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Chapter 3: Problem 67

Briefly describe how the percentiles are calculated for a data set.

Short Answer

Expert verified
Percentiles in a data set are calculated by first arranging the data in ascending order, then applying the formula \(P_k = \frac{{k (n+1)}}{100}\) where \(P_k\) is the \(k^{th}\) percentile and \(n\) is the number of values in the data set. The result might be interpolated between two values if it's not an integer.

Step by step solution

01

Understanding Percentiles

A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations falls. It divides the data into 100 equal parts, so the \(n^{th}\) percentile will have \(n\%\) of the observations below it.
02

Formula for Calculating Percentiles

The formula for calculating the \(k^{th}\) percentile (where \(k\) is any number between zero and one hundred) is given by: \[ P_k = \frac{{k (n+1)}}{100} \] where \(P_k\) is the \(k^{th}\) percentile and \(n\) is the number of values in the data set.
03

Calculation Process

First, arrange the values in the data set in ascending order. Then, substitute \(k\) and \(n\) into the formula. The result, \(P_k\), may not be an integer. If \(P_k\) is not an integer, round it off to the nearest whole number and find the corresponding value in the ordered data set. If the number is a decimal, then the \(k^{th}\) percentile is found by interpolating between the two nearest ranks.

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