How to use the standard deviation calculator
- Enter your numbers separated by commas, spaces or new lines in the data field.
- Choose Sample if your data is a subset of a larger group, or Population if it includes every value.
- Read the results instantly: count, mean, mean absolute deviation, standard deviation, variance and RSD.
How the mean absolute deviation is calculated
The mean absolute deviation (MAD) measures how far each value lies from the mean on average. First, find the arithmetic mean of all values. Then subtract the mean from each value, take the absolute value of each difference, and average those absolute differences.
For the dataset 2, 4, 4, 4, 5, 5, 7, 9 the mean is 40 / 8 = 5. The absolute deviations are |2-5|=3, |4-5|=1, |4-5|=1, |4-5|=1, |5-5|=0, |5-5|=0, |7-5|=2, |9-5|=4. Their sum is 12, so the MAD is 12 / 8 = 1.5.
This mean absolute deviation calculator also computes standard deviation, which squares the differences instead of taking their absolute values. Squaring gives more weight to outliers, so standard deviation is always equal to or greater than MAD for the same dataset.
Standard deviation and variance formulas
Variance is the average of the squared deviations from the mean. For a population, divide by N. For a sample, divide by n-1 (Bessel's correction) to produce an unbiased estimate.
| Measure | Population formula | Sample formula |
|---|---|---|
| Variance | σ2 = Σ(xi - μ)2 / N | s2 = Σ(xi - x̄)2 / (n-1) |
| Std Dev | σ = √(σ2) | s = √(s2) |
Using the same eight numbers above in population mode: the squared deviations are 9, 1, 1, 1, 0, 0, 4, 16. Their sum is 32. Population variance is 32 / 8 = 4, and population standard deviation is √4 = 2.
What is relative standard deviation (RSD)?
Relative standard deviation, also called the coefficient of variation, expresses the standard deviation as a percentage of the mean. The formula is RSD = (standard deviation / |mean|) × 100%.
RSD is useful when comparing variability across datasets with different units or scales. A lab measuring chemical concentrations might report an RSD below 2% to demonstrate precision. If the mean is zero, RSD is undefined because division by zero has no meaningful result.
For our example with mean 5 and population standard deviation 2, the RSD is (2 / 5) × 100 = 40%. An rsd calculator built into this tool saves you from doing that final step manually.
When to use MAD versus standard deviation
Both metrics describe spread, but they behave differently around extreme values. Because standard deviation squares each deviation, a single outlier pulls the result upward much more than it affects the MAD. This makes MAD a robust measure of dispersion.
In finance, quality control and laboratory work, standard deviation is the conventional choice because many statistical models assume normally distributed data where variance has clean mathematical properties. In exploratory data analysis or when outliers are expected, MAD often gives a clearer picture of typical spread.
If you need a quick sense of central tendency alongside these spread measures, tools like the average calculator can help verify your mean before interpreting deviations.
Common mistakes to avoid
- Mixing up sample and population mode. Using population formulas on sample data underestimates the true variability. Always select sample mode unless you have measured every member of the group.
- Entering non-numeric text. Letters, currency symbols or units mixed into the data field will cause errors. Enter only plain numbers.
- Forgetting negative values. Negative numbers are valid inputs. The absolute value step in MAD handles them correctly, and squaring handles them in standard deviation.
- Too few data points. Sample standard deviation requires at least two values because dividing by n-1 with n=1 means dividing by zero.
Worked examples
Each result below was produced by the same code the tool runs, so you can check your own numbers against them.
| Example | Input | Result |
|---|---|---|
| Test scores | Data values: 85, 90, 78, 92, 88; Mode: sample | Count: 5; Mean: 86.6; MAD: 4.08; Std Dev (s): 5.45894; Variance (s²): 29.8; RSD: 6.30362% |
| Lab measurements | Data values: 10.2, 10.5, 10.1, 10.4, 10.3, 10.2; Mode: sample | Count: 6; Mean: 10.2833; MAD: 0.116667; Std Dev (s): 0.147196; Variance (s²): 0.0216667; RSD: 1.4314% |
| Full population | Data values: 2, 4, 4, 4, 5, 5, 7, 9; Mode: population | Count: 8; Mean: 5; MAD: 1.5; Std Dev (σ): 2; Variance (σ²): 4; RSD: 40% |
| Two values | Data values: 10, 20; Mode: sample | Count: 2; Mean: 15; MAD: 5; Std Dev (s): 7.07107; Variance (s²): 50; RSD: 47.1405% |
FAQ
What is the difference between MAD and standard deviation?
MAD averages the absolute distances from the mean, while standard deviation averages the squared distances and then takes the square root. Standard deviation penalizes large outliers more heavily.
Should I choose sample or population mode?
Use population mode when your data includes every item in the group you care about. Use sample mode when your data is a subset drawn from a larger population, which is the more common scenario in statistics.
What does RSD stand for?
RSD stands for relative standard deviation. It is the standard deviation divided by the absolute value of the mean, expressed as a percentage. It lets you compare variability across datasets with different scales.
Can I enter negative numbers?
Yes. Both MAD and standard deviation handle negative values correctly. The absolute value operation in MAD and the squaring operation in standard deviation both eliminate the sign.
Why do I need at least two numbers for sample mode?
Sample variance divides by n-1. With only one number, n-1 equals zero, making the calculation undefined. You need at least two values to estimate sample variability.
Is my data sent to a server?
No. All calculations happen locally in your browser. Nothing you type is transmitted or stored.