## Quick Math Variance

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Example (dataset): $578910$

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### Variance in Statistics

Variance is a measure of the dispersion or spread of a set of data points. It quantifies how much the values in a data set differ from the mean (average) of the data set. The larger the variance, the more spread out the data points are.

#### Formula for Variance:

For a population, the variance \( \sigma^2 \) is given by:

\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2 \]

where:

– \( N \) is the number of data points.

– \( x_i \) represents each data point.

– \( \mu \) is the mean of the data set.

For a sample, the variance \( s^2 \) is given by:

\[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2 \]

where:

– \( n \) is the number of data points in the sample.

– \( x_i \) represents each data point.

– \( \bar{x} \) is the mean of the sample.

#### Detailed Example:

Consider the following sample data set of test scores: \( 5, 7, 8, 9, 10 \).

To find the variance of a set of numbers, we follow these steps:

1. Calculate the mean of the numbers.

2. Subtract the mean from each number and square the result.

3. Find the average of these squared differences (this is the variance).

Let’s calculate the variance for the numbers 5, 7, 8, 9, and 10.

To find the variance of a set of numbers, we follow these steps:

1. Calculate the mean of the numbers.

2. Subtract the mean from each number and square the result.

3. Find the average of these squared differences (this is the variance).

Let’s calculate the variance for the numbers 5, 7, 8, 9, and 10.

Step 1: Calculate the mean \(\mu\) of the numbers.

\[

\mu = \frac{5 + 7 + 8 + 9 + 10}{5} = \frac{39}{5} = 7.8

\]

Step 2: Subtract the mean from each number and square the result:

\[

(5-7.8)^2 = (-2.8)^2 = 7.84

\]

\[

(7-7.8)^2 = (-0.8)^2 = 0.64

\]

\[

(8-7.8)^2 = (0.2)^2 = 0.04

\]

\[

(9-7.8)^2 = (1.2)^2 = 1.44

\]

\[

(10-7.8)^2 = (2.2)^2 = 4.84

\]

Step 3: Find the average of these squared differences (variance):

\[

\text{Variance} = \frac{7.84 + 0.64 + 0.04 + 1.44 + 4.84}{5} = \frac{14.8}{5} = 2.96

\]

Therefore, the variance of the numbers 5, 7, 8, 9, and 10 is 2.96.

#### Interpretation:

A variance of 2.96 indicates that the test scores in the sample data set vary, on average, 2.96 units squared from the mean score of 7.8. A larger variance would indicate greater dispersion of scores, while a smaller variance would indicate that the scores are more tightly clustered around the mean.