Standard Deviation Calculator

Calculation Type

Data Values

#	Value	Actions

Count (n or N)

Mean (μ)

Variance (σ²)

Standard Deviation (σ)

Standard Deviation & Variance: The Complete Guide

Standard deviation and variance are the core measures of variability in data. If your dataset’s mean tells you the “center,” the variance and standard deviation quantify how tightly or loosely the values cluster around that center. Whether you’re analyzing quality metrics, A/B test results, financial returns, or lab measurements, understanding these two statistics helps you compare volatility, spot outliers, and make better decisions.

\( \displaystyle \textbf{Population:}\quad \mu = \frac{1}{N}\sum_{i=1}^{N} x_i,\quad \sigma^2 = \frac{1}{N}\sum_{i=1}^{N} (x_i-\mu)^2,\quad \sigma=\sqrt{\sigma^2} \)

\( \displaystyle \textbf{Sample:}\quad \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i,\quad s^2 = \frac{1}{n-1}\sum_{i=1}^{n} (x_i-\bar{x})^2,\quad s=\sqrt{s^2} \)

The difference between population and sample formulas is the denominator: divide by \(N\) for a full population; divide by \(n-1\) (Bessel’s correction) when using a sample to estimate a population’s variability.

When to Use Standard Deviation vs. Variance

Variance (\(\sigma^2\) or \(s^2\)) measures average squared deviation from the mean. It’s in squared units, which can be less intuitive but is fundamental for many statistical models.
Standard deviation (\(\sigma\) or \(s\)) is the square root of variance and returns to original units, making it easier to interpret and compare.
Population vs. Sample: If you measured the entire group, use population formulas (\(N\)). If you measured only a subset, use sample formulas (\(n-1\)).

Variables & Notation

\(x_i\): the \(i\)-th data value.
\(N\): number of values in a population (entire group).
\(n\): number of values in a sample (subset of a population).
\(\mu\): population mean, \(\mu = \frac{1}{N}\sum x_i\).
\(\bar{x}\): sample mean, \(\bar{x} = \frac{1}{n}\sum x_i\).
\(\sigma^2\), \(\sigma\): population variance and population standard deviation.
\(s^2\), \(s\): sample variance and sample standard deviation.

How to Calculate by Hand (Step-by-Step)

Compute the mean: \(\mu\) for population or \(\bar{x}\) for sample.
Find each deviation: \(d_i = x_i – \text{mean}\).
Square each deviation: \(d_i^2\).
Sum the squared deviations: \(\sum d_i^2\).
Divide by \(N\) (population) or \(n-1\) (sample) to get variance.
Take the square root to get standard deviation.

In practice, your calculator (above) performs these steps instantly and also shows intermediate values so you can audit the math.

Worked Examples

Example 1: Population Standard Deviation

Data (population): 4, 7, 9, 10, 10. Here \(N=5\).

\( \displaystyle \mu=\frac{4+7+9+10+10}{5}=\frac{40}{5}=8 \)

Deviations & squares:

#	\(x_i\)	\(x_i-\mu\)	\((x_i-\mu)^2\)
1	4	-4	16
2	7	-1	1
3	9	1	1
4	10	2	4
5	10	2	4

Sum of squares = \(16+1+1+4+4=26\).

\( \displaystyle \sigma^2=\frac{26}{5}=5.2 \quad\Rightarrow\quad \sigma=\sqrt{5.2}\approx 2.2804 \)

Example 2: Sample Standard Deviation

Data (sample): 12, 15, 18, 20. Here \(n=4\).

\( \displaystyle \bar{x}=\frac{12+15+18+20}{4}=\frac{65}{4}=16.25 \)

Deviations & squares:

#	\(x_i\)	\(x_i-\bar{x}\)	\((x_i-\bar{x})^2\)
1	12	-4.25	18.0625
2	15	-1.25	1.5625
3	18	1.75	3.0625
4	20	3.75	14.0625

Sum of squares = \(18.0625+1.5625+3.0625+14.0625=36.75\).

\( \displaystyle s^2=\frac{36.75}{n-1}=\frac{36.75}{3}=12.25 \quad\Rightarrow\quad s=\sqrt{12.25}=3.5 \)

Example 3: Weighted/Grouped Data (Optional)

Suppose you have distinct values with frequencies: value 2 occurs 3 times, value 5 occurs 4 times, value 8 occurs 3 times (total \(N=10\)).

\( \displaystyle \mu=\frac{(2)(3)+(5)(4)+(8)(3)}{10}=\frac{6+20+24}{10}=5 \)

For a population variance with grouped data:

\( \displaystyle \sigma^2=\frac{ \sum f_i(x_i-\mu)^2 }{N} =\frac{3(2-5)^2+4(5-5)^2+3(8-5)^2}{10} =\frac{3(9)+4(0)+3(9)}{10} =\frac{27+0+27}{10}=5.4 \)

Then \(\sigma=\sqrt{5.4}\approx 2.3238\).

How to Interpret Standard Deviation

Smaller standard deviation means observations are tightly clustered; your process is consistent.
Larger standard deviation means more spread; results are volatile or diverse.
For roughly bell-shaped (normal) data, about 68% of observations lie within \(\pm 1\) standard deviation of the mean, ~95% within \(\pm 2\), and ~99.7% within \(\pm 3\) (the empirical rule).

Common Use Cases

Finance: Compare volatility of returns across assets or portfolios.
Quality control: Track process variability and detect drifts or outliers.
Product analytics: Check consistency of load times, engagement metrics, or conversion rates.
Scientific measurement: Report uncertainty around experimental results.
Education & surveys: Compare spread in test scores or satisfaction ratings.

Limitations & Pitfalls

Outlier sensitivity: Squaring deviations makes variance and standard deviation sensitive to extreme values.
Non-normal data: If your distribution is highly skewed or heavy-tailed, the empirical rule won’t apply cleanly; consider robust measures like the median absolute deviation (MAD).
Units matter: Variance is in squared units, which can be hard to interpret directly; standard deviation fixes this but is still scale-dependent.
Population vs. sample: Using \(N\) instead of \(n-1\) on a sample underestimates true variability.

FAQ: Standard Deviation & Variance

What’s the difference between variance and standard deviation?

Variance averages the squared deviations from the mean and is expressed in squared units. Standard deviation is the square root of variance and returns to the original units, making it easier to interpret.

Should I divide by \(N\) or \(n-1\)?

Use \(N\) when you have the entire population. Use \(n-1\) (Bessel’s correction) when your data is a sample and you want an unbiased estimator of the population variance.

How many data points do I need?

You need at least two values to compute a sample standard deviation. More data generally leads to more stable estimates.

Does standard deviation assume normality?

The formula doesn’t, but many rules of thumb (like the 68–95–99.7% rule) rely on approximate normality. For skewed data, supplement with percentiles or robust statistics.

How do I compare variability across different scales?

Use the coefficient of variation (CV), defined as \( \text{CV} = \frac{\text{std dev}}{\text{mean}} \), to compare dispersion for variables measured in different units or scales (assuming means are positive and meaningfully comparable).

Quick Checklist for Accurate Results

Confirm you’re using sample vs population mode correctly.
Scan for outliers that can inflate the spread; consider reporting with and without them.
Ensure data are on a consistent scale (e.g., same units, no mixed currencies without conversion).
Include context: mean, standard deviation, and a plot (if available) provide a fuller picture.

Bottom Line

Variance and standard deviation turn raw numbers into insight about consistency and risk. Use the calculator above to compute both quickly, switch between population and sample modes, and review the step-by-step breakdown to verify every part of the calculation. When you pair these measures with visualizations and domain knowledge, you’ll make stronger, evidence-based decisions.