#1 Free Standard Deviation Calculator 2025

Formula:

Population Mean (μ) = Σxᵢ / N
Sample Mean (x̄) = Σxᵢ / n

Theory:

The mean is the average of all data points. It's calculated by summing all values and dividing by the count of values. It represents the central tendency of the data.

Step-by-step Calculation:

Formula:

Population Variance (σ²) = Σ(xᵢ - μ)² / N

Sample Variance (s²) = Σ(xᵢ - x̄)² / (n - 1)

Theory:

Variance measures the average squared difference of each data point from the mean. A larger variance indicates greater spread or dispersion in the data.

Population Variance (σ²): Used when your data includes every member of the group of interest. The denominator is N (the total number of data points).

Sample Variance (s²): Used when your data is a sample from a larger population. The denominator is n-1 (degrees of freedom), which provides an unbiased estimate of the population variance.

Step-by-step Calculation:

Formula:

Population Standard Deviation (σ) = √[Σ(xᵢ - μ)² / N]

Sample Standard Deviation (s) = √[Σ(xᵢ - x̄)² / (n - 1)]

Theory:

Standard Deviation is the square root of the variance. It measures the typical or average distance of data points from the mean, expressed in the original units of the data. A smaller standard deviation indicates data points are close to the mean; a larger one indicates they are spread out.

Population Standard Deviation (σ): The square root of the population variance. Describes the spread for the entire population.

Sample Standard Deviation (s): The square root of the sample variance. Estimates the spread for the population from which the sample was drawn.

Step-by-step Calculation: