Question

find the variance of the data set below. if necessary, round to the nearest hundredth.
9, 9, 11, 12, 15, 17, 17, 20, 20, 21, 22

question 16
2 pt
find the standard deviation of the data set below. if necessary, round tot he nearest hundredth.
64, 69, 73, 76, 77, 77, 87, 89

question 17
2 pts
find the standard deviation of the data set below. if necessary, round tot he nearest hundredth.
33, 36, 40, 43, 43, 45, 46, 50, 50, 51, 53, 53

Explanation:

Step1: Calculate the mean

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\).
For the data set \(9,9,11,12,15,17,17,20,20,21,22\), \(n = 11\).
\(\sum_{i=1}^{11}x_{i}=9 + 9+11+12+15+17+17+20+20+21+22=173\).
\(\bar{x}=\frac{173}{11}\approx15.73\).

Step2: Calculate the variance

The formula for the variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}\).
\((9 - 15.73)^{2}=(- 6.73)^{2}=45.2929\), \((9 - 15.73)^{2}=45.2929\), \((11 - 15.73)^{2}=(-4.73)^{2}=22.3729\), \((12 - 15.73)^{2}=(-3.73)^{2}=13.9129\), \((15 - 15.73)^{2}=(-0.73)^{2}=0.5329\), \((17 - 15.73)^{2}=(1.27)^{2}=1.6129\), \((17 - 15.73)^{2}=1.6129\), \((20 - 15.73)^{2}=(4.27)^{2}=18.2329\), \((20 - 15.73)^{2}=18.2329\), \((21 - 15.73)^{2}=(5.27)^{2}=27.7729\), \((22 - 15.73)^{2}=(6.27)^{2}=39.3129\).
\(\sum_{i = 1}^{11}(x_{i}-\bar{x})^{2}=45.2929+45.2929+22.3729+13.9129+0.5329+1.6129+1.6129+18.2329+18.2329+27.7729+39.3129 =234.19\).
\(s^{2}=\frac{234.19}{11}\approx21.29\).

For the data set \(64,69,73,76,77,77,87,89\), \(n = 8\).

Step1: Calculate the mean

\(\sum_{i=1}^{8}x_{i}=64 + 69+73+76+77+77+87+89=612\).
\(\bar{x}=\frac{612}{8}=76.5\).

Step2: Calculate the variance

\((64 - 76.5)^{2}=(-12.5)^{2}=156.25\), \((69 - 76.5)^{2}=(-7.5)^{2}=56.25\), \((73 - 76.5)^{2}=(-3.5)^{2}=12.25\), \((76 - 76.5)^{2}=(-0.5)^{2}=0.25\), \((77 - 76.5)^{2}=(0.5)^{2}=0.25\), \((77 - 76.5)^{2}=0.25\), \((87 - 76.5)^{2}=(10.5)^{2}=110.25\), \((89 - 76.5)^{2}=(12.5)^{2}=156.25\).
\(\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=156.25+56.25+12.25+0.25+0.25+0.25+110.25+156.25 =492\).
\(s^{2}=\frac{492}{8}=61.5\), and the standard deviation \(s=\sqrt{61.5}\approx7.84\).

For the data set \(33,36,40,43,43,45,46,50,50,51,53,53\), \(n = 12\).

Step1: Calculate the mean

\(\sum_{i=1}^{12}x_{i}=33 + 36+40+43+43+45+46+50+50+51+53+53=543\).
\(\bar{x}=\frac{543}{12}=45.25\).

Step2: Calculate the variance

\((33 - 45.25)^{2}=(-12.25)^{2}=150.0625\), \((36 - 45.25)^{2}=(-9.25)^{2}=85.5625\), \((40 - 45.25)^{2}=(-5.25)^{2}=27.5625\), \((43 - 45.25)^{2}=(-2.25)^{2}=5.0625\), \((43 - 45.25)^{2}=5.0625\), \((45 - 45.25)^{2}=(-0.25)^{2}=0.0625\), \((46 - 45.25)^{2}=(0.75)^{2}=0.5625\), \((50 - 45.25)^{2}=(4.75)^{2}=22.5625\), \((50 - 45.25)^{2}=22.5625\), \((51 - 45.25)^{2}=(5.75)^{2}=33.0625\), \((53 - 45.25)^{2}=(7.75)^{2}=60.0625\), \((53 - 45.25)^{2}=60.0625\).
\(\sum_{i = 1}^{12}(x_{i}-\bar{x})^{2}=150.0625+85.5625+27.5625+5.0625+5.0625+0.0625+0.5625+22.5625+22.5625+33.0625+60.0625+60.0625 =472.375\).
\(s^{2}=\frac{472.375}{12}\approx39.36\), and the standard deviation \(s = \sqrt{39.36}\approx6.27\).

Answer:

For the first data set (variance): \(21.29\).
For the second data set (standard deviation): \(7.84\).
For the third data set (standard deviation): \(6.27\).