The dataset shown in the picture shows the daily amount spent by a family during a 7-day vacation.
The sample size is n=7
To calculate the standard deviation (S) you have to calculate its variance first. To calculate the sample variance (S²) you have to use the following formula:
[tex]S^2=\frac{1}{n-1}\lbrack\Sigma x^2_i-\frac{(\Sigma x_i)^2}{n}\rbrack[/tex]
Before calculating the variance, you have to calculate the sum of the observations (∑xi) and the sum of squares of the observations (∑xi²)
[tex]\begin{gathered} \Sigma x_i=96+125+80+110+75+100+121 \\ \Sigma x_i=707 \end{gathered}[/tex][tex]\begin{gathered} \Sigma x^2_i=96^2+125^2+80^2+110^2+75^2+100^2+121^2 \\ \Sigma x^2_i=73607 \end{gathered}[/tex]
Replace both sums on the formula to determine the variance:
[tex]\begin{gathered} S^2=\frac{1}{7-1}\lbrack73607-\frac{(707)^2}{7}\rbrack \\ S^2=\frac{1}{6}\lbrack73607-71407\rbrack \\ S^2=\frac{1}{6}\cdot2200 \\ S^2=366.67 \end{gathered}[/tex]
Once you have calculated the sample variance, to determine the sample standard deviation you have to calculate the square root of the variance:
[tex]\begin{gathered} S=\sqrt[]{S^2} \\ S=\sqrt[]{366.67} \\ S=19.148 \\ S\approx19.1 \end{gathered}[/tex]
The standard deviation of the data set is S= $19.1
