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 largestnumber of decimal places given in the data.14, 12, 18, 10, 8, 6, 15, 9, 16, 20Copy DataAnswerKeypadHow to enter your answer (opens in new window)Keyboard ShortcutRange:Population Variance:Population Standard Deviation:Tables

Answer 1

Given the data below:

`14,12,18,10,8,6,15,9,16,20`

1) The range is the spread of data from the lowest to the highest value in the distribution.

In other words,

`range=highest\text{ value - lowest value}`

In this case,

`\begin{gathered} highest\text{ value = 20} \\ lowest\text{ value = 6} \\ thus, \\ range\text{ = 20-6=14} \\ range\approx\text{ 14.0\lparen 1 decimal place\rparen} \end{gathered}`

Thus, the range is 14.0 (1 decimal place)

2) Population variance:

The population variance is expressed as

`\begin{gathered} \sum_(i=1)^n\frac{\left(x_i-\bar{x}\right)^2}{n} \\ where \\ \bar{x}\text{ is the mean} \\ n\text{ is the numberof observation} \end{gathered}`

Thus, we have

Thus, the population variance is 18.8 (1 decimal place)

3) Standard deviation:

the standrard deviation is the square root of the variance.

Hence, we have

`\begin{gathered} standard\text{ deviation = }\sqrt{variance\text{ }}=√(18.76)=\text{ }4.3312816 \\ \implies standard\text{ deviation}\approx4.3\text{ \lparen1 decimal place\rparen} \end{gathered}`

Hence, the standard deviation is 4.3 (1 decimal place)