CALCULATING THE STANDARD DEVIATION
(a) Find the variance of these values:
60, 70, 50, 30, 40
Use the formula defining the variance as the average
squared deviation from the mean.

To calculate the variance of the given values, we use the following steps:

### Step 1: Find the mean
The mean (\(\mu\)) is the average of all the numbers. To calculate it:

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

### Step 2: Find the squared deviations from the mean
Now, subtract the mean from each value, square the result, and list the squared deviations:

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

### Step 3: Calculate the variance
The variance (\(\sigma^2\)) is the average of these squared deviations. So, sum them up and divide by the number of values (which is 5):

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

### Step 4: Interpret the result
The variance of the given data set is:

\[
\sigma^2 = 200
\]

Would you like to see the calculation of the standard deviation as well?

### Relative Questions:
1. How is standard deviation related to variance?
2. Can you explain what a high variance indicates in a data set?
3. How would adding more values to the set affect the variance?
4. What is the difference between population and sample variance?
5. Can variance ever be negative?

### Tip:
Variance gives a sense of how spread out the data points are from the mean; a larger variance means more dispersion.

CALCULATING THE STANDARD DEVIATION
(a) Find the variance of these values:
60, 70, 50, 30, 40
Use the formula defining the variance as the average squared deviation from the mean.

Learn how to calculate the variance of the data set {60, 70, 50, 30, 40} with a detailed step-by-step solution. This problem covers key statistical concepts, including the mean, variance, and standard deviation. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation