Question

question 2
consider the following data set. round your answers to the nearest hundredth as needed.
63 56 45 68 81
50 81 81 67 75

mean =

median =

mode =

range =

sample standard deviation =
question help: video 1 video 2 message

Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.
Here, $x_{1}=63,x_{2}=56,x_{3}=45,x_{4}=68,x_{5}=81,x_{6}=50,x_{7}=81,x_{8}=81,x_{9}=67,x_{10}=75$ and $n = 10$.
$\sum_{i=1}^{10}x_{i}=63 + 56+45+68+81+50+81+81+67+75=667$
$\bar{x}=\frac{667}{10}=66.7$

Step2: Calculate the median

First, order the data set: $45,50,56,63,67,68,75,81,81,81$
Since $n = 10$ (even), the median $M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}$
$x_{5}=67,x_{6}=68$
$M=\frac{67 + 68}{2}=67.5$

Step3: Calculate the mode

The mode is the most - frequently occurring value. In the data set $45,50,56,63,67,68,75,81,81,81$, the value $81$ occurs $3$ times. So the mode is $81$.

Step4: Calculate the range

The range $R=\text{Max}-\text{Min}$
$\text{Max}=81,\text{Min}=45$
$R=81 - 45=36$

Step5: Calculate the sample standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$
$\sum_{i=1}^{10}(x_{i}-\bar{x})^{2}=(63 - 66.7)^{2}+(56 - 66.7)^{2}+(45 - 66.7)^{2}+(68 - 66.7)^{2}+(81 - 66.7)^{2}+(50 - 66.7)^{2}+(81 - 66.7)^{2}+(81 - 66.7)^{2}+(67 - 66.7)^{2}+(75 - 66.7)^{2}$
$=(-3.7)^{2}+(-10.7)^{2}+(-21.7)^{2}+(1.3)^{2}+(14.3)^{2}+(-16.7)^{2}+(14.3)^{2}+(14.3)^{2}+(0.3)^{2}+(8.3)^{2}$
$=13.69+114.49+470.89+1.69+204.49+278.89+204.49+204.49+0.09+68.89$
$=1562.1$
$s=\sqrt{\frac{1562.1}{9}}\approx\sqrt{173.5667}\approx13.18$

Answer:

Mean = $66.7$
Median = $67.5$
Mode = $81$
Range = $36$
Sample Standard Deviation = $13.18$