Standard Deviation Calculator

The standard deviation is a measure of the variability in a dataset. In other words, the standard deviation describes how "spread-out" the data is around the mean. This calculator deals with separate data points, but we also have a dedicated grouped data standard deviation calculator for ranged data.

A high standard deviation indicates that a dataset is more spread out.

A low standard deviation indicates that the data is more tightly clustered around the mean or less spread out.

Can you imagine what a standard deviation looks like? While you can calculate the standard deviation for any dataset, it can be helpful to visualize the standard deviation for normally distributed data. The empirical rule states that for any dataset which approximates a normal distribution, about 68% of the data will fall within one standard deviation from the mean, shown in the figure below.

Not only is standard deviation a widely-used measure of variation, it forms the basis of other tools which characterize variation, including the quantities calculated by the relative standard deviation calculator and the confidence interval calculator.

Want to know more? Make sure to check out our article on Confidence Interval vs. Standard Deviation.

The mathematical definition for standard deviation (σ) is the positive square root of the variance (σ2σ2):

The standard deviation equation seems simple, but how to calculate variance?

Variance is defined as the average squared difference from the mean for all data points. It is written as:

where:

• σ2σ2 - Variance;
• μμ - Mean; and
• xi​xi​ - The i^th data point out of NN total data points.

You can calculate variance in three steps:

1. Find the difference from the mean for each point. Use the formula:
   xi​−μxi​−μ

2. Square the difference from the mean for each point:
   (xi​−μ)2(xi​−μ)2

3. Find the average of the squared differences from the mean which you found in step 2:
   N1​∑(xi​−μ)2N1​∑(xi​−μ)2
   This is the variance for population data. Note that this step is slightly different for sample data (see next section).

Now we recall that the standard deviation is the (positive) square root of variance, so the complete standard deviation equation (for population data) becomes:

In many scientific experiments, only a sample of a population is measured for practical reasons. This sample allows us to make inferences about the population. However, when sample data is used to estimate the variance of a population, the variance formula σ2=N1​∑(xi​−μ)2σ2=N1​∑(xi​−μ)2 underestimates the variance of the population.

To avoid underestimating the variance of a population (and consequently, the standard deviation), we replace NN with N−1N−1 in the formulas for variance and standard deviation, when sample data is used. This adjustment is known as Bessel's correction.

The sample variance formula becomes:

and the complete standard deviation formula becomes:

where:

• s2s2 - Estimate of variance;
• ss - Estimate of standard deviation; and
• xˉxˉ (pronounced as "x-bar") - Sample mean.

💡 Want to know more? Check our article: Variance vs. Standard Deviation: Understanding the Difference.

Let's say we have a sample dataset with seven numbers: 2, 4, 5, 6, 6, 9, 10. How do we calculate standard deviation? Follow these steps:

1. Calculate the mean

To calculate the mean (x̄), divide the sum of all numbers by the number of data points:
xˉ=72+4+5+6+6+9+10​=6xˉ=72+4+5+6+6+9+10​=6.

2. Calculate the squared differences from the mean

Now that we know the mean (x̄ = 6), we will calculate the squared difference from the mean for each data point:
(xi​−xˉ)2(xi​−xˉ)2.

For the first point with a value of 2, the calculation would be:
(2−6)2=(−4)2=16(2−6)2=(−4)2=16.

The calculated squared differences from the mean for all data points are shown in the table below:

3. Calculate the variance and standard deviation

Since we are using sample data, we calculate variance using the sample variance equation and the squared differences from the mean we found in step 2:

s2=N−11​∑(xi​−xˉ)2s2=N−11​∑(xi​−xˉ)2,
which gives
s2=7−116+4+1+0+0+9+16​=7.6667s2=7−116+4+1+0+0+9+16​=7.6667.

The standard deviation (s) is the square root of the variance, so our final step is:

s=7.6667​=2.7689s=7.6667​=2.7689.

The standard deviation of the sample dataset was 2.8. Now that you know how to find the standard deviation try calculating it yourself, then check your answer using our calculator!

🔎 Did you know? Standard deviation is one of the measures of dispersion and coefficient of dispersion, concepts that help us understand the spread of our data.

Using our standard deviation calculator is easy and intuitive. You just need to follow the steps below:

• Enter your dataset. You can add up to 30 values for the dataset.
• Choose between sample and population at the settings configuration. Then, you can define whether you want to find the sample or the population standard deviation.
• Check your results. Our tool also provides a step-by-step calculation of the standard deviation for your dataset.

If you are calculating standard deviation with a handheld calculator, there is an easier formula you should use to use to calculate variance. This alternative formula is mathematically equivalent but easier to type into a calculator.

The easy-to-type formula for variance (for population data) is:

The easy-to-type formula for sample variance is:

To find standard deviation, you would first calculate variance using either of the formulas above. Then, the standard deviation would be the square root of variance.

For example, with a sample dataset of 1, 2, 4, 6, the calculation for sample variance would be:
∑(xi2​)=(12+22+42+62)=57∑(xi2​)=(12+22+42+62)=57
(∑xi​)2=4(1+2+4+6)2​=4169​=42.25(∑xi​)2=4(1+2+4+6)2​=4169​=42.25

which give

σ2=4−157−42.25​=4.9167σ2=4−157−42.25​=4.9167.

The standard deviation would then be the square root of the variance:

4.9167​≈2.24.9167​≈2.2

Try it yourself, then check your answer with our standard deviation calculator!

Table 1. Variables for population data

Table 2. Variables for sample data