A researcher has collected the following sample data. 15 12 16 18 15 16 17 15 12 14 The 75th percentile is
A 16
B 13
C 14
D 15
The Correct Answer and Explanation is :
To find the 75th percentile (Q3) of the given data set:
Step 1: Arrange the Data in Ascending Order
The given data set is:
12, 12, 14, 15, 15, 15, 16, 16, 17, 18
Step 2: Determine the Position of the 75th Percentile
The 75th percentile (Q3) is found using the formula:
[
P = \frac{75}{100} \times (n + 1)
]
where n is the number of data points.
[
P = \frac{75}{100} \times (10 + 1) = 0.75 \times 11 = 8.25
]
This means the 75th percentile is located at the 8.25th position.
Step 3: Find the 8.25th Position
- The 8th value in the ordered dataset is 16.
- The 9th value is 17.
- Since 8.25 is slightly beyond the 8th position, we interpolate between the 8th and 9th values:
Using interpolation:
[
Q3 = 16 + 0.25 \times (17 – 16)
]
[
Q3 = 16 + 0.25 \times 1 = 16 + 0.25 = 16.25
]
Step 4: Choose the Closest Answer
The closest answer among the given options is 16.
Correct Answer:
A) 16
Explanation:
The 75th percentile (Q3) is the value that separates the highest 25% of the data. To find it, we use the formula P = 0.75 × (n + 1), determine its position, and interpolate if necessary. Here, the 8.25th position corresponds to 16.25, which rounds to 16, making option A the correct answer.