Question 1190757: The following data give the yearly numbers of law enforcement officers killed in the United
States over 10 years: 164, 165, 157, 164, 152, 147, 148, 131, 147, 155, calculate sample
variance and standard deviation
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
n = 10 = sample size
Add up the numbers and you should get 1530
Divide that over the n = 10 numbers. This yields the sample mean of 1530/10 = 153
xbar = 153 = sample mean
Next, subtract the sample mean from each item.
I recommend making a table to organize the data.
| x | x-xbar | | 164 | 11 | | 165 | 12 | | 157 | 4 | | 164 | 11 | | 152 | -1 | | 147 | -6 | | 148 | -5 | | 131 | -22 | | 147 | -6 | | 155 | 2 |
The second column tells us the error of each x value with respect to the center xbar = 153.
Square each item in the second column to form the third column.
| x | x-xbar | (x-xbar)^2 | | 164 | 11 | 121 | | 165 | 12 | 144 | | 157 | 4 | 16 | | 164 | 11 | 121 | | 152 | -1 | 1 | | 147 | -6 | 36 | | 148 | -5 | 25 | | 131 | -22 | 484 | | 147 | -6 | 36 | | 155 | 2 | 4 |
Add up everything in the third column to get 988. This is the sum of the squared errors (SSE). As the name implies, it's the result of adding up each error squared. Each x-xbar value is the error itself, and we square the item to avoid a negative number. This means any (x-xbar)^2 has a min value of 0.
Divide this SSE value over n-1 = 10-1 = 9 to compute the sample variance.
SSE/(n-1) = 988/9 = 109.7778 approximately
Apply the square root to the sample variance to get the sample standard deviation
sqrt(988/9) = 10.4775 approximately
You can use a graphing calculator to check the answers.
Or you can use a free dedicated online calculator such as this one
https://www.calculatorsoup.com/calculators/statistics/variance-calculator.php
Make sure the "sample" option is selected.
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In summary:
sample variance = 109.7778
sample standard deviation = 10.4775
both values are approximate
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