To determine the sample variance, [tex]\( s^2 \)[/tex], we need to follow a systematic approach. Here are the steps to compute it:
1. Identify the Data Points:
The given number of platinum albums for each artist or group is:
- Alabama: 49
- The Rolling Stones: 67
- Eminem: 62
- U2: 51
- Neil Diamond: 49
2. Calculate the Mean ([tex]\(\bar{x}\)[/tex]):
The mean is the average number of platinum albums. To get the mean, sum all the values and divide by the number of values.
[tex]\[
\bar{x} = \frac{49 + 67 + 62 + 51 + 49}{5} = \frac{278}{5} = 55.6
\][/tex]
3. Compute Each Deviation from the Mean and Square It:
For each data point, subtract the mean and square the result.
[tex]\[
(49 - 55.6)^2 = (-6.6)^2 = 43.56
\][/tex]
[tex]\[
(67 - 55.6)^2 = (11.4)^2 = 129.96
\][/tex]
[tex]\[
(62 - 55.6)^2 = (6.4)^2 = 40.96
\][/tex]
[tex]\[
(51 - 55.6)^2 = (-4.6)^2 = 21.16
\][/tex]
[tex]\[
(49 - 55.6)^2 = (-6.6)^2 = 43.56
\][/tex]
4. Sum These Squared Deviations:
Add up all the squared deviations.
[tex]\[
43.56 + 129.96 + 40.96 + 21.16 + 43.56 = 279.2
\][/tex]
5. Divide by the Number of Data Points Minus One (n-1) to Get the Sample Variance:
Since we are dealing with a sample, we divide by [tex]\( n-1 \)[/tex] (where [tex]\( n \)[/tex] is the number of data points).
[tex]\[
s^2 = \frac{279.2}{5 - 1} = \frac{279.2}{4} = 69.8
\][/tex]
Thus, the sample variance, [tex]\( s^2 \)[/tex], is 69.8.
