Variance and Standard Deviation Calculator

How to Calculate Variance and Standard Deviation

Variance and standard deviation are fundamental statistical tools that describe how data points differ from the mean. Here's how to calculate them:

Variance Calculation

Variance measures the average squared deviation of each data point from the mean. The formula is: \( \text{Variance} \ (\sigma^2) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \) Where:

• \( n \): Number of data points.
• \( x_i \): The \( i \)-th data point.
• \( \bar{x} \): The mean of the data set.
• \( (x_i - \bar{x})^2 \): The squared difference between each data point and the mean.

Standard Deviation Calculation

The standard deviation is the square root of the variance, representing the average deviation from the mean. The formula is: \( \text{Standard Deviation} \ (\sigma) = \sqrt{\text{Variance}} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2} \)

Calculation Steps

1. Find the Mean : Calculate the average of the data points \( \bar{x} \): \( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \)
2. Square the Deviations: For each data point, calculate the squared difference from the mean. \( (x_i - \bar{x})^2 \)
3. Calculate the Variance: Sum all squared differences and divide by the total number of data points \( n \).
4. Calculate the Standard Deviation: Take the square root of the variance.

For Example: Data Set = [ 3, 5, 6, 8, 9, 5], Calculate their variance and standard deviation.