Question

Calculate the range, population variance, and population standard deviation for the following data set. If necessary, round to one more decimal place than the largest
number of decimal places given in the data.
6,9,9,6,9, 4, 8, 5, 8, 27
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Answer 1

The standard deviation of a sample is the square root of the variance

• The range is 23
• The population variance is 5.5
• The population standard deviation is 6.20

How to calculate the range

From the dataset, the smallest is 4 and the highest is 27

So, the range is:

$Range = 27 - 4$

$Range = 23$

Hence, the range is 23

How to determine the variance

The sample is given as: 6,9,9,6,9, 4, 8, 5, 8, 27

Start by calculating the mean

$\bar x = \frac{\sum x}{n}$

So, we have:

$\bar x = \frac{6+9+9+6+9+ 4+ 8+ 5+ 8+ 27}{10}$

$\bar x = \frac{91}{10}$

$\bar x = 9.1$

The population variance is then calculated as:

$\sigma^2 = \frac{\sum(x - \bar x)^2}{n }$

This gives

$\sigma^2 = \frac{(6 - 9.1)^2+(9- 9.1)^2+(9- 9.1)^2+(6- 9.1)^2+(9- 9.1)^2+(4- 9.1)^2+(8- 9.1)^2+(5- 9.1)^2+(8- 9.1)^2+(27- 9.1)^2}{10}$

$\sigma^2 = 38.49$

Hence, the population variance is 5.5

How to calculate the standard deviation

In (a), we have:

$\sigma^2 = 38.49$

Take the square roots of both sides

$\sigma = 6.20$

Hence, the population standard deviation is 6.20

Read more about variance and standard deviation at:

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