Question

find each measure for the given set of data: 11, 13, 17, 20, 22, 25, 27, 31, 31, 33
mean =
median =
range =
interquartile range =

Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Here, $n = 10$, and $\sum_{i=1}^{10}x_{i}=11 + 13+17+20+22+25+27+31+31+33=230$. So, $\bar{x}=\frac{230}{10}=23$.

Step2: Find the median

Since $n = 10$ (an even - numbered data set), the median $M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}$. Here, $x_{5}=22$ and $x_{6}=25$. So, $M=\frac{22 + 25}{2}=23.5$.

Step3: Calculate the range

The range $R=x_{\max}-x_{\min}$. Here, $x_{\max}=33$ and $x_{\min}=11$. So, $R = 33-11=22$.

Step4: Calculate the inter - quartile range

First, find the first quartile $Q_{1}$. For $n = 10$, the position of $Q_{1}$ is $\frac{n + 1}{4}=\frac{10+1}{4}=2.75$. So, $Q_{1}=x_{2}+0.75(x_{3}-x_{2})=13+0.75(17 - 13)=13 + 3=16$.
The position of the third quartile $Q_{3}$ is $3\times\frac{n + 1}{4}=3\times\frac{10 + 1}{4}=8.25$. So, $Q_{3}=x_{8}+0.25(x_{9}-x_{8})=31+0.25(31 - 31)=31$.
The inter - quartile range $IQR=Q_{3}-Q_{1}=31 - 16=15$.

Answer:

Mean = 23, Median = 23.5, Range = 22, Interquartile range = 15