What is a standard deviation calculator?
A standard deviation calculator measures how spread out a set of numbers is around its average. Paste your data and the calculator returns the sample standard deviation and the population standard deviation, both variances, the mean, the sum of squares, the standard error and the coefficient of variation (RSD). It also builds a standard deviation calculator with steps table – every value, its deviation from the mean and the squared deviation – so you can check the working or copy it into an assignment.
What is standard deviation in simple words?
Standard deviation is the typical distance of the values from their mean. A small standard deviation means the numbers are bunched close to the average; a large one means they are scattered. Two classes can both average 70 marks, but if one has a standard deviation of 5 and the other 20, the first class is much more consistent. In statistics it is written σ (sigma) for a population and s for a sample.
How to use this calculator
- Type or paste your data set. Commas, spaces, tabs and new lines all work, so you can copy a column from Excel.
- Choose whether your main result should be the sample (n − 1) or population (N) standard deviation. Both are always shown.
- Read the result and open the step-by-step table to see how it was worked out.
Standard deviation formula
Population: σ = √( Σ(x − μ)² ÷ N ) · Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
- x = each value in the data set
- μ or x̄ = the mean (population mean or sample mean)
- N or n = the number of values
- Σ(x − mean)² = the sum of squared deviations, also called the sum of squares
The variance is the same formula without the square root: σ² = Σ(x − μ)² ÷ N, or s² = Σ(x − x̄)² ÷ (n − 1). So to calculate standard deviation from variance, simply take the square root – a variance of 25 gives a standard deviation of 5.
How to calculate standard deviation – step by step with example
Take the default data set: 10, 12, 23, 23, 16, 23, 21, 16.
- Find the mean. Sum = 144, n = 8, so the mean = 144 ÷ 8 = 18.
- Subtract the mean from each value: −8, −6, 5, 5, −2, 5, 3, −2.
- Square each deviation: 64, 36, 25, 25, 4, 25, 9, 4.
- Add the squares: Σ(x − x̄)² = 192.
- Divide by n − 1 = 7 for the sample variance (27.4286) or by N = 8 for the population variance (24).
- Take the square root: sample standard deviation s = √27.4286 = 5.2372; population standard deviation σ = √24 = 4.899.
Sample vs population standard deviation
| Population (σ) | Sample (s) | |
|---|---|---|
| Use when | Your data covers every member of the group | Your data is a sample used to estimate a larger group |
| Divide by | N | n − 1 (Bessel's correction) |
| Example | Marks of all 40 students in your class | Heights of 100 people to estimate India's average height |
| Excel function | STDEV.P | STDEV.S |
| Default data result | 4.899 | 5.2372 |
Why n − 1? A sample's values sit closer to their own mean than to the true population mean, so dividing by n would underestimate the spread. Dividing by n − 1 corrects that bias. For a large sample the two results are almost identical; for small samples the difference matters. If you are unsure, use the sample standard deviation – it is the default in most research, lab work and school statistics.
Interpreting standard deviation: the empirical rule
For data that follows a normal distribution (a bell curve), the empirical rule (68–95–99.7 rule) tells you where the values fall:
- About 68% of values lie within 1 standard deviation of the mean.
- About 95% lie within 2 standard deviations.
- About 99.7% lie within 3 standard deviations.
If exam scores have a mean of 70 and a standard deviation of 8, roughly two-thirds of students scored between 62 and 78, and almost all between 46 and 94. A single value's distance from the mean in SD units is its z-score: z = (x − mean) ÷ SD. A score of 86 has z = 2, which is higher than about 97.7% of scores.
Coefficient of variation (RSD) and standard error
The coefficient of variation, also called relative standard deviation (RSD), expresses the standard deviation as a percentage of the mean: CV = SD ÷ mean × 100. It lets you compare spread between data sets with different units or sizes, and it is widely used in chemistry, lab analysis and quality control, where an RSD under 2% often signals good precision. The standard error of the mean (SEM) = s ÷ √n shows how precisely the sample mean estimates the population mean – it shrinks as the sample size grows.
Standard deviation in mutual funds and stocks
In investing, standard deviation measures volatility – how much a fund's or stock's returns swing around their average. A mutual fund fact sheet in India may show an annualised standard deviation of 12% for a large-cap fund and 20% or more for a small-cap fund. Higher standard deviation means a wider range of likely returns, both up and down. To calculate it yourself, enter a series of monthly returns here, then multiply the monthly standard deviation by √12 (about 3.46) to annualise it.
Grouped data and frequency tables
For grouped data with a frequency table, the formula becomes σ = √( Σf(x − μ)² ÷ Σf ), where x is each value (or class midpoint) and f its frequency. With this calculator you can type each value as many times as its frequency – for example "5 5 5 7 7" for 5 with frequency 3 and 7 with frequency 2 – and the answer is exactly the same.
How to find standard deviation on a calculator or in Excel
- Excel / Google Sheets:
=STDEV.S(A1:A20)for a sample,=STDEV.P(A1:A20)for a population,=VAR.Sand=VAR.Pfor variance, and=AVERAGEfor the mean. - Scientific calculator (Casio fx-991): press MODE → STAT → 1-VAR, enter the data, then use the statistics menu: σx is the population SD and sx is the sample SD.
- TI-84: enter data in L1 with STAT → EDIT, then STAT → CALC → 1-Var Stats. Sx is the sample standard deviation and σx the population one.
To find the mean, median and mode of the same data, use the average calculator. For square roots of any variance by hand, the square root calculator helps.
Standard deviation calculator formula and symbol
The standard deviation calculator formula for a population is σ = √(Σ(x − μ)² ÷ N); for a sample it is s = √(Σ(x − x̄)² ÷ (n − 1)). The symbol σ (sigma) means population standard deviation, s means sample standard deviation, and μ or x̄ is the mean.
Standard deviation calculator using the mean: worked steps
For the default data 10, 12, 23, 23, 16, 23, 21, 16, the mean is 144 ÷ 8 = 18. The squared deviations from the mean add up to 192. Population variance = 192 ÷ 8 = 24, so σ = 4.899. Sample variance = 192 ÷ 7 = 27.43, so s = 5.237.
Standard deviation equation vs variance
Variance is the average squared distance from the mean; standard deviation is its square root. Variance is in squared units (cm², ₹²), while standard deviation is back in the original units, which is why it is easier to read.
Standard deviation calculator: how to use and input data
To use the standard deviation calculator, input your numbers separated by commas, spaces or new lines, pick sample or population and read the result. You can paste a column straight from Excel. No app is needed; it works in the browser on any phone.
Standard deviation calculator with x and y values
If you have paired x and y data, find the standard deviation of each list separately by entering the x values, then the y values. To measure how x and y move together you need covariance or correlation, which use both standard deviations.
Standard deviation calculator for grouped data with a frequency table
For grouped data with frequency, use each class midpoint x and its frequency f: mean = Σfx ÷ Σf and σ = √(Σf(x − mean)² ÷ Σf). For the standard deviation calculator with a frequency table, you can also type each value as many times as its frequency: 5 appearing 3 times becomes 5, 5, 5.
| Class | Midpoint x | f | fx |
|---|---|---|---|
| 0–10 | 5 | 2 | 10 |
| 10–20 | 15 | 5 | 75 |
| 20–30 | 25 | 3 | 75 |
Mean = 160 ÷ 10 = 16. Σf(x − 16)² = 2 × 121 + 5 × 1 + 3 × 81 = 490, so σ = √49 = 7.
Standard deviation calculator for a probability distribution
For a probability distribution, weight each value by its probability: μ = Σx·P(x) and σ = √(Σ(x − μ)²·P(x)). For a fair die, μ = 3.5 and the variance is 35 ÷ 12 ≈ 2.917, so the standard deviation with probability weights is about 1.708.
Standard deviation calculator and the normal distribution
In a normal distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. If exam marks have mean 60 and SD 10, about 95% of students score between 40 and 80.
Standard deviation calculator and z score
A z score tells you how many standard deviations a value is from the mean: z = (x − mean) ÷ SD. A mark of 75 with mean 60 and SD 10 has z = 1.5.
Standard deviation calculator graph
This standard deviation calculator shows numbers, not a graph. To see the spread, plot a histogram of your data or a bell curve with the mean in the middle and lines at ±1, ±2 and ±3 SD; Excel and Google Sheets draw both.
Standard deviation calculator in percentage
To express standard deviation in percentage terms, divide it by the mean: this is the coefficient of variation (relative standard deviation). An SD of 4 with a mean of 50 is 8%. For stock returns that are already percentages, the SD itself is quoted in percent.
Standard deviation calculator for stocks
Enter a stock's monthly returns, for example 2, −1.5, 3.2, 0.8, −2.4 (in %), to get its monthly volatility. Multiply a monthly SD by √12 to estimate yearly volatility.
Standard deviation in quality control, pediatrics and reloading
- Quality control: control charts put limits at the mean ± 3 standard deviations; a point outside suggests the process has changed.
- Pediatric growth: growth charts report a child's height or weight as an SD score (z score). WHO standards treat below −2 SD as low.
- Reloading: shooters enter chronograph velocities to get the SD of muzzle velocity; a lower SD means more consistent loads.
How to find standard deviation on a TI-84 calculator
On a TI-84, press STAT, choose 1:Edit and type your data into L1. Then press STAT, move to CALC and pick 1:1-Var Stats. Sx is the sample standard deviation and σx the population value. On a Casio, the SD button appears in STAT mode as σx or sx.
Frequently asked questions
How do you calculate standard deviation?
Find the mean, subtract it from each value, square the results, add them up, divide by n − 1 for a sample (or N for a population), and take the square root.
What is the difference between sample and population standard deviation?
Population standard deviation divides the sum of squares by N and is used when you have data for the whole group. Sample standard deviation divides by n − 1 and is used when your data is a sample from a larger group.
What is the difference between standard deviation and variance?
Variance is the average squared deviation from the mean. Standard deviation is the square root of the variance, so it is in the same units as the data and easier to interpret.
What is a good or low standard deviation?
It depends on the data. Compare the standard deviation with the mean using the coefficient of variation: a CV under about 10% usually means values are tightly grouped, while a much higher CV means they vary widely.
What does standard deviation mean in mutual funds?
It measures volatility: how far a fund’s returns typically move away from its average return. A higher standard deviation means higher risk and a wider range of possible returns.
How do I calculate standard deviation for grouped data with frequency?
Use σ = √(Σf(x − μ)² ÷ Σf). In this calculator, enter each value as many times as its frequency and the result is the same.
Is this standard deviation calculator free?
Yes. It is a free online standard deviation calculator with steps, no sign-up and no limit on the size of your data set.
What is the symbol for standard deviation?
σ (sigma) for a population and s for a sample.
How do I find standard deviation on a TI-84?
Enter data in L1 with STAT → Edit, then STAT → CALC → 1-Var Stats. Sx is the sample SD and σx the population SD.
Is there a standard deviation calculator app?
You do not need one: this page works in any mobile browser. On a handheld calculator, the standard deviation calculator button is σx or sx in STAT mode.
Can I use it as a standard deviation calculator for pediatric growth or quality control?
Yes, for the arithmetic. Enter measurements to get the mean and SD. Clinical z scores use reference tables such as WHO charts, and a standard deviation calculator for quality control work usually sets limits at mean ± 3 SD.
Does it work as a standard deviation calculator for reloading?
Yes. Paste your chronograph velocities and read the sample SD; this standard deviation calculator reloading use is common among shooters.
Can this be a standard deviation calculator using mean and variance together?
Yes. It shows the mean, variance and standard deviation for the same data, so it works as a standard deviation calculator and variance tool in one.
Is there a standard deviation calculator with frequency table input?
Type each value as many times as its frequency, or use the grouped-data formula above. That is how to use this standard deviation calculator with frequency data.
Is it a standard deviation calculator online free of charge?
Yes. It is free, with no sign-up.
Last updated 2026-09-18