Certainly! Let's find the range and the standard deviation for the given set of data: 31, 25, 40, 10, 37, 13, 19.
### (a) The Range
The range of a data set is the difference between the maximum and minimum values.
1. Identify the maximum value in the data set: 40.
2. Identify the minimum value in the data set: 10.
3. Subtract the minimum value from the maximum value to find the range:
[tex]\[
\text{Range} = \text{Maximum} - \text{Minimum} = 40 - 10 = 30
\][/tex]
So, the range of the data set is [tex]\( 30 \)[/tex].
### (b) The Standard Deviation
The standard deviation is a measure of the amount of variation or dispersion in a set of values.
1. Find the mean (average) of the data set:
[tex]\[
\text{Mean} = \frac{\sum \text{Data values}}{\text{Number of values}} = \frac{31 + 25 + 40 + 10 + 37 + 13 + 19}{7} = \frac{175}{7} = 25
\][/tex]
2. Calculate the variance:
- Subtract the mean from each data value and square the result:
[tex]\[
\begin{align*}
(31 - 25)^2 & = 6^2 = 36 \\
(25 - 25)^2 & = 0^2 = 0 \\
(40 - 25)^2 & = 15^2 = 225 \\
(10 - 25)^2 & = (-15)^2 = 225 \\
(37 - 25)^2 & = 12^2 = 144 \\
(13 - 25)^2 & = (-12)^2 = 144 \\
(19 - 25)^2 & = (-6)^2 = 36 \\
\end{align*}
\][/tex]
- Sum these squared differences:
[tex]\[
\sum (\text{Data value} - \text{Mean})^2 = 36 + 0 + 225 + 225 + 144 + 144 + 36 = 810
\][/tex]
- Divide by the number of data values to find the variance:
[tex]\[
\text{Variance} = \frac{\sum (\text{Data value} - \text{Mean})^2}{\text{Number of values}} = \frac{810}{7} \approx 115.71
\][/tex]
3. Take the square root of the variance to get the standard deviation:
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{115.71} \approx 10.76
\][/tex]
So, the standard deviation of the data set is approximately [tex]\( 10.76 \)[/tex] when rounded to the nearest hundredth.
### Summary
(a) The range is [tex]\( 30 \)[/tex].
(b) The standard deviation is approximately [tex]\( 10.76 \)[/tex].
