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For the following set of data, find the sample standard deviation, to the nearest thousandth.
59,68,40,56,36,70,20

For the following set of data, find the sample standard deviation, to the nearest thousandth. $\newline$ $59,68,40,56,36,70,20$

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Find values of normal variables

Full solution

Q. For the following set of data, find the sample standard deviation, to the nearest thousandth.

\newline

59,68,40,56,36,70,20

List data set and calculate mean: List the data set and calculate the mean (average). $\newline$ Data set: $59, 68, 40, 56, 36, 70, 20$ $\newline$ Mean $(\mu) = \frac{59 + 68 + 40 + 56 + 36 + 70 + 20}{7}$ $\newline$ Mean $(\mu) = \frac{349}{7}$ $\newline$ Mean $(\mu) = 49.857$
Find squared differences: Subtract the mean from each data point and square the result to find the squared differences. $\newline$ Squared differences: $\newline$ $(59 - 49.857)^2 = 83.796$ $\newline$ $(68 - 49.857)^2 = 329.796$ $\newline$ $(40 - 49.857)^2 = 97.796$ $\newline$ $(56 - 49.857)^2 = 37.796$ $\newline$ $(36 - 49.857)^2 = 193.796$ $\newline$ $(70 - 49.857)^2 = 406.796$ $\newline$ $(20 - 49.857)^2 = 889.796$
Sum squared differences: Sum the squared differences. $\newline$ Sum of squared differences = $83.796 + 329.796 + 97.796 + 37.796 + 193.796 + 406.796 + 889.796$ $\newline$ Sum of squared differences = $2039.572$
Calculate variance: Divide the sum of squared differences by the sample size minus one to find the variance. $\newline$ Sample size $n$ = $7$ $\newline$ Variance $s^2$ = $2039.572 / (7 - 1)$ $\newline$ Variance $s^2$ = $2039.572 / 6$ $\newline$ Variance $s^2$ = $339.929$
Find sample standard deviation: Take the square root of the variance to find the sample standard deviation. $\newline$ Sample standard deviation $s$ = $\sqrt{339.929}$ $\newline$ Sample standard deviation $s$ $\approx 18.440$

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