Find the variance 20.2
33.1
22
58.5
23.4
110.7
30.4
24.9
75
60.1
40
45.9
65.4
76.4

To calculate the variance, we follow these steps:

1. **Find the mean (average) of the data.**
2. **Subtract the mean from each data point and square the result** (this is the squared deviation).
3. **Find the average of these squared deviations** (this is the variance).

Here are the given numbers:

20.2, 33.1, 22, 58.5, 23.4, 110.7, 30.4, 24.9, 75, 60.1, 40, 45.9, 65.4, 76.4

### Step 1: Find the mean
\[
\text{Mean} = \frac{\sum x_i}{n} = \frac{20.2 + 33.1 + 22 + 58.5 + 23.4 + 110.7 + 30.4 + 24.9 + 75 + 60.1 + 40 + 45.9 + 65.4 + 76.4}{14} 
\]
\[
= \frac{686}{14} = 49
\]

### Step 2: Calculate the squared deviations
For each number \(x_i\), we compute \((x_i - \text{mean})^2\).

| Data point (\(x_i\)) | Deviation (\(x_i - 49\)) | Squared deviation (\((x_i - 49)^2\)) |
|----------------------|--------------------------|--------------------------------------|
| 20.2                 | -28.8                    | 829.44                               |
| 33.1                 | -15.9                    | 252.81                               |
| 22                   | -27                      | 729                                  |
| 58.5                 | 9.5                      | 90.25                                |
| 23.4                 | -25.6                    | 655.36                               |
| 110.7                | 61.7                     | 3806.89                              |
| 30.4                 | -18.6                    | 345.96                               |
| 24.9                 | -24.1                    | 580.81                               |
| 75                   | 26                       | 676                                  |
| 60.1                 | 11.1                     | 123.21                               |
| 40                   | -9                      | 81                                   |
| 45.9                 | -3.1                     | 9.61                                 |
| 65.4                 | 16.4                     | 268.96                               |
| 76.4                 | 27.4                     | 750.76                               |

### Step 3: Find the variance
Now, sum the squared deviations and divide by the number of data points.

\[
\text{Variance} = \frac{\sum (x_i - 49)^2}{n} = \frac{8199.1}{14} \approx 585.65
\]

Thus, the variance is approximately **585.65**.

Would you like a more detailed explanation or have any questions?

### Related questions:
1. What is the difference between variance and standard deviation?
2. How is variance used in statistical analysis?
3. What happens to the variance if an outlier is added?
4. Can the variance ever be negative?
5. How does the number of data points affect the variance?

### Tip: 
The standard deviation is simply the square root of the variance, which provides a measure of how spread out the data is in the same units as the original data.

Find the variance 20.2, 33.1, 22, 58.5, 23.4, 110.7, 30.4, 24.9, 75, 60.1, 40, 45.9, 65.4, 76.4

Learn how to calculate the variance for the data set 20.2, 33.1, 22, 58.5, 23.4, 110.7, 30.4, 24.9, 75, 60.1, 40, 45.9, 65.4, 76.4. This problem involves statistical concepts such as mean, deviation, and variance, with step-by-step calculations. Suitable for students in Grades 9-12.

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