Question

sabine records the daily heights of a random sample of bamboo stalks, in inches. they are: 20, 19, 17, 16, 18, 15, 20, 21. consider the formulas: a: $s^{2}=\frac{(x_{1}-overline{x})^{2}+(x_{2}-overline{x})^{2}+cdots+(x_{n}-overline{x})^{2}}{n - 1}$ b: $s=sqrt{\frac{(x_{1}-overline{x})^{2}+(x_{2}-overline{x})^{2}+cdots+(x_{n}-overline{x})^{2}}{n - 1}}$ c: $sigma^{2}=\frac{(x_{1}-mu)^{2}+(x_{2}-mu)^{2}+cdots+(x_{n}-mu)^{2}}{n}$ d: $sigma=sqrt{\frac{(x_{1}-mu)^{2}+(x_{2}-mu)^{2}+cdots+(x_{n}-mu)^{2}}{n}}$. which formula should you use for variance? which formula should you use for standard deviation? the mean of the data is 18.25. find the variance. find the standard deviation.

Explanation:

Step1: Recall variance formula

We use formula A $s^{2}=\frac{(x_1 - \bar{x})^{2}+(x_2 - \bar{x})^{2}+\cdots+(x_n - \bar{x})^{2}}{n - 1}$ as we have a sample. Given $\bar{x}=18.25$ and data points $x_1 = 20,x_2 = 19,\cdots,x_8=21$, $n = 8$.

Step2: Calculate numerator

$(20 - 18.25)^{2}+(19 - 18.25)^{2}+(17 - 18.25)^{2}+(16 - 18.25)^{2}+(18 - 18.25)^{2}+(15 - 18.25)^{2}+(20 - 18.25)^{2}+(21 - 18.25)^{2}$
$=1.75^{2}+0.75^{2}+(- 1.25)^{2}+(-2.25)^{2}+(-0.25)^{2}+(-3.25)^{2}+1.75^{2}+2.75^{2}$
$=3.0625 + 0.5625+1.5625 + 5.0625+0.0625 + 10.5625+3.0625+7.5625$
$=31.5$

Step3: Calculate variance

$s^{2}=\frac{31.5}{8 - 1}=\frac{31.5}{7}=4.5$

Step4: Recall standard - deviation formula

Standard deviation $s=\sqrt{s^{2}}$. Since $s^{2}=4.5$, $s=\sqrt{4.5}\approx2.12$

Answer:

Variance: 4.5
Standard deviation: 2.12