Question

for the sample data shown, answer the questions. round to 2 decimal places. x 6.2 10.3 10.4 14.2 19.3 20.5 20.9 26.6 find the mean: find the median: find the sample standard deviation: question help: message instructor post to forum

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 8$ and $x_{i}$ are the data - points. $\sum_{i=1}^{8}x_{i}=6.2 + 10.3+10.4 + 14.2+19.3+20.5+20.9+26.6=128.4$. So, $\bar{x}=\frac{128.4}{8}=16.05$.

Step2: Calculate the median

Since $n = 8$ (an even number), the median is the average of the $\frac{n}{2}$ - th and $(\frac{n}{2}+1)$ - th ordered data - points. First, the data is already in ascending order. The $\frac{n}{2}=4$ - th value is $14.2$ and the $(\frac{n}{2}+1)=5$ - th value is $19.3$. The median $M=\frac{14.2 + 19.3}{2}=16.75$.

Step3: Calculate the sample standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
$(x_1-\bar{x})^2=(6.2 - 16.05)^2=(-9.85)^2 = 97.0225$
$(x_2-\bar{x})^2=(10.3 - 16.05)^2=(-5.75)^2 = 33.0625$
$(x_3-\bar{x})^2=(10.4 - 16.05)^2=(-5.65)^2 = 31.9225$
$(x_4-\bar{x})^2=(14.2 - 16.05)^2=(-1.85)^2 = 3.4225$
$(x_5-\bar{x})^2=(19.3 - 16.05)^2=(3.25)^2 = 10.5625$
$(x_6-\bar{x})^2=(20.5 - 16.05)^2=(4.45)^2 = 19.8025$
$(x_7-\bar{x})^2=(20.9 - 16.05)^2=(4.85)^2 = 23.5225$
$(x_8-\bar{x})^2=(26.6 - 16.05)^2=(10.55)^2 = 111.3025$
$\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=97.0225+33.0625+31.9225+3.4225+10.5625+19.8025+23.5225+111.3025 = 320.62$
$s=\sqrt{\frac{320.62}{7}}\approx6.78$

Answer:

Find the mean: 16.05
Find the median: 16.75
Find the sample standard deviation: 6.78