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

Explanation:

Step1: Calculate the mean

The mean $\bar{x}$ of a data - set $x_1,x_2,\cdots,x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. 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: Calculate the median

First, arrange the data in ascending order (which it already is). Since $n = 10$ (an even - numbered data set), the median $M$ is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered data values. $\frac{n}{2}=5$ and $\frac{n}{2}+1 = 6$. The 5th value is 22 and the 6th value is 25. So, $M=\frac{22 + 25}{2}=23.5$.

Step3: Calculate the range

The range $R$ of a data - set is given by $R=\text{max}-\text{min}$. Here, $\text{max}=33$ and $\text{min}=11$. So, $R=33 - 11=22$.

Step4: Calculate the inter - quartile range

First, find the lower half and upper half of the data. The lower half is $11,13,17,20,22$ and the upper half is $25,27,31,31,33$. The median of the lower half ($Q_1$) is the 3rd value of the lower - half data set, so $Q_1 = 17$. The median of the upper half ($Q_3$) is the 3rd value of the upper - half data set, so $Q_3=31$. The inter - quartile range $IQR=Q_3 - Q_1=31 - 17 = 14$.

Answer:

Mean = 23 Median = 23.5 Range = 22 Interquartile range = 14