Question 57.
Give the following data sets:
• 0, 0, 10, 10
,• 0, 1, 9, 10
,• 2, 3, 5, 7
,• 2, 4, 6, 8
,• 5, 5, 5, 5
Let's determine the data set with the largest standard deviation.
Now, let's find the standard deviation for each data set.
• Data set A:
First find the mean
[tex]mean=\frac{0+0+10+10}{4}=\frac{20}{4}=5[/tex]The standard deviation will be:
[tex]\begin{gathered} s=\sqrt{\frac{(0-5)^2+(0-5)^2+(10-5)^2+(10-5)^2}{4}} \\ \\ s=\sqrt{\frac{100}{4}}=\sqrt{25}=5 \end{gathered}[/tex]The standard deviation for data set A is 5.
• Data set B;
Let's find the mean and use the mean to find the standard deviation:
[tex]\begin{gathered} mean=\frac{0+1+9+10}{4}=\frac{20}{4}=5 \\ \\ s=\sqrt{\frac{(0-5)^2+(1-5)^2+(9-5)^2+(10-5)^2}{4}} \\ \\ s=\sqrt{\frac{25+16+16+25}{4}}=\sqrt{\frac{82}{4}}=\sqrt{20.5}=4.5 \end{gathered}[/tex]The standard deviation of set B is 4.5
• Data set C:
Let's find the mean and use the mean to find the standard deviation:
[tex]\begin{gathered} mean=\frac{2+3+5+7}{4}=\frac{17}{4}=4.25 \\ \\ s=\sqrt{\frac{(2-4.25)^2+(3-4.25)^2+(5-4.25)^2+(7-4.25)^2}{4}} \\ \\ s=\sqrt{\frac{5.0625+1.5625+1.5625+7.5625}{4}} \\ \\ s=\sqrt{\frac{15.75}{4}}=1.98 \end{gathered}[/tex]The standard deviation of data set C is 1.98
• Data set D:
[tex]\begin{gathered} mean=\frac{2+4+6+8}{4}=\frac{20}{4}=5 \\ \\ s=\sqrt{\frac{(2-5)^2+(4-5)^2+(6-5)^2+(8-5)^2}{4}} \\ \\ s=\sqrt{\frac{9+1+1+9}{4}}=\sqrt{\frac{20}{4}}=\sqrt{5}=2.24 \end{gathered}[/tex]The standard deviation of data set D is 2.24.
• Data set E.
[tex]\begin{gathered} mean=\frac{5+5+5+5}{4}=\frac{20}{4}=5 \\ \\ s=\sqrt{\frac{(5-5)^2+(5-5)^2+(5-5)^2+(5-5)^2}{4}} \\ \\ s=\sqrt{\frac{0}{4}}=0 \end{gathered}[/tex]The standard deviation of data set E is 0.
Therefore, the data set that has the largest standard deviation is data set A because it has the largest standard deviation of 5.
ANSWER:
A. 0, 0, 10, 10
