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 the top five artists or groups along with the number of platinum albums in a recent year. Determine the population variance,
σ
2
, representing the spread of the data. Round your final answer to the nearest tenth if necessary.

Top Five Artists or Groups with Platinum Albums
Artist or Group
Number of

Platinum Albums

The Beatles
183

Garth Brooks
157

Elvis Presley
147

The Eagles
120

Led Zeppelin
113

To determine the population variance (\(\sigma^2\)) for the number of platinum albums among the top five artists or groups, we can follow these steps:

### Step 1: List the Data
The data set includes the following values:

- The Beatles: 183
- Garth Brooks: 157
- Elvis Presley: 147
- The Eagles: 120
- Led Zeppelin: 113

### Step 2: Calculate the Mean (\(\mu\))
The mean (\(\mu\)) is the average number of platinum albums.

\[
\mu = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{183 + 157 + 147 + 120 + 113}{5}
\]

Let's calculate this:

\[
\mu = \frac{183 + 157 + 147 + 120 + 113}{5} = \frac{720}{5} = 144
\]

### Step 3: Calculate Each Value's Deviation from the Mean and Square It
For each data point, subtract the mean and square the result:

\[
\text{Deviation for The Beatles} = (183 - 144)^2 = 39^2 = 1521
\]
\[
\text{Deviation for Garth Brooks} = (157 - 144)^2 = 13^2 = 169
\]
\[
\text{Deviation for Elvis Presley} = (147 - 144)^2 = 3^2 = 9
\]
\[
\text{Deviation for The Eagles} = (120 - 144)^2 = (-24)^2 = 576
\]
\[
\text{Deviation for Led Zeppelin} = (113 - 144)^2 = (-31)^2 = 961
\]

### Step 4: Calculate the Population Variance (\(\sigma^2\))
The population variance is the average of these squared deviations.

\[
\sigma^2 = \frac{1521 + 169 + 9 + 576 + 961}{5} = \frac{3236}{5} = 647.2
\]

### Final Answer
The population variance (\(\sigma^2\)) is **647.2** (rounded to the nearest tenth).

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Do you want details or have any questions?

Here are 5 related questions:
1. How does the standard deviation relate to the population variance?
2. What does a high population variance indicate about the data set?
3. How is population variance different from sample variance?
4. Can you calculate the standard deviation from this variance?
5. What are some other statistical measures used to describe data spread?

**Tip:** Variance provides insight into the data's spread, but the standard deviation (the square root of variance) is often more intuitive as it is in the same units as the data.

Learn how to calculate the population variance of platinum albums for the top five artists or groups, including The Beatles, Garth Brooks, Elvis Presley, The Eagles, and Led Zeppelin. This problem involves statistical analysis using the population variance formula to determine the spread of data. Suitable for college-level students and statistics enthusiasts.

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