In the United States, the Recording Industry Association of America (RIAA) awards artists with a platinum album certification when an album has sold one million copies. The following data set includes a random sample of five artists or groups along with the number of platinum albums in a recent year. Determine the sample variance,
s
2
. Round your final answer to the nearest tenth if necessary.

Random Sample of Artists or Groups with Platinum Albums
Artist or Group
Number of

Platinum Albums

Alabama
49

The Rolling Stones
67

Eminem
62

U2
51

Neil Diamond
50

Provide your answer below:

The sample variance, $s^2$ , is
platinum albums.

To calculate the sample variance $ s^2 $, we can use the formula:

\[
s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2
\]

Where:
- $ n $ is the number of observations (in this case, 5).
- $ x_i $ represents each individual value in the sample.
- $ \bar{x} $ is the sample mean.

Let's go step by step:

### 1. Find the Sample Mean $ \bar{x} $
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{49 + 67 + 62 + 51 + 50}{5}
\]

\[
\bar{x} = \frac{279}{5} = 55.8
\]

### 2. Find Each $ (x_i - \bar{x})^2 $
Now, subtract the mean from each value, square the result, and then sum these squared differences.

\[
(x_1 - \bar{x})^2 = (49 - 55.8)^2 = (-6.8)^2 = 46.24
\]
\[
(x_2 - \bar{x})^2 = (67 - 55.8)^2 = (11.2)^2 = 125.44
\]
\[
(x_3 - \bar{x})^2 = (62 - 55.8)^2 = (6.2)^2 = 38.44
\]
\[
(x_4 - \bar{x})^2 = (51 - 55.8)^2 = (-4.8)^2 = 23.04
\]
\[
(x_5 - \bar{x})^2 = (50 - 55.8)^2 = (-5.8)^2 = 33.64
\]

### 3. Calculate the Sum of Squared Differences
\[
\sum (x_i - \bar{x})^2 = 46.24 + 125.44 + 38.44 + 23.04 + 33.64 = 266.8
\]

### 4. Calculate the Sample Variance $ s^2 $
Finally, divide the sum of squared differences by $ n-1 = 5-1 = 4 $:

\[
s^2 = \frac{266.8}{4} = 66.7
\]

### Final Answer
The sample variance $ s^2 $ is **66.7** platinum albums.

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Would you like to see further details or have any questions?

Here are some related questions you might consider:
1. How is sample variance different from population variance?
2. What is the standard deviation for this data set?
3. How would the variance change if an additional artist with 55 platinum albums was added?
4. What does a high variance indicate in the context of this data?
5. How would you interpret the variance if the number of platinum albums were much closer to each other?

**Tip:** When calculating variance by hand, it's important to keep your intermediate calculations as precise as possible to avoid rounding errors in the final result.

Learn how to calculate the sample variance of platinum albums sold by artists in the United States. This problem involves using statistical methods to determine the variability in album sales data. Suitable for college-level students studying statistics.

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