Standard deviation is one of the most common ways to measure the spread of values in a dataset.
It turns out that there are two different types of standard deviations you can calculate, depending on the type of data you’re working with.
1. Population standard deviation
You should calculate the population standard deviation when the dataset you’re working with represents an entire population, i.e. every value that you’re interested in.
The formula to calculate a population standard deviation, denoted as σ, is:
σ = √Σ(xi – μ)2 / N
where:
- Σ: A symbol that means “sum”
- xi: The ith value in a dataset
- μ: The population mean
- N: The population size
2. Sample standard deviation
You should calculate the sample standard deviation when the dataset you’re working with represents a a sample taken from a larger population of interest.
The formula to calculate a sample standard deviation, denoted as s, is:
s = √Σ(xi – x̄)2 / (n – 1)
where:
- Σ: A symbol that means “sum”
- xi: The ith value in a dataset
- x̄: The sample mean
- n: The sample size
Population vs. Sample Standard Deviation: The Difference
From the formulas above, we can see that there is one tiny difference between the population and the sample standard deviation: When calculating the sample standard deviation, we divided by n-1 instead of N.
The reason for this is because when we calculate the sample standard deviation, we tend to underestimate the true variability in the population. In other words, our estimate of the true population standard deviation is biased.*
To correct this bias, we divide by n-1. This has been shown to make the sample standard deviation an unbiased estimate of the population standard deviation.
*Proof of this is beyond the scope of this article. For a mathematical proof, refer to this post from Stack Exchange.
Population vs. Sample Standard Deviation: When to Use Each
Use the following practice problems to gain a better understanding of when you should use population vs sample standard deviation.
Practice Problem 1: Sports
Suppose a basketball coach wants to summarize the mean and standard deviation of points scored by the 12 players on his team.
When calculating the standard deviation of points scored, should he use the population or sample standard deviation formula?
Answer: He should use the population standard deviation because he is only interested in the points scored by his players and not any other players on any other team.
Practice Problem 2: Height
Suppose a gym teacher wants to summarize the mean and standard deviation of heights of students in his class.
When calculating the standard deviation of height, should he use the population or sample standard deviation formula?
Answer: He should use the population standard deviation because he is only interested in the height of students in this one particular class.
Practice Problem 3: Biology
Suppose a biologist wants to summarize the mean and standard deviation of the weight of a particular species of turtles. She decides to go out and collect a simple random sample of 20 turtles from the population.
When calculating the standard deviation of weights, should she use the population or sample standard deviation formula?
Answer: She should use the sample standard deviation because she is interested in the weights of the entire population of turtles, not just the weights of the turtles in her sample.
Practice Problem 4: Manufacturing
Suppose an inspector wants to summarize the mean and standard deviation of the weight of tires produced at a certain factory. He decides to collect a simple random sample of 40 tires from the factory and weighs each of them.
When calculating the standard deviation of weights, should he use the population or sample standard deviation formula?
Answer: He should use the sample standard deviation because he is interested in the weights of all tires produced at this factory, not just the weights of the tires in his sample.
Additional Resources
The following tutorials provide additional information about the standard deviation:
Why is Standard Deviation Important?
What is Considered a Good Standard Deviation?
6 Examples of Using Standard Deviation in Real Life
Coefficient of Variation vs. Standard Deviation: The Difference