Question

for the data shown, answer the questions. round to 2 decimal places.

|x|
|----|
|16.2|
|10.2|
|17.5|
|1.9|
|22.7|
|2.1|

compute the sample mean:

compute the median:

compute the sample standard deviation:

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Explanation:

Step1: Calculate sample mean

The formula for the sample mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 6$ and $x_{1}=16.2,x_{2}=10.2,x_{3}=17.5,x_{4}=1.9,x_{5}=22.7,x_{6}=2.1$.
$\sum_{i=1}^{6}x_{i}=16.2 + 10.2+17.5 + 1.9+22.7+2.1=70.6$
$\bar{x}=\frac{70.6}{6}\approx11.77$

Step2: Calculate median

First, order the data: $1.9,2.1,10.2,16.2,17.5,22.7$. Since $n = 6$ (an even - numbered data set), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered values.
The $\frac{n}{2}=3$rd value is $10.2$ and the $(\frac{n}{2}+1) = 4$th value is $16.2$.
Median$=\frac{10.2 + 16.2}{2}=\frac{26.4}{2}=13.20$

Step3: Calculate sample standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
$(x_{1}-\bar{x})^{2}=(16.2-11.77)^{2}=(4.43)^{2}=19.6249$
$(x_{2}-\bar{x})^{2}=(10.2-11.77)^{2}=(-1.57)^{2}=2.4649$
$(x_{3}-\bar{x})^{2}=(17.5-11.77)^{2}=(5.73)^{2}=32.8329$
$(x_{4}-\bar{x})^{2}=(1.9-11.77)^{2}=(-9.87)^{2}=97.4169$
$(x_{5}-\bar{x})^{2}=(22.7-11.77)^{2}=(10.93)^{2}=119.4649$
$(x_{6}-\bar{x})^{2}=(2.1-11.77)^{2}=(-9.67)^{2}=93.5089$
$\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=19.6249+2.4649+32.8329+97.4169+119.4649+93.5089 = 365.3134$
$s=\sqrt{\frac{365.3134}{6 - 1}}=\sqrt{\frac{365.3134}{5}}=\sqrt{73.06268}\approx8.55$

Answer:

Sample mean: $11.77$
Median: $13.20$
Sample standard deviation: $8.55$