Question

for the sample data shown, answer the questions. round to 2 decimal places.
|x|
|----|
|12.2|
|12.6|
|16.6|
|16.8|
|19.8|
|19.9|
|20.7|
|23.2|
|29.3|
|30|
find the mean:
find the median:
find the sample standard deviation:
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Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 10$ and $x_{i}$ are the data - points.
$\sum_{i=1}^{10}x_{i}=12.2 + 12.6+16.6+16.8+19.8+19.9+20.7+23.2+29.3+30=197.9$
$\bar{x}=\frac{197.9}{10}=19.79$

Step2: Calculate the median

First, order the data: $12.2,12.6,16.6,16.8,19.8,19.9,20.7,23.2,29.3,30$.
Since $n = 10$ (an even number), the median is the average of the $\frac{n}{2}$ - th and $(\frac{n}{2}+1)$ - th ordered values.
The 5 - th value is $19.8$ and the 6 - th value is $19.9$.
Median $=\frac{19.8 + 19.9}{2}=19.85$

Step3: Calculate the sample standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
$(x_1-\bar{x})^2=(12.2 - 19.79)^2=(-7.59)^2 = 57.6081$
$(x_2-\bar{x})^2=(12.6 - 19.79)^2=(-7.19)^2 = 51.6961$
$(x_3-\bar{x})^2=(16.6 - 19.79)^2=(-3.19)^2 = 10.1761$
$(x_4-\bar{x})^2=(16.8 - 19.79)^2=(-2.99)^2 = 8.9401$
$(x_5-\bar{x})^2=(19.8 - 19.79)^2=(0.01)^2 = 0.0001$
$(x_6-\bar{x})^2=(19.9 - 19.79)^2=(0.11)^2 = 0.0121$
$(x_7-\bar{x})^2=(20.7 - 19.79)^2=(0.91)^2 = 0.8281$
$(x_8-\bar{x})^2=(23.2 - 19.79)^2=(3.41)^2 = 11.6281$
$(x_9-\bar{x})^2=(29.3 - 19.79)^2=(9.51)^2 = 90.4401$
$(x_{10}-\bar{x})^2=(30 - 19.79)^2=(10.21)^2 = 104.2441$
$\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=57.6081+51.6961+10.1761+8.9401+0.0001+0.0121+0.8281+11.6281+90.4401+104.2441 = 335.573$
$s=\sqrt{\frac{335.573}{9}}\approx6.09$

Answer:

Mean: $19.79$
Median: $19.85$
Sample standard deviation: $6.09$