Consider a sample with data values of 28, 25, 26, 15, 30, 34, 24, and 25.
Compute the sample variance.

To compute the sample variance, follow these steps:

1. **Find the mean (average)** of the data values.
2. **Calculate the squared differences** between each data value and the mean.
3. **Sum the squared differences**.
4. **Divide by \( n - 1 \)** (where \( n \) is the number of data points) to get the sample variance.

### Step 1: Find the mean
The data values are: 28, 25, 26, 15, 30, 34, 24, 25.

The mean \( \bar{x} \) is calculated as:
\[
\bar{x} = \frac{28 + 25 + 26 + 15 + 30 + 34 + 24 + 25}{8} = \frac{207}{8} = 25.875
\]

### Step 2: Calculate the squared differences from the mean
For each data point, we calculate \( (x_i - \bar{x})^2 \):

- \( (28 - 25.875)^2 = (2.125)^2 = 4.5156 \)
- \( (25 - 25.875)^2 = (-0.875)^2 = 0.7656 \)
- \( (26 - 25.875)^2 = (0.125)^2 = 0.0156 \)
- \( (15 - 25.875)^2 = (-10.875)^2 = 118.5156 \)
- \( (30 - 25.875)^2 = (4.125)^2 = 17.0156 \)
- \( (34 - 25.875)^2 = (8.125)^2 = 66.0156 \)
- \( (24 - 25.875)^2 = (-1.875)^2 = 3.5156 \)
- \( (25 - 25.875)^2 = (-0.875)^2 = 0.7656 \)

### Step 3: Sum of squared differences
Now, sum the squared differences:
\[
4.5156 + 0.7656 + 0.0156 + 118.5156 + 17.0156 + 66.0156 + 3.5156 + 0.7656 = 211.1404
\]

### Step 4: Calculate the sample variance
Finally, divide the sum by \( n - 1 = 8 - 1 = 7 \):
\[
\text{Sample Variance} = \frac{211.1404}{7} = 30.1629
\]

Thus, the **sample variance** is approximately \( 30.16 \).

Would you like more details or have any questions?

Here are 5 related questions for further understanding:
1. How does sample variance differ from population variance?
2. What is the standard deviation for this sample?
3. How does variance help in understanding the spread of data?
4. What would happen to the variance if one data point was an outlier?
5. Can variance ever be negative? Why or why not?

**Tip**: Variance is a useful measure of data spread, but it is in squared units. The standard deviation, its square root, is in the same units as the data and is often more intuitive to interpret.

Consider a sample with data values of 28, 25, 26, 15, 30, 34, 24, and 25. Compute the sample variance.

Learn how to calculate the sample variance for a data set with values 28, 25, 26, 15, 30, 34, 24, and 25. This step-by-step guide covers key statistical concepts like mean, squared differences, and sample variance. Suitable for students in Grades 9-12.

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