Questions: 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. 8, 10, 7, 3, 7, 6, 4, 8, 4, 31

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.
8, 10, 7, 3, 7, 6, 4, 8, 4, 31

Transcript text: 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. \[ 8,10,7,3,7,6,4,8,4,31 \] Copy Data

Solution

Solution Steps

Step 1: Calculate the Range

To calculate the range, we identify the minimum and maximum values in the data set.

Minimum (\(min\)) = 3
Maximum (\(max\)) = 31 Then, we calculate the range as \(Range = max - min = 31 - 3 = 28\).

Step 2: Calculate the Population Variance (\(\sigma^2\))

First, we compute the mean (\(\mu\)) of the data set as \(\mu = \frac{1}{n}\sum_{i=1}^{n}x_i = 8.8\). Then, for each value in the data set, we calculate the square of its difference from the mean and sum all the squared differences. Finally, we divide by the number of observations to get the population variance: \(\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2 = 59\).

Step 3: Calculate the Population Standard Deviation (\(\sigma\))

We take the square root of the population variance to get the population standard deviation: \(\sigma = \sqrt{\sigma^2} = 7.7\).