The standard deviation is a statistic that measures the data variability. It is derived from the square root of the distances between each value in the population and the population's mean squared.
We can calculate the population variance when we have the entire data
| σ2 = | Σ(𝑥i - x̄)2 |
| n |
σ = √(population variance)
In statistics, you usually do not know the true values, only the estimations base on samples. The standard deviation is important because it helps as to calculate ranges and probabilities.
For example:
What is the Confidence interval of the mean? What is the probability to get a value smaller than 2?
When we can not have the data of the entire population, we calculate the sample variance from the sampled data.
Unlike the variance, when calculating the sample variance we divide by (n - 1), in this case, the result statistic is not biased.
Following the sample variance formula
| S2 = | Σ(𝑥i - x̄)2 |
| n-1 |
When we can not have the data of the entire population, we calculate the sample standard deviation from the sampled data.
The sample standard deviation is the square root of the sample variance.
Unlike the sample variance, the sample standard deviation is biased, but the is the best simple formula.
The sample standard deviation formula is the square-root of the sample varaince.
S = √(sample variance)
The sample average is a random variable. Each time you calculate the sample average you will get a different result.
The standard deviation of the sample average is called SEM, the Standard Error of the Mean.
It is smaller than the standard error of the population.
| SEM = | S |
| √n |
The Median Absolute Deviation (MAD), is a statistic that measures the data variability.
The MAD is the average absolute distance from the arithmetic mean.
It is similar to the standard deviation, but instead of the addition of squares differences, it uses the absolute differences, and obviously, there is no need to take a square root.
| MAD = | Σ|𝑥i - x̄| |
| n |
Use the standard deviation (σ) if the data contains the entire population, otherwise use the sample standard deviation (S).
The standard deviation (σ) is the square root of the variance (Var), and the sample standard deviation (S) is the square root of the sample variance (S2). For sample variance we use n-1 instead of n, to correct the biased estimation of the variance (partially correct the estimation of the standard deviation) (Bessel's correction).
The SD calculator calculates both the population standard deviation and the sample standard deviation.