3.2 measures of spread: standard deviation and variance\nfor the sample data shown, answer the questions…

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 7$, $x_1=9.2,x_2 = 9.8,x_3=9.9,x_4 = 12,x_5=14.6,x_6=22.7,x_7=28.6$. $\bar{x}=\frac{9.2 + 9.8+9.9+12+14.6+22.7+28.6}{7}=\frac{106.8}{7}\approx15.26$

Step2: Calculate the median

Arrange the data in ascending - order: $9.2,9.8,9.9,12,14.6,22.7,28.6$. Since $n = 7$ (odd), the median is the $\left(\frac{n + 1}{2}\right)$-th value. $\frac{7+1}{2}=4$ - th value, so the median is $12$.

Step3: Calculate the sample standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2}{n - 1}}$. $(9.2-15.26)^2=(-6.06)^2 = 36.7236$ $(9.8-15.26)^2=(-5.46)^2 = 29.8116$ $(9.9-15.26)^2=(-5.36)^2 = 28.7296$ $(12-15.26)^2=(-3.26)^2 = 10.6276$ $(14.6-15.26)^2=(-0.66)^2 = 0.4356$ $(22.7-15.26)^2=(7.44)^2 = 55.3536$ $(28.6-15.26)^2=(13.34)^2 = 177.9556$ $\sum_{i = 1}^{7}(x_{i}-15.26)^2=36.7236 + 29.8116+28.7296+10.6276+0.4356+55.3536+177.9556=339.6372$ $s=\sqrt{\frac{339.6372}{7 - 1}}=\sqrt{\frac{339.6372}{6}}\approx7.53$

Answer:

Mean: $15.26$ Median: $12$ Sample standard deviation: $7.53$