Q 3.3. - The Practice of Statistics for AP

Question

The heights (in inches) of the five starters on a basketball team are $67, 72, 76, 76, \mathrm{and} 84.$

Show how to calculate the variance and standard deviation from the values in your table. 

Step-by-Step Solution

Answer

The sample variance is $\mathbf{39}$ and standard deviation is$\mathbf{ }\mathbf{6}\mathbf{.}\mathbf{2449}$

Step 1: Given information

Given data : 

The heights five starters are : $67, 72, 76, 76, \mathrm{and} 84.$

Step 2: Concept

Standard deviation is the spread of a group of numbers from the mean. 

$SV=\frac{\Sigma {(xi-\overline{x})}^2}{N-1}$

$\sigma = \sqrt{sample variance}$

Step 3: Calculation

For the given data 67,72,76,84 table is, 

x	$\left(x-\overline{x}\right)$	${\left(x-\overline{x}\right)}^2$
67	-8	64
72	-3	9
76	1	1
76	1	1
84	9	81
Total=375	0	156

The mean is calculated as 75

The sample variance is computed using the formula:

$$\begin{gathered} =\frac{\Sigma {(xi-\overline{x}) }^2}{n-1} \\ =\frac{156}{4} \\ =39 \end{gathered}$$

Standard deviation is calculated as,

σ = $\sqrt{sample variance}$

σ = $\sqrt{39}$

$= 6.2449$

The standard deviation is a metric that indicates how far a set of data deviates from its mean. It's a measurement of distance's absolute variability. The standard deviation increases as the dispersion increases. The standard deviation is $6.2449$and the sample variance is $39$