Question

calculate the standard deviation and variance of the data shown, to two decimal places.
x
24.6
4.6
25.4
17.8
9.3
28.1
1.3
standard deviation:
variance:
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Explanation:

Step1: Calculate sample mean

First, sum all data points and divide by the count $n=7$.
$\bar{x} = \frac{24.6 + 4.6 + 25.4 + 17.8 + 9.3 + 28.1 + 1.3}{7} = \frac{111.1}{7} \approx 15.8714$

Step2: Find squared deviations

Compute $(x_i - \bar{x})^2$ for each data point:
$(24.6-15.8714)^2 \approx 76.188$,
$(4.6-15.8714)^2 \approx 127.044$,
$(25.4-15.8714)^2 \approx 90.794$,
$(17.8-15.8714)^2 \approx 3.719$,
$(9.3-15.8714)^2 \approx 43.183$,
$(28.1-15.8714)^2 \approx 149.538$,
$(1.3-15.8714)^2 \approx 212.325$

Step3: Calculate sample variance

Sum squared deviations, divide by $n-1=6$.
$s^2 = \frac{76.188+127.044+90.794+3.719+43.183+149.538+212.325}{6} = \frac{702.791}{6} \approx 117.13$

Step4: Calculate standard deviation

Take square root of the variance.
$s = \sqrt{117.13} \approx 10.82$

Answer:

Standard deviation: $10.82$
Variance: $117.13$