Question

find the variance of the data set below. if necessary, round to the nearest hundredth.
26, 33, 43, 25, 35, 46, 37, 31, 28, 37

question 13
2 pts
find the variance of the data set below. if necessary, round to the nearest hundredth.
33, 36, 40, 43, 43, 45, 46, 50, 50, 51, 53, 53

question 14
2 pts
find the variance of the data set below. if necessary, round to the nearest hundredth.
64, 69, 73, 76, 77, 77, 87, 89

Explanation:

Step1: Calculate the mean

For the first data set \(26,33,43,25,35,46,37,31,28,37\):
The mean \(\bar{x}=\frac{26 + 33+43+25+35+46+37+31+28+37}{10}=\frac{341}{10}=34.1\)
For the second data set \(33,36,40,43,43,45,46,50,50,51,53,53\):
The mean \(\bar{x}=\frac{33 + 36+40+43+43+45+46+50+50+51+53+53}{12}=\frac{543}{12}=45.25\)
For the third data set \(64,69,73,76,77,77,87,89\):
The mean \(\bar{x}=\frac{64 + 69+73+76+77+77+87+89}{8}=\frac{612}{8}=76.5\)

Step2: Calculate the variance

The formula for variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}\)

• First data set:

\(\sum_{i=1}^{10}(x_{i}-\bar{x})^{2}=(26 - 34.1)^{2}+(33 - 34.1)^{2}+(43 - 34.1)^{2}+(25 - 34.1)^{2}+(35 - 34.1)^{2}+(46 - 34.1)^{2}+(37 - 34.1)^{2}+(31 - 34.1)^{2}+(28 - 34.1)^{2}+(37 - 34.1)^{2}\)
\(=(-8.1)^{2}+(-1.1)^{2}+(8.9)^{2}+(-9.1)^{2}+(0.9)^{2}+(11.9)^{2}+(2.9)^{2}+(-3.1)^{2}+(-6.1)^{2}+(2.9)^{2}\)
\(=65.61+1.21 + 79.21+82.81+0.81+141.61+8.41+9.61+37.21+8.41\)
\(=434.9\)
\(s^{2}=\frac{434.9}{10}=43.49\)

• Second data set:

\(\sum_{i = 1}^{12}(x_{i}-\bar{x})^{2}=(33 - 45.25)^{2}+(36 - 45.25)^{2}+(40 - 45.25)^{2}+(43 - 45.25)^{2}+(43 - 45.25)^{2}+(45 - 45.25)^{2}+(46 - 45.25)^{2}+(50 - 45.25)^{2}+(50 - 45.25)^{2}+(51 - 45.25)^{2}+(53 - 45.25)^{2}+(53 - 45.25)^{2}\)
\(=(-12.25)^{2}+(-9.25)^{2}+(-5.25)^{2}+(-2.25)^{2}+(-2.25)^{2}+(-0.25)^{2}+(0.75)^{2}+(4.75)^{2}+(4.75)^{2}+(5.75)^{2}+(7.75)^{2}+(7.75)^{2}\)
\(=150.0625+85.5625+27.5625+5.0625+5.0625+0.0625+0.5625+22.5625+22.5625+33.0625+60.0625+60.0625\)
\(=472.25\)
\(s^{2}=\frac{472.25}{12}\approx39.35\)

• Third data set:

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

Answer:

43.49, 39.35, 61.5