Find the deviations from the mean

Oct 13, 2025 by talynt.org 45 views

Understanding the Problem

Fiona recorded the number of miles she biked each day last week as shown below: $4, 7, 4, 10, 5$ . The mean is given by $m = 6$ . We are asked to find the equation that shows the variance for the number of miles Fiona biked last week.

What is Variance?

Variance is a measure of the spread or dispersion of a set of data from its mean value. It is calculated by finding the average of the squared differences from the mean. In other words, it measures how much each data point deviates from the mean value.

Calculating Variance

To calculate the variance, we need to follow these steps:

Find the mean: The mean is given as $m = 6$ .
Find the deviations from the mean: Subtract the mean from each data point to find the deviations.
Square the deviations: Square each deviation to find the squared deviations.
Find the average of the squared deviations: Add up the squared deviations and divide by the number of data points to find the average.

Calculating the Variance

Let's calculate the variance using the given data.

Day	Miles Biked	Deviation from Mean	Squared Deviation
1	4	-2	4
2	7	1	1
3	4	-2	4
4	10	4	16
5	5	-1	1

Step 1: Find the mean

The mean is given as $m = 6$ .

Step 2: Find the deviations from the mean

Subtract the mean from each data point to find the deviations.

Day	Miles Biked	Deviation from Mean
1	4	-2
2	7	1
3	4	-2
4	10	4
5	5	-1

Step 3: Square the deviations

Square each deviation to find the squared deviations.

Day	Miles Biked	Deviation from Mean	Squared Deviation
1	4	-2	4
2	7	1	1
3	4	-2	4
4	10	4	16
5	5	-1	1

Step 4: Find the average of the squared deviations

Add up the squared deviations and divide by the number of data points to find the average.

Average of squared deviations = (4 + 1 + 4 + 16 + 1) / 5 = 26 / 5 = 5.2

The Variance Equation

The variance equation is given by:

σ^2 = Σ(xi - μ)^2 / N

where σ^2 is the variance, xi is each data point, μ is the mean, and N is the number of data points.

Plugging in the values, we get:

σ^2 = (4 + 1 + 4 + 16 + 1) / 5 σ^2 = 26 / 5 σ^2 = 5.2

Conclusion

In conclusion, the variance equation that shows the variance for the number of miles Fiona biked last week is:

σ^2 = (4 + 1 + 4 + 16 + 1) / 5 σ^2 = 26 / 5 σ^2 = 5.2

This equation represents the average of the squared deviations from the mean, which is a measure of the spread or dispersion of the data.

Final Answer

$The final answer is: $\boxed{5.2}$<br/> **Frequently Asked Questions (FAQs)**$

Q: What is variance?

A: Variance is a measure of the spread or dispersion of a set of data from its mean value. It is calculated by finding the average of the squared differences from the mean.

Q: How is variance calculated?

A: To calculate variance, you need to follow these steps:

Find the mean of the data.
Find the deviations from the mean by subtracting the mean from each data point.
Square each deviation to find the squared deviations.
Find the average of the squared deviations by adding them up and dividing by the number of data points.

Q: What is the formula for variance?

A: The formula for variance is:

σ^2 = Σ(xi - μ)^2 / N

where σ^2 is the variance, xi is each data point, μ is the mean, and N is the number of data points.

Q: What is the difference between variance and standard deviation?

A: Variance is a measure of the spread or dispersion of a set of data from its mean value, while standard deviation is the square root of the variance. Standard deviation is a more intuitive measure of spread, as it is measured in the same units as the data.

Q: Why is variance important?

A: Variance is important because it helps to understand the spread or dispersion of a set of data. It can be used to identify outliers, detect changes in the data, and make predictions about future data.

Q: Can variance be negative?

A: No, variance cannot be negative. The squared deviations are always positive, so the average of the squared deviations will also be positive.

Q: Can variance be zero?

A: Yes, variance can be zero. This occurs when all the data points are the same, so there is no spread or dispersion.

Q: How is variance used in real-world applications?

A: Variance is used in many real-world applications, including:

Finance: to measure the risk of investments
Engineering: to measure the spread of data in quality control
Medicine: to measure the spread of data in clinical trials
Business: to measure the spread of data in market research

Q: What are some common mistakes to avoid when calculating variance?

A: Some common mistakes to avoid when calculating variance include:

Not using the correct formula
Not squaring the deviations
Not dividing by the correct number of data points
Not checking for outliers

Q: How can I calculate variance using a calculator or computer?

A: You can calculate variance using a calculator or computer by using a statistical software package or a spreadsheet program such as Excel. These programs will allow you to enter the data and calculate the variance automatically.

Q: Can I calculate variance by hand?

A: Yes, you can calculate variance by hand, but it can be time-consuming and prone to errors. It is recommended to use a calculator or computer to calculate variance.