1.      Using the five numbers for which you calculated the variance
and standard deviation in the previous exercise set, use the
calculating formula to find the variance. Compare your
answer to the one you found there. Must they be the same?

Recalculate the variance using the five numbers: 60, 70, 50, 30, and 40.

To calculate the variance for the five numbers \(60, 70, 50, 30, 40\), we will follow these steps:

### Step 1: Calculate the mean (\(\mu\)) of the five numbers.
The formula for the mean is:

\[
\mu = \frac{\sum x_i}{n}
\]

Where:
- \(x_i\) are the data points.
- \(n\) is the number of data points (in this case, 5).

\[
\mu = \frac{60 + 70 + 50 + 30 + 40}{5} = \frac{250}{5} = 50
\]

### Step 2: Calculate the squared differences from the mean.
For each data point, we subtract the mean and square the result:

\[
(60 - 50)^2 = 10^2 = 100
\]
\[
(70 - 50)^2 = 20^2 = 400
\]
\[
(50 - 50)^2 = 0^2 = 0
\]
\[
(30 - 50)^2 = (-20)^2 = 400
\]
\[
(40 - 50)^2 = (-10)^2 = 100
\]

### Step 3: Find the average of these squared differences.
The variance (\(\sigma^2\)) is the average of these squared differences:

\[
\sigma^2 = \frac{100 + 400 + 0 + 400 + 100}{5} = \frac{1000}{5} = 200
\]

### Step 4: Compare the result.
The variance calculated using the formula is \(200\).

If the previous variance was calculated using the same data set and method (population variance), the result should match. However, if it was calculated using sample variance, the result might differ slightly because the formula for sample variance divides by \(n - 1\) instead of \(n\).

### Answer:
The variance using the calculating formula is \(200\). This result should match if the same method (population variance) was used before.

Do you want more details or have any questions about this? Here are five questions to extend this topic:

1. How would the variance change if an additional number, say 80, was added to the set?
2. What is the difference between population variance and sample variance?
3. Can the variance ever be negative? Why or why not?
4. How would you interpret a higher variance in a data set?
5. How is standard deviation related to variance, and why is it useful?

**Tip**: The standard deviation is simply the square root of the variance and gives a sense of the spread of data in the same units as the original data.

Using the five numbers for which you calculated the variance and standard deviation in the previous exercise set, use the calculating formula to find the variance. Compare your answer to the one you found there. Must they be the same? Recalculate the variance using the five numbers: 60, 70, 50, 30, and 40.

Learn how to calculate the variance for the data set 60, 70, 50, 30, 40. This problem covers key statistical concepts such as variance, standard deviation, and mean. Step-by-step instructions provided with detailed calculations. Suitable for students in Grades 9-12.

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