 7, 8, 9, 10, 12, 12, 13, 13, 16, 16, 16, 16, 17, 19, 19, 20, 23, 24, 24, 24, 25, 26, 26, 27, 28, 28, 29, 31, 32, 34, 34, 36, 37, 38, 42, 44, 45, 46, 47, 47, 48, 50, 52, 53, 53, 54, 55, 56, 56, 57, 58 \]
Finding Percentiles:
1. 80th Percentile
\[ P = \frac{60 \cdot 80}{100} = 48 \]
The 80th percentile is the value at the 48th position in the sorted dataset.
From the dataset:
- 48th value = 56
2. 43rd Percentile
\[ P = \frac{60 \cdot 43}{100} = 25.8 \]
We round up to 26 to find the value at the 26th position.
From the dataset:
- 26th value = 26
3. 18th Percentile
\[ P = \frac{60 \cdot 18}{100} = 10.8 \]
We round up to 11.
From the dataset:
- 11th value = 13
4. 66th Percentile
\[ P = \frac{60 \cdot 66}{100} = 39.6 \]
We round up to 40.
From the dataset:
- 40th value = 47
Summary of Results:
- 80th Percentile: 56
- 43rd Percentile: 26
- 18th Percentile: 13
- 66th Percentile: 47