Step-by-step solution
How to Use the Standard Deviation Calculator
This standard deviation calculator can find either the sample or population standard deviation of a set of values in one place. To use the calculator:
- Choose Sample if your data are part of a larger population, or Population if the data include the entire group you want to study.
- Enter or paste your values into the data box. You can separate values with commas, spaces, tabs, or line breaks.
- Click Calculate
The calculator will return the standard deviation, related statistics, and a clear, step-by-step solution using the shortcut formula.
Example
Suppose a researcher records the number of hours spent exercising during a week for 7 people: 4, 6, 7, 9, 12, 14, 18. Find the standard deviation.
Solution
If the 7 people were selected from a larger population, we would calculate the sample standard deviation. To find it using the calculator:
- Select the Sample data option.
- Copy-paste the values into the data input field.
- Click Calculate.
The calculator returns: sample standard deviation, s = 4.9329.
However, if the 7 people consist of the entire group you want to study, we would treat the data as population data and follow these steps:
- Select the Population data option.
- Copy-paste the values into the calculator input field.
- Click Calculate.
The calculator gives: population standard deviation, σ = 4.5670.
As you can see, the results are different. As such, always read and re-read the question to determine whether you’re working with sample or population data. In many research settings, you’ll be working with sample data unless otherwise stated.
Sample or Population: Which Should I Choose?
The formula you should use depends on what your data represent.
Choose Sample when the observations are only part of a larger population. For example, suppose a researcher records the study hours of 40 students selected from a university with several thousand students. The 40 students are a sample.
Choose Population when your data include every observation in the group you want to describe. For example, if a class contains 25 students and you have the scores of all 25 students, the scores can be treated as a population if that class is the entire group of interest.
If you already know which type of data you have, you can use our sample standard deviation calculator or population standard deviation calculator instead.
Formulas Used by the Calculator
This calculator uses the shortcut, or computational, standard deviation formula. The formula changes depending on whether the data are a sample or a population.
For sample data, the calculator uses:
s = √[(Σxi2 − nx̄2) / (n − 1)]
Where:
- s is the sample standard deviation
- Σxi2 is the sum of the squared values
- x̄ is the sample mean
- n is the sample size
However, for population data, the calculator uses:
σ = √[(ΣXi2 − Nμ²) / N]
Where:
- σ = population standard deviation
- ΣXi2 = sum of the squared values
- μ = population mean
- N = population size
The calculator finds the required mean automatically. However, if you only need to calculate a mean, you can use the sample mean calculator or population mean calculator.
Standard Formula vs. Shortcut Formula
Standard deviation can also be calculated by finding each value’s deviation from the mean, squaring the deviations, and then adding them.
The standard and shortcut formulas give the same standard deviation when applied correctly. The difference is in how the calculation is organized.
This calculator uses the shortcut formula, which works directly with the sum of squared values. This keeps the calculation compact, especially when working with several observations.
However, if you want to see the deviation-from-the-mean method instead, our sample standard deviation calculator and population standard deviation calculator show that method step by step.
Understanding Your Standard Deviation Result
Standard deviation shows how spread out the values are around their mean.
A smaller standard deviation means the observations tend to be closer to the mean. A larger standard deviation means the observations are more spread out.
For example, two groups can have the same mean but very different standard deviations if one group has values that are much more dispersed.
Standard deviation is also expressed in the same units as the original data. If the observations are measured in kilograms, for example, the standard deviation is also measured in kilograms.
There is no single value that makes a standard deviation automatically “high” or “low.” The result should be interpreted in relation to the scale of the data and the values being studied.
Related
Frequently Asked Questions
This is an online tool that measures how spread out a set of values is around its mean. This calculator can find both sample and population standard deviation and shows the calculation steps used to obtain the answer.
Yes. Select Sample when your observations represent part of a larger population. Select Population when the observations include the entire group you want to describe.
The main difference is the denominator used in the calculation. Population standard deviation divides by N, while sample standard deviation divides by n − 1. The two methods therefore give slightly different results for the same data.
The shortcut formula calculates standard deviation using the sum of squared values and the mean. It gives the same result as the standard deviation-from-the-mean formula but provides another convenient way to organize the calculation.
Yes. You can paste a row or column of numeric values directly into the calculator. Values may be separated by commas, spaces, tabs, or line breaks.
