Question

find the sample variance and standard deviation.
6, 49, 12, 50, 36, 25, 29, 32, 34, 29

choose the correct answer below. fill in the answer box to complete your choice.
(round to two decimal places as needed.)
a. $\sigma^2 = \square$

b. $s^2 = \square$

Explanation:

Step1: Calculate sample mean

First, sum all values and divide by sample size $n=10$.
$\bar{x} = \frac{6+49+12+50+36+25+29+32+34+29}{10} = \frac{302}{10} = 30.2$

Step2: Compute squared deviations

Find $(x_i - \bar{x})^2$ for each value:
$(6-30.2)^2=585.64$, $(49-30.2)^2=353.44$, $(12-30.2)^2=331.24$, $(50-30.2)^2=392.04$, $(36-30.2)^2=33.64$, $(25-30.2)^2=27.04$, $(29-30.2)^2=1.44$, $(32-30.2)^2=3.24$, $(34-30.2)^2=14.44$, $(29-30.2)^2=1.44$

Step3: Sum squared deviations

Add all the squared results:
$585.64+353.44+331.24+392.04+33.64+27.04+1.44+3.24+14.44+1.44 = 1743.6$

Step4: Calculate sample variance

Divide the sum by $n-1=9$ (sample variance uses $n-1$):
$s^2 = \frac{1743.6}{9} \approx 193.73$

Step5: Calculate sample standard deviation

Take the square root of sample variance:
$s = \sqrt{193.73} \approx 13.92$

Answer:

B. $s^2 = 193.73$
Sample standard deviation: $13.92$