Question

The 10th and 50th percentiles of the observation 32, 49, 23, 29, 118 respectively are

• A. 21, 32
• B. 23, 32
• C. 23, 33
• D. 22, 31

Hint:

To find percentiles, always start by arranging the data in ascending order. The 50th percentile is just another name for the median, which is the middle value of the dataset. For small datasets, percentiles near the ends (like the 10th) often correspond to the first or last value.

Answer 1

The Correct Option is B

First, arrange the observations in ascending order.
The observations are: 32, 49, 23, 29, 118.
In ascending order: 23, 29, 32, 49, 118.
There are \( n = 5 \) observations. The \( k \)-th percentile is the value below which \( k \)% of the observations may be found. The position of the \( k \)-th percentile is given by \( P = \frac{k}{100}(n+1) \). 

For the 10th percentile (\( P_{10} \)):
Position = \( \frac{10}{100}(5+1) = 0.1 \times 6 = 0.6 \). This means the value is between the 0th and 1st term. Since we start from the 1st term, it's very close to the start. In most simple definitions for small datasets, this would be the first value. Let's try another common formula \( P = \frac{k}{100} \times n \).
Position = \( \frac{10}{100} \times 5 = 0.5 \). This also indicates the first value.
The 10th percentile is the first value in the ordered list, which is 23.
 

For the 50th percentile (\( P_{50} \)): The 50th percentile is the median.
Position = \( \frac{50}{100}(5+1) = 0.5 \times 6 = 3 \).
The value at the 3rd position in the ordered list is the median.
The ordered list is: 23, 29, 32, 49, 118.
The 3rd value is 32.
So, the 10th and 50th percentiles are 23 and 32, respectively.