|Standard deviation (s):||0|
The standard deviation calculator shows you how to calculate the mean and standard deviation of a dataset. If you are learning statistics, it is essential to learn how to find standard deviation, because it is very widely used.
You'll love the special features of our standard deviation calculator:
- It works as a population or sample standard deviation calculator
- We show you the steps for easy understanding
- It's excellent as a learning tool, or as a calculator for small datasets
- The definition and formula for standard deviation are explained below
Read on to get started!
What is standard deviation?
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.
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 on the figure below.
Standard deviation formula
The mathematical definition for standard deviation (σ) is the positive square root of the variance (σ2):
variance = σ2
standard deviation = √(σ2) = σ
The standard deviation equation seems simple, but how do you calculate variance?
Variance is defined as the average squared difference from the mean for all data points. It is written as:
σ2 = ∑(xi - μ)2 / N
where σ2 is the variance, μ is the mean, and xᵢ represents the ith data point out of N total data points.
You can calculate variance in three steps:
Find the difference from the mean for each point. Use the formula:
xi - μ
Square the difference from the mean for each point:
(xi - μ)2
Find the average of the squared differences from the mean which you found in step 2:
∑(xi - μ)2 / N. 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:
σ = √(∑(xi - μ)2 / N)
Population vs. sample standard deviation formula
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 = ∑(xi - μ)2 / N underestimates the variance of the population.
To avoid underestimating the variance of a population (and consequently, the standard deviation), we replace
N - 1 in the formulas for variance and standard deviation, when sample data is used. This adjustment is known as Bessels' correction.
The sample variance formula becomes:
s2 = ∑(xi - x̄)2 / (N - 1)
and the complete standard deviation formula becomes:
s = √(∑(xi - x̄)2 / (N - 1))
where s2 is the estimate of variance, s is the estimate of standard deviation, and x̄ (pronounced as "x-bar") is the sample mean.
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:
x̄ = (2 + 4 + 5 + 6 + 6 + 9 + 10) / 7
x̄ = 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 - x̄)2
For the first point with a value of 2, the calculation would be:
(2 - 6)2 = (-4)2 = 16
The calculated squared differences from the mean for all data points are shown in the table below:
|xi||(xi - x̄)2|
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 = ∑(xi - x̄)2 / (N - 1)
s2 = (16 + 4 + 1 + 0 + 0 + 9 + 16) / (7 - 1)
s2 = 7.6667
The standard deviation (s) is the square root of the variance, so our final step is:
s = √7.6667
s = 2.7689
The standard deviation of the sample dataset was 2.8. Now that you know how to find standard deviation, try calculating it yourself, then check your answer using our calculator!
How to find standard deviation by hand?
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:
σ2 = ( ∑(xi2) - (∑xi)2/N ) / N
The easy-to-type formula for sample variance is:
s2 = ( ∑(xi2) - (∑xi)2/N ) / (N - 1)
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:
s2 = (( 12 + 22 + 42 + 62) - (1 + 2 + 4 + 6)2/4 ) / (4 - 1)
= (57 - (169 / 4)) / 3 = 4.9167
The standard deviation would then be the square root of the variance:
√4.9167 = 2.2
Try it yourself, then check your answer with our standard deviation calculator!
Summary of variables and equations
Table 1. Variables for population data
|Number of observations||N|
|Population mean||μ||∑(xi) / N|
|Sum of squares||SS||∑(xi - μ)2|
|Variance||σ2||SS / N|
Table 2. Variables for sample data
|Sample mean||x̄||∑(xi) / N|
|Sum of squares||SS||∑(xi - x̄)2|
|Sample variance||s2||SS / (N - 1)|