Question

for the sample data shown, answer the questions. round to 2 decimal places.
find the mean:
find the median:
find the sample standard deviation:

next question

Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}$ of a sample $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here $n = 8$, and $\sum_{i=1}^{8}x_i=2.6 + 8.9+13.4 + 18.7+20.2+21.4+21.8+25.9=132.9$. So $\bar{x}=\frac{132.9}{8}=16.6125\approx16.61$.

Step2: Calculate the median

First, order the data set. Since $n = 8$ (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 ordered data set is $2.6,8.9,13.4,18.7,20.2,21.4,21.8,25.9$. The $\frac{n}{2}=4$-th value is $18.7$ and the $(\frac{n}{2}+1) = 5$-th value is $20.2$. So the median is $\frac{18.7 + 20.2}{2}=\frac{38.9}{2}=19.45$.

Step3: Calculate the sample standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$.

1. Calculate $(x_i-\bar{x})^2$ for each $i$:

• $(2.6-16.61)^2=(-14.01)^2 = 196.2801$
• $(8.9-16.61)^2=(-7.71)^2 = 59.4441$
• $(13.4-16.61)^2=(-3.21)^2 = 10.3041$
• $(18.7-16.61)^2=(2.09)^2 = 4.3681$
• $(20.2-16.61)^2=(3.59)^2 = 12.8881$
• $(21.4-16.61)^2=(4.79)^2 = 22.9441$
• $(21.8-16.61)^2=(5.19)^2 = 26.9361$
• $(25.9-16.61)^2=(9.29)^2 = 86.3041$

1. Then $\sum_{i = 1}^{n}(x_i-\bar{x})^2=196.2801+59.4441+10.3041+4.3681+12.8881+22.9441+26.9361+86.3041 = 419.479$
2. And $s=\sqrt{\frac{419.479}{8 - 1}}=\sqrt{\frac{419.479}{7}}\approx\sqrt{59.9256}\approx7.74$.

Answer:

Mean: $16.61$
Median: $19.45$
Sample standard deviation: $7.74$