Question

determine the range and standard deviation of the set of data, round to the nearest hundredth when applicable.

1. 10, 10, 14, 16, 8, 8
2. 3, 5, 7, 9, 11, 13, 15
3. 1, 3, 71, 8, 92, 63
4. 5, 10, 20, 200, 300, 400
5. 0, 3, 5, 9, 9, 10, 11, 13

Explanation:

Step1: Calculate the range

The range is the difference between the maximum and minimum values in the data set.
For the data set \(10,10,14,16,8,8\):
Maximum value \(M = 16\), minimum value \(m=8\)
Range \(R = M - m=16 - 8=8\)

Step2: Calculate the mean

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n\) is the number of data points and \(x_{i}\) are the data values.
\(n = 6\), \(\sum_{i=1}^{6}x_{i}=10 + 10+14 + 16+8 + 8=66\)
\(\bar{x}=\frac{66}{6}=11\)

Step3: Calculate the variance

The variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\)
\((10 - 11)^{2}=1\), \((10 - 11)^{2}=1\), \((14 - 11)^{2}=9\), \((16 - 11)^{2}=25\), \((8 - 11)^{2}=9\), \((8 - 11)^{2}=9\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=1+1 + 9+25+9+9=54\)
\(s^{2}=\frac{54}{6 - 1}=\frac{54}{5}=10.8\)

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^{2}}\)
\(s=\sqrt{10.8}\approx3.29\)

Answer:

Range: \(8\), Standard deviation: \(3.29\)

(Note: The above steps are for the first data - set \(10,10,14,16,8,8\). For other data - sets, the process is similar:

1. For data set \(3,5,7,9,11,13,15\):

• Range: \(15 - 3=12\)
• Mean \(\bar{x}=\frac{3 + 5+7+9+11+13+15}{7}=\frac{63}{7}=9\)
• \(\sum_{i = 1}^{7}(x_{i}-\bar{x})^{2}=(3 - 9)^{2}+(5 - 9)^{2}+(7 - 9)^{2}+(9 - 9)^{2}+(11 - 9)^{2}+(13 - 9)^{2}+(15 - 9)^{2}=36+16 + 4+0+4+16+36 = 112\)
• Variance \(s^{2}=\frac{112}{7 - 1}=\frac{112}{6}\approx18.67\)
• Standard deviation \(s=\sqrt{\frac{112}{6}}\approx4.32\)

1. For data set \(1,3,71,8,92,63\):

• Range: \(92 - 1=91\)
• Mean \(\bar{x}=\frac{1+3 + 71+8+92+63}{6}=\frac{238}{6}\approx39.67\)
• \(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=(1 - 39.67)^{2}+(3 - 39.67)^{2}+(71 - 39.67)^{2}+(8 - 39.67)^{2}+(92 - 39.67)^{2}+(63 - 39.67)^{2}\approx(- 38.67)^{2}+(-36.67)^{2}+(31.33)^{2}+(-31.67)^{2}+(52.33)^{2}+(23.33)^{2}=1495.37+1344.79+981.57+1002.99+2738.43+544.29 = 8107.44\)
• Variance \(s^{2}=\frac{8107.44}{6 - 1}=\frac{8107.44}{5}=1621.49\)
• Standard deviation \(s=\sqrt{1621.49}\approx40.27\)

1. For data set \(5,10,20,200,300,400\):

• Range: \(400 - 5=395\)
• Mean \(\bar{x}=\frac{5+10+20+200+300+400}{6}=\frac{935}{6}\approx155.83\)
• \(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=(5 - 155.83)^{2}+(10 - 155.83)^{2}+(20 - 155.83)^{2}+(200 - 155.83)^{2}+(300 - 155.83)^{2}+(400 - 155.83)^{2}\approx(-150.83)^{2}+(-145.83)^{2}+(-135.83)^{2}+(44.17)^{2}+(144.17)^{2}+(244.17)^{2}=22749.69+21268.39+18450.79+1950.99+20784.99+59619.09 = 144823.94\)
• Variance \(s^{2}=\frac{144823.94}{6 - 1}=\frac{144823.94}{5}=28964.79\)
• Standard deviation \(s=\sqrt{28964.79}\approx170.20\)

1. For data set \(0,3,5,9,9,10,11,13\):

• Range: \(13 - 0=13\)
• Mean \(\bar{x}=\frac{0+3+5+9+9+10+11+13}{8}=\frac{60}{8}=7.5\)
• \(\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=(0 - 7.5)^{2}+(3 - 7.5)^{2}+(5 - 7.5)^{2}+(9 - 7.5)^{2}+(9 - 7.5)^{2}+(10 - 7.5)^{2}+(11 - 7.5)^{2}+(13 - 7.5)^{2}=56.25+20.25+6.25+2.25+2.25+6.25+12.25+30.25 = 136\)
• Variance \(s^{2}=\frac{136}{8 - 1}=\frac{136}{7}\approx19.43\)
• Standard deviation \(s=\sqrt{\frac{136}{7}}\approx4.41\))