What is the difference between sample and population standard deviation?

Population standard deviation measures the dispersion of an entire population and divides the sum of squared deviations by N. Sample standard deviation estimates the dispersion based on a sample and divides by N-1 (Bessel's correction) to provide an unbiased estimate.

What does standard deviation tell you?

Standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation indicates that values are clustered close to the mean, while a high standard deviation indicates that values are spread out over a wider range.

How do I calculate standard deviation?

To calculate it manually: 1) Find the mean of the data set. 2) Subtract the mean from each data point and square the result. 3) Find the mean of those squared differences (this is the variance). 4) Take the square root of the variance.

Why is the sample formula divided by N-1?

Dividing by N-1, known as Bessel's correction, corrects the bias in the estimation of the population variance. It mathematically adjusts the estimate upward, preventing the sample from underestimating the true population spread.

Standard Deviation Calculator

Instantly calculate standard deviation, variance, mean, and sum of squares for any numerical dataset. Compare population and sample variance with full step-by-step mathematical working.

Enter Data Set (comma or space separated)

Data Type

Use Sample (N-1) when your data is a subset of a larger population.

monitoringResults

Standard Deviation (s)

5.2372

Variance (s²)

27.4286

Mean (μ)

Count (N)

Sum (Σx)

144

functionsStep-by-Step Calculation

Value (x)	Mean (μ)	Deviation (x - μ)	Squared Dev (x - μ)²
10	18.00	-8.00	64.0000
12	18.00	-6.00	36.0000
23	18.00	5.00	25.0000
23	18.00	5.00	25.0000
16	18.00	-2.00	4.0000
23	18.00	5.00	25.0000
21	18.00	3.00	9.0000
16	18.00	-2.00	4.0000
Sum = 144.00			Sum of Squares = 192.0000

1. Calculate the Mean: Sum of all values (144.00) ÷ 8 = 18.0000

2. Find Squared Deviations: Subtract the mean from each value and square the result, then sum them up = 192.0000

3. Calculate Variance: Sum of Squares ÷ (N-1) i.e., 7 = 27.4286

4. Calculate Standard Deviation: √Variance = 5.2372

psychologyQuick Answer: What is Standard Deviation?

Standard deviation is a statistical measure that calculates the dispersion of a dataset relative to its mean. If the standard deviation is low, the data points tend to be very close to the mean. If it is high, the data points are spread out over a large range of values.

Symbol: σ (Sigma) for Population, s for Sample.
Variance: Standard deviation is exactly the square root of the variance.
Normal Distribution: In a normal curve, 68% of values fall within 1 standard deviation, 95% within 2, and 99.7% within 3.

The Formulas

Population Standard Deviation (σ)

Use when your data represents the entire population of interest.

σ = √( Σ(x - μ)² / N )

μ = Population Mean
N = Size of the population
x = Each individual value

Sample Standard Deviation (s)

Use when your data represents a sample subset of a larger population.

s = √( Σ(x - x̄)² / (n - 1) )

x̄ = Sample Mean
n = Size of the sample
n - 1 = Bessel's correction

Population vs. Sample Comparison

Feature	Population (σ)	Sample (s)
Use Case	When analyzing the entire group.	When estimating a larger group from a subset.
Denominator	N (Total Count)	n - 1 (Bessel's Correction)
Mean Symbol	μ (Mu)	x̄ (x-bar)
Result Magnitude	Always smaller (tighter)	Always slightly larger (more conservative)

How to Calculate Standard Deviation By Hand

Calculate the Mean

Add up all the data points and divide by the total number of data points.

Calculate the Deviations

Subtract the mean from each individual data point.

Square the Deviations

Multiply each deviation by itself. This removes negative numbers and exponentially penalizes outliers.

Calculate the Variance

Find the sum of all the squared deviations. Divide this sum by N (for population) or N-1 (for a sample).

Take the Square Root

Finally, take the square root of the variance to scale the number back down to the original units of your data.

Frequently Asked Questions

What does a standard deviation of 0 mean?expand_more

A standard deviation of zero means there is absolutely no variation in the dataset. Every single data point is exactly the same, which means they are all identical to the mean.

Can standard deviation be negative?expand_more

No, standard deviation can never be negative. Because you square the deviations before averaging them and then take the principal (positive) square root, the result is always a positive number or zero.

Why do we square the deviations instead of taking absolute values?expand_more

Squaring the differences not only makes all negative values positive, but it also gives more weight (or penalty) to data points that are further from the mean (outliers). This mathematical property makes standard deviation highly sensitive to outliers, which is often desirable in statistics.

What is the Empirical Rule (68-95-99.7 rule)?expand_more

If your data follows a normal distribution (a bell curve), the empirical rule states that approximately 68% of your data will fall within one standard deviation of the mean, 95% will fall within two standard deviations, and 99.7% will fall within three standard deviations.

Methodology & Accuracy

This calculator is designed to provide clinical-grade statistical outputs utilizing established mathematical formulas (Bessel's Correction for sample sets and standard division for population sets). It is capable of handling heavy floating-point precision and large comma-separated arrays.

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updateUpdated: Sept 2026

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