Standard Deviation Calculator

The standard deviation calculator computes the standard deviation, variance, mean, sum, SEM and MAD, with step by step calculation.
Enter comma , space or Enter after each data value.
The tool doesn't count empty cells or non-numeric cells.

Variance and standard deviation calculator

Calculate sample standard deviation, population standard deviation, sample variance, population variance, average, sample size.
Present histogram chart and average range chart

What is the standard deviation (SD)?

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.

Population variance formula

We can calculate the population variance when we have the entire data

σ2 =Σ(𝑥i - x̄)2
n

Population standard deviation formula

σ = √(population variance)

Why is standard deviation important?

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?

What is the sample variance? (S2)

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.

Sample variance formula

Following the sample variance formula

S2 =Σ(𝑥i - x̄)2
n-1

What is the sample standard deviation? (S)

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 it is the best simple formula.

Sample standard deviation formula

The sample standard deviation formula is the square-root of the sample varaince.

S = √(sample variance)

What is the standard deviation of the average (mean)?

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.

Standard Error of the Mean (SEM) formula
SEM =S
√n

What is MAD?

In statistics MAD may be the acronym for Median Absolute Deviation or Mean Absolute Deviation

What is Mean Absolute Deviation?

The Mean Absolute Deviation (MAD), is a statistic that measures the data variability.
The MAD is the average absolute distances 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.

Mean Absolute Deviation (MAD) formula

MAD =Σ|𝑥i - x̄|
n

What is Median Absolute Deviation?

The Median Absolute Deviation (MAD), is a statistic that measures the data variability.
The MAD is the median of the absolute distances from the median.

Median Absolute Deviation (MAD) formula

MAD = median(|Xi - Mdn|)
Mdn = median(Xi)

Standard deviation or sample standard deviation

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.


Statistics

n - Sample size, the number of values.
Mean - Average.
S - Sample standard deviation, if the list of values you entered is only a sample from the entire population, this is the best estimation for the Population Standard Deviation.
S2 - Sample variance, if the list of values you entered is only a sample from the entire population, this is the best estimation for the Population Variance.
σ - Population standard deviation, if the list of values you entered is the entire population, this is the exact Standard deviation.
σ2 - Population variance, if the list of values you entered is the entire population, this is the exact Variance.
MAD (Mean) - Mean absolute deviation, the average of the absolute distances from the average.
MAD (Median) - Median absolute deviation, the median of the absolute distances from the median.
SEM - Standard Error of the Mean, The standard deviation of the mean.