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Standard Deviation Calculator

Standard Deviation Calculator

First, select "population" or "sample", input two or more numbers, then it will calculate the "total", "average", "variance", and "standard deviation"

What is Standard Deviation

Standard deviation is a measure of dispersion within a data set commonly used in statistics, represented by σ, indicating the deviation of each data point from the mean. A lower standard deviation means data points are closer to the mean, and a higher standard deviation indicates a wider distribution of data points from the mean.

Standard deviation is often used in fields such as finance and scientific research to quantify the variability or risk level.

Difference between Population Standard Deviation and Sample Standard Deviation

Standard deviation usually refers to the population standard deviation unless specified

Population standard deviation refers to the standard deviation calculated for the entire data set (population), whereas sample standard deviation is based on a subset (sample) randomly selected from the population. The main difference lies in the adjustment of degrees of freedom: population standard deviation calculates the average deviation of all data directly, while sample standard deviation decreases the number of data points by one (N-1) when calculating variance. This adjustment corrects potential biases when estimating population parameters using only a subset of data, making the sample standard deviation closer to the true population standard deviation.

Standard Deviation Formulas

Population Standard Deviation

σ=1Ni=1N(xiμ)2\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2}

  • NN: Population size
  • xix_i: Each value
  • μ\mu: Population mean

Sample Standard Deviation

s=1n1i=1n(xixˉ)2s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}

  • nn: Sample size
  • xix_i: Each value
  • xˉ\bar{x}: Sample mean

What is Variance

Variance is a statistical measure that represents the average of the squared differences from the mean. It measures how far each number in the set is from the mean.

Standard deviation is the square root of variance