calculating the variance and standard deviation\nadimas found the mean of her 11 math test scores for the…

Explanation:

Step1: Calculate squared - differences

The formula for variance $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}$, where $x_{i}$ are the data - points, $\bar{x}$ is the mean, and $n$ is the number of data - points. For $x_1 = 76$, $(76 - 81)^{2}=(-5)^{2}=25$; for $x_2 = 87$, $(87 - 81)^{2}=6^{2}=36$; for $x_3 = 65$, $(65 - 81)^{2}=(-16)^{2}=256$; for $x_4 = 88$, $(88 - 81)^{2}=7^{2}=49$; for $x_5 = 67$, $(67 - 81)^{2}=(-14)^{2}=196$; for $x_6 = 84$, $(84 - 81)^{2}=3^{2}=9$; for $x_7 = 77$, $(77 - 81)^{2}=(-4)^{2}=16$; for $x_8 = 82$, $(82 - 81)^{2}=1^{2}=1$; for $x_9 = 91$, $(91 - 81)^{2}=10^{2}=100$; for $x_{10}=85$, $(85 - 81)^{2}=4^{2}=16$; for $x_{11}=90$, $(90 - 81)^{2}=9^{2}=81$.

Step2: Calculate the sum of squared - differences

$\sum_{i = 1}^{11}(x_{i}-81)^{2}=25 + 36+256+49+196+9+16+1+100+16+81 = 785$.

Step3: Calculate the variance

$\sigma^{2}=\frac{\sum_{i = 1}^{11}(x_{i}-81)^{2}}{11}=\frac{785}{11}\approx71.36$.

Step4: Calculate the standard deviation

The standard deviation $\sigma=\sqrt{\sigma^{2}}$. Since $\sigma^{2}\approx71.36$, then $\sigma=\sqrt{71.36}\approx8.45$.

Answer:

$\sigma^{2}\approx71.36$ $\sigma\approx8.45$