Question

1. the data below represents the total number of points each hockey team of the nhl eastern conference earned this past hockey season. find the standard deviation. round to the nearest tenth. {52, 66, 84, 88, 93, 100, 101, 117}

Explanation:

Step1: Calculate the mean ($\mu$)

First, find the sum of the data set. The data set is $\{52, 66, 84, 88, 93, 100, 101, 117\}$. The number of data points $n = 8$.

Sum $S = 52 + 66 + 84 + 88 + 93 + 100 + 101 + 117$.

Calculating the sum: $52+66 = 118$; $118 + 84 = 202$; $202 + 88 = 290$; $290 + 93 = 383$; $383 + 100 = 483$; $483 + 101 = 584$; $584 + 117 = 701$. Wait, no, let's recalculate:

$52+66 = 118$; $118+84 = 202$; $202+88 = 290$; $290+93 = 383$; $383+100 = 483$; $483+101 = 584$; $584+117 = 701$? Wait, 52+66=118, 118+84=202, 202+88=290, 290+93=383, 383+100=483, 483+101=584, 584+117=701? Wait, 52+66=118, 118+84=202, 202+88=290, 290+93=383, 383+100=483, 483+101=584, 584+117=701. Wait, but 52+66=118, 118+84=202, 202+88=290, 290+93=383, 383+100=483, 483+101=584, 584+117=701. Then the mean $\mu=\frac{S}{n}=\frac{701}{8}=87.625$.

Step2: Calculate the squared differences from the mean

For each data point $x_i$, calculate $(x_i - \mu)^2$:

• For $x_1 = 52$: $(52 - 87.625)^2 = (-35.625)^2 = 1269.140625$
• For $x_2 = 66$: $(66 - 87.625)^2 = (-21.625)^2 = 467.640625$
• For $x_3 = 84$: $(84 - 87.625)^2 = (-3.625)^2 = 13.140625$
• For $x_4 = 88$: $(88 - 87.625)^2 = (0.375)^2 = 0.140625$
• For $x_5 = 93$: $(93 - 87.625)^2 = (5.375)^2 = 28.890625$
• For $x_6 = 100$: $(100 - 87.625)^2 = (12.375)^2 = 153.140625$
• For $x_7 = 101$: $(101 - 87.625)^2 = (13.375)^2 = 179.015625$
• For $x_8 = 117$: $(117 - 87.625)^2 = (29.375)^2 = 862.890625$

Step3: Calculate the variance ($\sigma^2$)

Variance is the average of the squared differences. So sum up the squared differences and divide by $n$.

Sum of squared differences: $1269.140625 + 467.640625 + 13.140625 + 0.140625 + 28.890625 + 153.140625 + 179.015625 + 862.890625$.

Calculating the sum:

$1269.140625+467.640625 = 1736.78125$;

$1736.78125 + 13.140625 = 1749.921875$;

$1749.921875 + 0.140625 = 1750.0625$;

$1750.0625 + 28.890625 = 1778.953125$;

$1778.953125 + 153.140625 = 1932.09375$;

$1932.09375 + 179.015625 = 2111.109375$;

$2111.109375 + 862.890625 = 2974$.

So variance $\sigma^2=\frac{2974}{8}=371.75$.

Step4: Calculate the standard deviation ($\sigma$)

Standard deviation is the square root of the variance. So $\sigma=\sqrt{371.75}\approx19.3$.

Answer:

19.3