mean and standard deviation of 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

To calculate the **mean** and **standard deviation** for the numbers \(4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9\):

### 1. **Mean**:
The formula for the mean (\(\mu\)) is:

\[
\mu = \frac{1}{n} \sum_{i=1}^{n} x_i
\]

Where \(x_i\) are the data points and \(n\) is the number of data points.

Given:
\[
n = 9, \quad \sum x_i = 4.1 + 4.2 + 4.3 + 4.4 + 4.5 + 4.6 + 4.7 + 4.8 + 4.9
\]

\[
\mu = \frac{4.1 + 4.2 + 4.3 + 4.4 + 4.5 + 4.6 + 4.7 + 4.8 + 4.9}{9}
\]

\[
\mu = \frac{40.5}{9} = 4.5
\]

### 2. **Standard Deviation**:
The formula for the standard deviation (\(\sigma\)) is:

\[
\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}
\]

Using the mean \(\mu = 4.5\):

Let's compute the variance first:

\[
\sigma^2 = \frac{1}{9} \left( (4.1 - 4.5)^2 + (4.2 - 4.5)^2 + (4.3 - 4.5)^2 + (4.4 - 4.5)^2 + (4.5 - 4.5)^2 + (4.6 - 4.5)^2 + (4.7 - 4.5)^2 + (4.8 - 4.5)^2 + (4.9 - 4.5)^2 \right)
\]

\[
\sigma^2 = \frac{1}{9} \left( (-0.4)^2 + (-0.3)^2 + (-0.2)^2 + (-0.1)^2 + 0^2 + (0.1)^2 + (0.2)^2 + (0.3)^2 + (0.4)^2 \right)
\]

\[
\sigma^2 = \frac{1}{9} \left( 0.16 + 0.09 + 0.04 + 0.01 + 0 + 0.01 + 0.04 + 0.09 + 0.16 \right) = \frac{1}{9} \times 0.6 = 0.0667
\]

Finally, the standard deviation:

\[
\sigma = \sqrt{0.0667} \approx 0.258
\]

### Summary:
- **Mean**: \(4.5\)
- **Standard deviation**: \(0.258\)

Would you like any more details or have questions on this?

Here are 5 related questions:
1. How does standard deviation relate to data spread?
2. What is the difference between population and sample standard deviation?
3. How would you interpret a small versus large standard deviation?
4. How can outliers affect the standard deviation?
5. How does standard deviation help in comparing datasets?

**Tip**: When dealing with larger data sets, you can use statistical software or a calculator to compute mean and standard deviation efficiently.

Calculate the mean and standard deviation of the data set: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, and 4.9. This solution covers essential statistical concepts such as central tendency and dispersion, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation