Mean and Standard Deviation Calculator

What are Mean and Standard Deviation?

The mean is a measure of central tendency, representing the average value of a dataset. Standard deviation is a measure of variability or dispersion, indicating how much the data points typically deviate from the mean.

Formulas and Their Meanings

1. Mean (\(\bar{x}\)): \[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\] Where \(x_i\) are individual values and \(n\) is the number of values.

2. Variance (\(s^2\)): \[s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}\] This measures the average squared deviation from the mean.

3. Standard Deviation (\(s\)): \[s = \sqrt{s^2}\] The square root of the variance, giving a measure of spread in the same units as the original data.

4. Population Variance (\(\sigma^2\)): \[\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}\] Similar to sample variance, but divided by (n-1) instead of n.

5. Population Standard Deviation (\(\sigma\)): \[\sigma = \sqrt{\sigma^2}\] The square root of the population variance.

Calculation Steps

1. Calculate the mean by summing all values and dividing by the count.
2. Subtract the mean from each value and square the result.
3. Sum these squared differences.
4. Divide by n (or n-1 for population) to get the variance.
5. Take the square root of the variance to get the standard deviation.

Example Calculation

Let's calculate for the dataset: 2, 4, 4, 4, 5, 5, 7, 9

1. Mean: \(\bar{x} = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5\)
2. Squared differences: \((2-5)^2, (4-5)^2, ..., (9-5)^2\)
3. Sum of squared differences: \(9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32\)
4. Variance: \(s^2 = \frac{32}{8} = 4\)
5. Standard Deviation: \(s = \sqrt{4} = 2\)

Visual Representation

This scatter plot represents the example dataset. The red dashed line indicates the mean (5), and the spread of points illustrates the standard deviation.