Find variance for 172,188,190,194,201,205,210,215,225,230,232,260 x² method

To calculate the variance of a data set using the \(x^2\) method (also known as the computational formula for variance), follow these steps:

### Step-by-step Process:

1. **List the Data:**
   The data set is:  
   \(172, 188, 190, 194, 201, 205, 210, 215, 225, 230, 232, 260\)

2. **Find the Mean (\(\bar{x}\)):**  
   The formula for the mean is:  
   \[
   \bar{x} = \frac{\sum x_i}{n}
   \]
   where \(x_i\) represents each data point and \(n\) is the number of data points.

3. **Find the Sum of Squares (\(\sum x_i^2\)):**  
   Calculate the square of each data point, then sum them up.

4. **Apply the Variance Formula:**
   The computational formula for variance is:
   \[
   \text{Variance} = \frac{\sum x_i^2}{n} - \left( \frac{\sum x_i}{n} \right)^2
   \]
   where \(\sum x_i^2\) is the sum of the squares of the data points, and \(\sum x_i\) is the sum of the data points.

Let's calculate this step-by-step:

### Step 1: Data Summary
- The data set is:  
  \(172, 188, 190, 194, 201, 205, 210, 215, 225, 230, 232, 260\)
- \(n = 12\) (12 data points)

### Step 2: Calculate the Mean (\(\bar{x}\))

\[
\bar{x} = \frac{172 + 188 + 190 + 194 + 201 + 205 + 210 + 215 + 225 + 230 + 232 + 260}{12}
\]

### Step 3: Calculate the Sum of Squares (\(\sum x_i^2\))

Now, I will perform the calculations.### Step 2: Calculate the Mean (\(\bar{x}\))
\[
\bar{x} = 210.17 \quad \text{(rounded to two decimal places)}
\]

### Step 3: Calculate the Variance
Using the formula for variance:

\[
\text{Variance} = \frac{\sum x_i^2}{n} - \left( \frac{\sum x_i}{n} \right)^2 = 526.97
\]

Thus, the variance of the data set is approximately **526.97**.

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How would the variance change if we added a new data point to the set?
2. What is the difference between variance and standard deviation?
3. Why is variance important in statistics?
4. How does variance differ from mean absolute deviation?
5. Can variance ever be negative, and why or why not?

**Tip**: The larger the variance, the more spread out your data is from the mean.

Find variance for 172, 188, 190, 194, 201, 205, 210, 215, 225, 230, 232, 260 using x² method.

Discover how to calculate the variance for the data set 172, 188, 190, 194, 201, 205, 210, 215, 225, 230, 232, 260 using the computational formula (x² method). Learn the step-by-step process, covering statistical concepts like variance, mean, and sum of squares. Suitable for Grades 9-12.

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