Standard Deviation Calculator
Calculate standard deviation for your dataset instantly. Supports both population and sample standard deviation calculations.
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How to Use This Calculator
Enter Data
Input your numerical values separated by commas or spaces
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Select population or sample standard deviation
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View standard deviation, variance, and mean instantly
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Browse All Math CalculatorsFrequently Asked Questions
What is standard deviation?
Standard deviation is a measure of how spread out numbers are from their average (mean). A low standard deviation means data points are close to the mean, while a high standard deviation indicates data points are spread out over a wider range.
What's the difference between population and sample standard deviation?
Population standard deviation is used when you have data for the entire population and divides by N. Sample standard deviation is used when you have a sample from a larger population and divides by N-1 for better estimation.
How do you calculate standard deviation?
To calculate standard deviation: 1) Find the mean of your data, 2) Calculate the squared differences from the mean, 3) Find the average of those squared differences (variance), 4) Take the square root of the variance.
How to Use This Standard Deviation Calculator
This calculator computes the standard deviation, variance, and mean of any numerical dataset in seconds. Here is how to use it:
- Enter your data in the text area. Separate numbers with commas, spaces, or new lines. For example:
4, 8, 6, 5, 3, 7, 2, 9. - Select the type: check the "Population Standard Deviation" box if your data represents an entire population. Leave it unchecked for sample standard deviation (the more common choice when working with a subset of data).
- View results instantly. The calculator updates in real-time as you type, showing the standard deviation, variance, mean, and count of values.
- Interpret your results: a smaller standard deviation means the data points are clustered closely around the mean, while a larger value indicates more spread.
Understanding Standard Deviation
What Is Standard Deviation?
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. It tells you, on average, how far each data point is from the mean (average) of the dataset. A low standard deviation indicates that data points tend to be very close to the mean, while a high standard deviation indicates the data is spread out over a wide range of values.
Variance vs. Standard Deviation
Variance is the average of the squared differences from the mean. Standard deviation is simply the square root of the variance. While variance is useful in mathematical derivations, standard deviation is more intuitive because it is expressed in the same units as the original data. For example, if your data is in centimeters, the standard deviation is also in centimeters, but the variance would be in square centimeters.
Population vs. Sample Standard Deviation
The key difference lies in the denominator of the variance formula. Population standard deviation divides by N (the total number of values) and is used when your dataset includes every member of the population. Sample standard deviation divides by N-1 (known as Bessel's correction) and is used when your data is a sample drawn from a larger population. The N-1 correction produces a less biased estimate of the true population variance.
The 68-95-99.7 Rule (Empirical Rule)
For data that follows a normal (bell-shaped) distribution, the standard deviation has a powerful interpretation:
- 68% of data falls within 1 standard deviation of the mean.
- 95% of data falls within 2 standard deviations of the mean.
- 99.7% of data falls within 3 standard deviations of the mean.
This rule is widely used in quality control, finance, and science to identify outliers and set confidence intervals.
Standard Deviation Formulas
Population Standard Deviation
Mean: mu = (x1 + x2 + ... + xN) / N
Variance: sigma^2 = SUM((xi - mu)^2) / N
Std Dev: sigma = sqrt(sigma^2)
Sample Standard Deviation
Mean: x_bar = (x1 + x2 + ... + xN) / N
Variance: s^2 = SUM((xi - x_bar)^2) / (N - 1)
Std Dev: s = sqrt(s^2)
Notice the only difference is the denominator: N for population, N-1 for sample. This small change (Bessel's correction) has a significant impact on accuracy when working with samples.
Worked Example: Calculate Sample Standard Deviation
Find the sample standard deviation of the dataset: 4, 8, 6, 5, 3.
Step 1: Find the mean. Mean = (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2
Step 2: Calculate each squared difference from the mean.
(4 - 5.2)^2 = (-1.2)^2 = 1.44
(8 - 5.2)^2 = (2.8)^2 = 7.84
(6 - 5.2)^2 = (0.8)^2 = 0.64
(5 - 5.2)^2 = (-0.2)^2 = 0.04
(3 - 5.2)^2 = (-2.2)^2 = 4.84
Step 3: Sum the squared differences. 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8
Step 4: Divide by (N - 1) for sample variance. Variance = 14.8 / (5 - 1) = 14.8 / 4 = 3.7
Step 5: Take the square root. Standard Deviation = sqrt(3.7) = 1.9235
The sample standard deviation is approximately 1.92. This means the data points deviate from the mean by about 1.92 units on average.
Tips and Tricks for Standard Deviation
- Choose the right type. Use population standard deviation only when you have data for every single member of the group. For surveys, experiments, or any subset, use sample standard deviation.
- Standard deviation cannot be negative. Since it is derived from squared differences, the result is always zero or positive. A standard deviation of zero means all values are identical.
- Outliers have a large effect. Because differences are squared, extreme values disproportionately increase the standard deviation. Consider whether outliers should be removed before analysis.
- Compare using the coefficient of variation. To compare variability between datasets with different units or means, divide the standard deviation by the mean (CV = s / x_bar). This gives a relative measure of spread.
- Use the empirical rule for quick estimates. If your data is approximately normally distributed, you can quickly estimate that about 68% of values lie within one standard deviation of the mean.
- More data generally means more accuracy. Larger sample sizes produce more reliable estimates of the true population standard deviation.