Calculate the variance for the data set. Show all of your steps. {4, 16, 21, 32, 11, 12}

To calculate the variance of the data set \{4, 16, 21, 32, 11, 12\}, we follow these steps:

### Step 1: Find the Mean (Average)
The mean is calculated by summing all the numbers and dividing by the number of data points.

\[
\text{Mean} = \frac{4 + 16 + 21 + 32 + 11 + 12}{6} = \frac{96}{6} = 16
\]

### Step 2: Calculate the Squared Differences from the Mean
Now, we subtract the mean from each data point and square the result.

- For \(4\): \((4 - 16)^2 = (-12)^2 = 144\)
- For \(16\): \((16 - 16)^2 = 0^2 = 0\)
- For \(21\): \((21 - 16)^2 = 5^2 = 25\)
- For \(32\): \((32 - 16)^2 = 16^2 = 256\)
- For \(11\): \((11 - 16)^2 = (-5)^2 = 25\)
- For \(12\): \((12 - 16)^2 = (-4)^2 = 16\)

### Step 3: Find the Average of the Squared Differences
Sum the squared differences:

\[
144 + 0 + 25 + 256 + 25 + 16 = 466
\]

Now, divide by the number of data points (for population variance) or by \(n - 1\) (for sample variance).

#### Population Variance
If this is the entire population, divide by \(n = 6\):

\[
\text{Variance} = \frac{466}{6} \approx 77.67
\]

#### Sample Variance
If this is a sample from a larger population, divide by \(n - 1 = 5\):

\[
\text{Variance} = \frac{466}{5} = 93.2
\]

### Final Answer:
- **Population Variance**: \(77.67\)
- **Sample Variance**: \(93.2\)

Would you like to know more details or have any other questions?

Here are some related questions:
1. What is the relationship between variance and standard deviation?
2. How do you calculate standard deviation from variance?
3. What is the significance of dividing by \(n - 1\) in sample variance?
4. What are the implications of a higher variance in a data set?
5. How does variance change if one data point is significantly larger than the others?

**Tip:** The square root of variance gives the standard deviation, a useful measure of data dispersion.

Learn how to calculate the variance for the data set {4, 16, 21, 32, 11, 12} with a detailed step-by-step solution. This example includes finding the mean, calculating squared differences, and determining both population and sample variance. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation