The following data set represents the salaries of randomly selected teachers in six different states. Determine the sample variance, s2. Round your answer to the nearest tenth, if necessary.

47421,44731,62849,54085,44968,47884

To determine the sample variance (\(s^2\)) of the salaries, we can follow these steps:

1. **Calculate the mean (\(\bar{x}\))** of the salary values.
2. **Subtract the mean** from each salary value and **square the result**.
3. **Sum** all the squared differences.
4. **Divide** by \(n-1\) (where \(n\) is the number of data points) since it's the sample variance.

Let's go through each step.

### Step 1: Calculate the Mean (\(\bar{x}\))
The salary values are: 47421, 44731, 62849, 54085, 44968, 47884.

\[
\bar{x} = \frac{47421 + 44731 + 62849 + 54085 + 44968 + 47884}{6} 
\]

\[
\bar{x} = \frac{301938}{6} = 50323
\]

### Step 2: Calculate the Squared Differences
Next, we'll subtract the mean from each value, square the result, and sum these squared differences.

\[
\text{Squared Differences} = (47421 - 50323)^2, (44731 - 50323)^2, (62849 - 50323)^2, \ldots, (47884 - 50323)^2
\]

\[
(47421 - 50323)^2 = (-2902)^2 = 8427684
\]

\[
(44731 - 50323)^2 = (-5792)^2 = 33540464
\]

\[
(62849 - 50323)^2 = (12526)^2 = 156452676
\]

\[
(54085 - 50323)^2 = (3762)^2 = 14158404
\]

\[
(44968 - 50323)^2 = (-5355)^2 = 28676425
\]

\[
(47884 - 50323)^2 = (-2440)^2 = 5953600
\]

### Step 3: Sum the Squared Differences

\[
\text{Sum} = 8427684 + 33540464 + 156452676 + 14158404 + 28676425 + 5953600 = 248931253
\]

### Step 4: Calculate the Sample Variance

Since this is a sample variance, divide the sum of squared differences by \(n-1\) where \(n = 6\).

\[
s^2 = \frac{\text{Sum of Squared Differences}}{n-1} = \frac{248931253}{5} = 49786250.6
\]

Rounded to the nearest tenth:

\[
s^2 \approx 49786250.6
\]

So, the sample variance, \(s^2\), is **49,786,250.6**.

Would you like any further details or explanations?

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Here are five related questions to expand your understanding:
1. How is the calculation of sample variance different from population variance?
2. What would the standard deviation be for this data set?
3. How does sample variance change with larger sample sizes?
4. What are the implications of high variance in a dataset like salaries?
5. How would the sample variance change if one of the salaries were significantly higher or lower?

**Tip:** When calculating sample variance, remember to divide by \(n-1\) instead of \(n\) to account for the degrees of freedom.

Learn how to calculate the sample variance of teacher salaries using statistical methods. Step-by-step solution provided with specific data points and rounding to the nearest tenth. Suitable for Grades 10-12 students studying statistics.

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