Question

question #eight
give the data set 29 38 49 50 53 54 55 60 65 67 68 68. calculate the following:
a) $sum x^{2}=371498$
b) variance = $s^{2}=$
c) standard deviation = $s=$

Explanation:

Step1: Calculate the mean ($\bar{x}$)

The data - set is $29,38,49,50,53,54,55,60,65,67,68$. The sum of the data - set $\sum x=29 + 38+49+50+53+54+55+60+65+67+68 = 588$, and $n = 11$. So, $\bar{x}=\frac{\sum x}{n}=\frac{588}{11}\approx53.4545$.

Step2: Calculate the variance ($s^{2}$)

The formula for the sample variance is $s^{2}=\frac{\sum x^{2}-\frac{(\sum x)^{2}}{n}}{n - 1}$. We are given that $\sum x^{2}=371498$, $\sum x = 588$ and $n = 11$.
First, calculate $\frac{(\sum x)^{2}}{n}=\frac{588^{2}}{11}=\frac{345744}{11}\approx31431.2727$.
Then, $s^{2}=\frac{371498-31431.2727}{11 - 1}=\frac{371498 - 31431.2727}{10}=\frac{340066.7273}{10}=34006.67273\approx157.97$.

Step3: Calculate the standard deviation ($s$)

The standard deviation is the square - root of the variance. So, $s=\sqrt{s^{2}}=\sqrt{157.97}\approx12.569$.

Answer:

a) $\sum x^{2}=371498$
b) $s^{2}\approx157.97$
c) $s\approx12.57$