For the following distribution, use the computational variance formula to find the variance and standard deviation: 10, 15, 15, 17, 17, 17, 18, 18, 19, 22, 25, 27, 33, 35, 39, 50, 75, 76 show work

To find the **variance** and **standard deviation** of the given distribution using the **computational variance formula**, follow these steps:

### Given Data:
The data is:
\[
10, 15, 15, 17, 17, 17, 18, 18, 19, 22, 25, 27, 33, 35, 39, 50, 75, 76
\]

### Step 1: Calculate the Mean (\(\mu\))
The formula for the mean is:
\[
\mu = \frac{\sum x_i}{n}
\]
where \( x_i \) are the individual values, and \( n \) is the number of data points.

First, find the sum of all the data points:
\[
\sum x_i = 10 + 15 + 15 + 17 + 17 + 17 + 18 + 18 + 19 + 22 + 25 + 27 + 33 + 35 + 39 + 50 + 75 + 76 = 508
\]
The number of data points, \( n = 18 \).

Now calculate the mean:
\[
\mu = \frac{508}{18} \approx 28.22
\]

### Step 2: Use the Computational Formula for Variance
The computational variance formula is:
\[
\sigma^2 = \frac{\sum x_i^2}{n} - \mu^2
\]
where \( \sum x_i^2 \) is the sum of the squares of the individual data points.

Let's calculate \( \sum x_i^2 \):
\[
\sum x_i^2 = 10^2 + 15^2 + 15^2 + 17^2 + 17^2 + 17^2 + 18^2 + 18^2 + 19^2 + 22^2 + 25^2 + 27^2 + 33^2 + 35^2 + 39^2 + 50^2 + 75^2 + 76^2
\]
\[
\sum x_i^2 = 100 + 225 + 225 + 289 + 289 + 289 + 324 + 324 + 361 + 484 + 625 + 729 + 1089 + 1225 + 1521 + 2500 + 5625 + 5776 = 19219
\]

### Step 3: Calculate the Variance
Now, apply the values into the variance formula:
\[
\sigma^2 = \frac{19219}{18} - (28.22)^2
\]
First, calculate the square of the mean:
\[
(28.22)^2 \approx 796.34
\]
Now calculate the variance:
\[
\sigma^2 = \frac{19219}{18} - 796.34 \approx 1067.72 - 796.34 = 271.38
\]

### Step 4: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[
\sigma = \sqrt{271.38} \approx 16.47
\]

### Final Results:
- **Variance**: \( \sigma^2 \approx 271.38 \)
- **Standard Deviation**: \( \sigma \approx 16.47 \)

Would you like more details or further clarifications?

Here are 5 related questions:
1. How is the computational formula for variance derived from the basic definition?
2. What is the difference between population and sample variance?
3. Can you explain how the variance changes if outliers are present in the dataset?
4. How do you interpret the standard deviation in the context of data distribution?
5. How would you calculate the variance for a grouped frequency distribution?

**Tip**: Variance gives a measure of how spread out your data is, while standard deviation, being in the same units as the data, is easier to interpret.

Learn how to calculate the variance and standard deviation for the distribution: 10, 15, 15, 17, 17, 17, 18, 18, 19, 22, 25, 27, 33, 35, 39, 50, 75, 76. Follow step-by-step calculations using the computational variance formula, including the mean, sum of squares, and final standard deviation. Suitable for students in Grades 9-12.

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