Calculate sample and population standard deviation, variance, mean, median, quartiles, outliers, z-scores, standard error, confidence interval, and full descriptive statistics from your dataset. Paste a column from a spreadsheet, read every formula and step, see box-plot and histogram charts, and download a 12-tab Excel workbook.
Transparent assumptions Sample & population · every formula Charts: number line, box plot, histogram z-scores, outliers, CI, empirical rule 12-tab Excel workbook Educational — verify important figures
Educational tool — exact to the precision you choose. Verify figures for coursework, research, or professional reports.
Standard deviation measures how far values typically sit from the mean. Find the mean, square each value’s deviation, average the squares (÷ N for a population, ÷ n−1 for a sample), then take the square root. For 10, 12, 8, 15, 11 the mean is 11.2, the population SD is 2.3152, and the sample SD is 2.5884.
Enter your dataset
Common datasets & examples
Quick Standard Deviation
5 values detected. Zeros and negatives are kept.
Sample standard deviation (s)
2.59
Divides by n − 1. Use for a sample.
Population standard deviation (σ)
2.32
Divides by N. Use for a full population.
Mean (x̄)
11.20
Σx ÷ n = 56.00 ÷ 5.
Sample variance (s²)
6.70
SD squared (n − 1).
Population variance (σ²)
5.36
SD squared (N).
Median
11.00
Middle value.
Mode
No mode
Every value appears once.
Count (n)
5
Sum = 56.00.
Range
7.00
Min 8.00 · Max 15.00.
The values show moderate spread around the mean. Compare the standard deviation with the mean (or use the coefficient of variation) to judge whether that spread is large for your context. The sample standard deviation (2.59) is slightly larger than the population standard deviation (2.32) because it divides by n − 1 to correct for sample bias.
The mean and median are close, so the distribution is approximately symmetric — the mean and standard deviation describe it well.
Mean ± standard deviation
Each value plotted against the mean and ±1, ±2, ±3 standard deviation bands (sample SD).
Minimum, Q1, median, Q3, maximum, with the 1.5×IQR whiskers and any outliers.
Min
8.00
Q1
10.00
Median
11.00
Q3
12.00
Max
15.00
Add more values to see a meaningful distribution — the histogram appears once there are at least 8 values, where the bin counts start to mean something.
Deviation table
Each value vs the mean, with its z-score (using the sample SD).
Deviation and z-score table
#
Value
xᵢ − mean
(xᵢ − mean)²
|xᵢ − mean|
z-score
Outlier?
1
10.00
-1.20
1.44
1.20
-0.46
—
2
12.00
0.80
0.64
0.80
0.31
—
3
8.00
-3.20
10.24
3.20
-1.24
—
4
15.00
3.80
14.44
3.80
1.47
—
5
11.00
-0.20
0.04
0.20
-0.08
—
Σ
56.00
≈ 0
26.80
9.20
n = 5
Quick answers
What do the calculation steps look like on a real dataset?
Find the mean, subtract it from each value, square the deviations, add them, divide by N (population) or n−1 (sample), then take the square root. For 10,12,8,15,11: mean 11.2, Σ(x−mean)² = 26.8, population SD = √(26.8/5) = 2.3152, sample SD = √(26.8/4) = 2.5884.
Sample vs population standard deviation
Population SD divides by N and is used when the data is the entire group. Sample SD divides by n−1 (Bessel’s correction) and is used when the data is a sample of a larger group. Sample SD is always the larger of the two.
How do I read the z-score column in my results?
z = (x − mean) ÷ SD. It is how many standard deviations a value sits above (positive) or below (negative) the mean. A z of 0 is exactly average; |z| > 2 or 3 is unusual under a normal model.
My data has outliers — which spread measure should I trust?
Outliers hit the SD hard: because deviations are squared, a single far-away value can inflate it sharply. For skewed or outlier-heavy data, the median, IQR, and median absolute deviation are more robust.
How to use this standard deviation calculator
Paste or type your data. Drop in values separated by commas, spaces, tabs, or new lines — including a column copied straight from Excel or Google Sheets. Zeros and negatives are kept; non-numeric text is ignored with a count.
Pick a mode. Quick SD for the essentials, Full descriptive statistics for everything, Compare for two datasets, Frequency table or Grouped data for summarised data, Confidence interval, Outlier analysis, or the Normal-distribution helper.
Choose sample or population. Show both, or pick sample (÷ n−1) or population (÷ N). Set the decimal precision (0–6) and, if you like, sort the values.
Read the results and charts. You get every statistic, a plain-English interpretation, a deviation/z-score table, and lightweight charts — a mean ± SD number line, a box plot, and a histogram.
Review outliers — don’t delete them. The calculator flags potential outliers by the IQR fence and the z-score threshold and shows the SD with and without them. Outliers may be errors, rare-but-valid values, or signals — review before removing.
Download the workbook. Export the 12-tab Excel workbook built from your dataset: live formulas (STDEV, QUARTILE, T.INV.2T…), a deviation table, outlier analysis, confidence interval, comparison, frequency/grouped tabs, charts, a formula reference, and 25 practice questions.
What is standard deviation?
Standard deviation measures how spread out a set of values is around the mean. A small standard deviation means the values are close to the average — consistent and predictable. A large standard deviation means the values are more spread out — variable and harder to pin down. It is the most widely used measure of spread because it uses every value and shares the data’s units.
The idea is simple: for each value, measure how far it is from the mean, square that distance (so positive and negative gaps don’t cancel), average the squares, and take the square root to return to the original units. The result is, roughly, the typical distance of a value from the mean.
Standard deviation formula
Population standard deviation
σ = √(Σ(xᵢ − μ)² ÷ N)
Use when the data is the entire population.
Sample standard deviation
s = √(Σ(xᵢ − x̄)² ÷ (n − 1))
Use when the data is a sample of a larger group.
Population / sample variance
σ² = Σ(xᵢ − μ)² ÷ N · s² = Σ(xᵢ − x̄)² ÷ (n − 1)
Variance is SD squared.
Mean
x̄ = Σx ÷ n
The arithmetic average.
What each symbol means
xᵢ — each individual value in the dataset.
μ (mu) — the population mean; x̄ (x-bar) — the sample mean.
N — the number of values in a population; n — the number in a sample.
Σ (sigma) — “sum of”, add up the terms that follow.
(xᵢ − mean)² — the squared deviation of one value from the mean.
Sample vs population standard deviation
Use the population standard deviation when your dataset includes every member of the group you care about — for example, the test scores of every student in one class. Use the sample standard deviation when your dataset is a sample meant to represent a larger group — for example, survey responses from 500 people chosen to represent a city.
The only difference in the formula is the denominator: population divides by N, sample divides by n−1. That n−1 is Bessel’s correction. Because the sample mean is estimated from the same data, a sample tends to underestimate the true population spread; dividing by n−1 instead of n corrects most of that bias, which is why the sample standard deviation is always a little larger than the population standard deviation for the same numbers.
Variance vs standard deviation
Variance is the average squared deviation from the mean; standard deviation is its square root. They carry the same information, but standard deviation is easier to interpret because it is in the same units as the data. If your data is in dollars, the variance is in “dollars squared” — hard to reason about — while the standard deviation is back in dollars. Variance is still useful: it adds neatly across independent sources of variation, which is why it appears throughout statistics even though standard deviation is what you usually report.
Standard error vs standard deviation
These are easy to confuse but answer different questions. Standard deviation describes the spread of the individual data values. Standard error describes the uncertainty in the estimated mean — how much the sample mean would bounce around if you repeated the sampling.
The formula is SE = s ÷ √n. Notice the √n: as your sample grows, the standard error shrinks even though the standard deviation does not. With an SD of 10 and n of 100, the standard error is 10 ÷ √100 = 1 — the mean is estimated ten times more precisely than any single value is spread. The confidence interval on this page is built from the standard error and a t critical value.
Coefficient of variation
The coefficient of variation expresses the standard deviation relative to the mean: CV = (SD ÷ mean) × 100. Because it is unit-free, it lets you compare variability across datasets with different averages or different units — daily returns of two stocks, the consistency of two machines, or the spread of prices at very different price points.
A caution: the CV is not meaningful when the mean is zero or close to zero, because dividing by a tiny mean produces a huge, unstable number. The calculator returns “undefined” in that case rather than a misleading figure.
Outliers and standard deviation
Outliers can have an outsized effect on the standard deviation because deviations are squared — a value far from the mean contributes its distance times itself. One extreme point can dominate the result. This calculator flags potential outliers two ways:
IQR method. Anything below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is flagged. This is robust because the quartiles ignore the extremes.
z-score method. Anything with |z| above your threshold (default 3) is flagged. This assumes roughly normal data and can be misleading for small or skewed datasets.
Do not delete outliers automatically. A flagged value may be a data-entry error, a rare but genuine value, or the most important signal in the data. Review each one in context. When outliers are present, the median, IQR, and median absolute deviation often describe the data more faithfully than the mean and standard deviation.
When standard deviation can be misleading
Small datasets. With only a handful of values, the standard deviation is itself uncertain and can swing widely with one more data point.
Skewed data. When the distribution has a long tail, the mean and SD are pulled toward it; the median and IQR are steadier.
Heavy-tailed data. Occasional extreme values inflate the SD far beyond the typical spread.
Outliers. A single mistake or rare value can dominate, as above.
Mixed populations. Combining two different groups can produce an SD that describes neither.
Non-numeric categories. Standard deviation has no meaning for labels or rankings without a true numeric scale.
Comparing across different means. Use the coefficient of variation, not the raw SD, to compare relative variability.
Empirical rule on non-normal data. The 68–95–99.7 percentages only hold for roughly bell-shaped distributions.
The empirical rule (68–95–99.7)
For data that is roughly bell-shaped (normal), the empirical rule says that about 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. It is a quick way to judge whether a value is typical or unusual.
It is a rule of thumb, not a law. For skewed or heavy-tailed data the actual percentages can be very different, so the calculator shows the theoretical share alongside the share your data actually has in each band — a large gap is a sign the data is not normal and the rule should not be applied.
Standard deviation measures how spread out values are around the mean. A small standard deviation means the values are tightly clustered near the average; a large one means they are widely spread.
How do I calculate standard deviation?
Find the mean, subtract it from each value, square the deviations, average them using N (population) or n−1 (sample), then take the square root. The deviation table on this page shows each step.
What is the difference between sample and population standard deviation?
Population SD divides the sum of squared deviations by N and is used for a full population. Sample SD divides by n−1 and is used when the data is a sample of a larger group. Sample SD is always slightly larger.
Why does sample standard deviation divide by n−1?
It applies Bessel’s correction. Because the sample mean is itself estimated from the data, dividing by n would systematically underestimate the population spread; dividing by n−1 corrects most of that bias.
What is variance?
Variance is the average squared distance from the mean. Standard deviation is the square root of variance, which brings the measure back into the same units as the data.
Why is standard deviation easier to interpret than variance?
Standard deviation is in the same units as the original data, while variance is in squared units. Saying values are “about 2.6 points from the mean” is clearer than “6.7 points-squared”.
What is standard error?
Standard error is the sample standard deviation divided by the square root of n (SE = s ÷ √n). It measures how precisely the sample mean estimates the true population mean — not the spread of the data itself.
What is a z-score?
A z-score is (x − mean) ÷ SD — the number of standard deviations a value lies above or below the mean. It lets you compare values from different datasets on a common scale.
How do outliers affect standard deviation?
Outliers can increase the standard deviation sharply because deviations are squared, giving far-away values extra weight. A single extreme value can change the SD substantially.
What is IQR?
The interquartile range is Q3 − Q1, the spread of the middle 50% of the data. It ignores the smallest and largest quarter of values, which makes it robust to outliers.
When should I use IQR or MAD instead of standard deviation?
Use the IQR or the median absolute deviation when the data is skewed, heavy-tailed, or affected by outliers. Both are far more stable than the standard deviation in those situations.
What is the coefficient of variation?
The coefficient of variation is the standard deviation divided by the mean, expressed as a percentage. It compares relative variability across datasets with different averages or units. It is undefined when the mean is 0.
Can standard deviation be negative?
No. Standard deviation is the square root of variance, and variance is an average of squared numbers, so it is never negative. The smallest possible value is 0.
What does a standard deviation of zero mean?
Every value in the dataset is identical, so there is no spread at all.
What is a good standard deviation?
There is no universal “good” standard deviation — it depends entirely on the context and the units. Compare it with the mean (or use the coefficient of variation) and with comparable datasets to judge whether the spread is large.
What is the empirical rule?
For roughly bell-shaped (normal) data, about 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. It is a rough guide, not a guarantee, and does not hold for skewed data.
Is standard deviation useful for finance?
It is widely used to measure the volatility of returns, but it does not capture every risk — it treats upside and downside moves the same and is sensitive to outliers and fat tails. Use it alongside other measures.
Can I paste data from Excel or Google Sheets?
Yes. The calculator accepts comma-separated, space-separated, new-line, and tab-separated values, so you can paste a column straight from a spreadsheet. Zeros and negative numbers are kept.
How many values can I enter?
Hundreds of values work smoothly; the deviation table shows the first several hundred rows and the statistics use them all. There is no fixed small cap — paste your whole column.
Is this calculator suitable for formal research?
It is educational. It shows every formula and step and validates its workbook against its own engine, but for formal research, coursework, or professional use you should confirm the required formula, quartile convention, outlier treatment, and rounding rules.
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Sources & methodology
The calculator applies the standard descriptive-statistics definitions — population and sample standard deviation and variance, the mean, median, mode, quartiles (type-7, matching Excel QUARTILE.INC), the IQR and its 1.5×IQR outlier fences, mean and median absolute deviation, the coefficient of variation, the standard error, z-scores, a Student-t confidence interval, and the empirical rule — as published in the references below. The exported workbook reproduces these as live, editable Excel formulas (STDEV, STDEVP, VAR, VARP, QUARTILE, AVEDEV, SKEW, T.INV.2T), validated cell-by-cell against this page's engine. Calculator Matters is an independent project, not affiliated with any source listed. Links open in a new tab.
Last reviewed: 15 June 2026. Formula and assumptions reviewed for accuracy. First published 13 June 2026.
Educational use disclaimer
This standard deviation calculator and its Excel workbook are for education and informational use only. They are not financial, medical, legal, academic, statistical-consulting, or professional advice. Results are mathematically exact to the precision you choose, but only as correct as the data you enter and the method your context requires — quartile conventions, the choice of sample vs population, outlier treatment, confidence level, and rounding all affect the answer. Standard deviation is sensitive to outliers and can mislead on skewed, heavy-tailed, or mixed data; the empirical rule applies only to roughly normal distributions. For graded coursework, research papers, regulatory reporting, or professional analysis, confirm the required formula, the treatment of outliers, the confidence level, and the rounding method with the appropriate source or professional.
Built and maintained by Calculator Matters, an independent calculator project. Formulas reviewed against the published sources above · Last reviewed 15 June 2026 · How we calculate · Found an error? [email protected]
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