Standard Deviation Calculator (Free)
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A standard deviation calculator measures how far values typically sit from the mean. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squared distances from it add up to 32. Divide by n − 1 = 7 and take the square root: the sample standard deviation is s ≈ 2.14. Divide by N = 8 instead and the population value is σ = 2.
| Within | Normal curve | Your data |
|---|---|---|
| x̄ ± 1s2.8619 to 7.1381 | 68% | 6 of 8 (75%) |
| x̄ ± 2s0.7238 to 9.2762 | 95% | 8 of 8 (100%) |
| x̄ ± 3s−1.4143 to 11.4143 | 99.7% | 8 of 8 (100%) |
Show steps and the deviation table
- Count the values: n = 8.
- Find the mean: x̄ = Σx ÷ n = 40 ÷ 8 = 5.
- Subtract the mean from each value and square the result (table below). Squaring makes every distance positive.
- Add the squared deviations: SS = Σ(x − x̄)² = 32.
- Divide by n − 1 = 7 (a sample): s² = 32 ÷ 7 = 4.5714. This is the sample variance. Dividing by n − 1 instead of n corrects for a sample usually being less spread out than the whole population (Bessel’s correction).
- Take the square root: s = √4.5714 = 2.1381.
- For comparison, a population divides by N = 8: σ² = 4 and σ = 2.
- Standard error of the mean = s ÷ √n = 2.1381 ÷ 2.8284 = 0.7559. Coefficient of variation = s ÷ |x̄| × 100% = 42.76%.
| # | x | x − x̄ | (x − x̄)² | z-score |
|---|---|---|---|---|
| 1 | 2 | −3 | 9 | −1.4 |
| 2 | 4 | −1 | 1 | −0.47 |
| 3 | 4 | −1 | 1 | −0.47 |
| 4 | 4 | −1 | 1 | −0.47 |
| 5 | 5 | 0 | 0 | 0 |
| 6 | 5 | 0 | 0 | 0 |
| 7 | 7 | 2 | 4 | 0.94 |
| 8 | 9 | 4 | 16 | 1.87 |
| Σ | 40 | 0 | 32 |
How to use the standard deviation calculator
- Choose how your data looks: a Data list of numbers, a Frequency table (each value and how often it appears), or Grouped data in class intervals such as 10–20.
- Choose Sample or Population (see the next section if you are not sure).
- Type or paste the numbers. Results update as you type, or press Calculate.
- Read the standard deviation, then the mean, variance, standard error and coefficient of variation. Open the steps to see the deviation table with a z-score for every value.
Below the numbers, the empirical-rule chart shows your data against a bell curve, and the outlier check tests every value two ways.
Population vs. sample standard deviation
Use the population standard deviation (σ) only when your numbers include every member of the group you want to describe, such as the test scores of all 25 students in one class when that class is all you care about. Divide by N.
Use the sample standard deviation (s) when your numbers are a part of a larger group and you want to describe that larger group, such as 30 students surveyed from a whole school. Divide by n − 1. This is the usual choice in homework, science labs and research, and it is what most calculators label Sx.
Dividing by n − 1 (Bessel’s correction) makes s slightly larger than σ. A sample tends to cluster around its own mean, so it understates the true spread a little; the correction makes up for that. With large data sets the two values are nearly equal.
Standard deviation formulas
| Data type | Sample | Population |
|---|---|---|
| Data list | s = √[Σ(x − x̄)² ÷ (n − 1)] | σ = √[Σ(x − μ)² ÷ N] |
| Frequency table | s = √[Σf(x − x̄)² ÷ (n − 1)], n = Σf | σ = √[Σf(x − μ)² ÷ N] |
| Grouped data | same, with x replaced by each class midpoint m | same, with midpoints |
Here x is a value, x̄ (or μ) is the mean, f is a frequency and Σ means “add them all up”. The part under the square root is the variance, so the standard deviation is the square root of the variance.
How to find standard deviation by hand
Example 1: a data list
Data: 2, 4, 4, 4, 5, 5, 7, 9.
- Mean: 40 ÷ 8 = 5.
- Deviations from the mean: −3, −1, −1, −1, 0, 0, 2, 4.
- Squares: 9, 1, 1, 1, 0, 0, 4, 16. Their sum is 32.
- Sample: s² = 32 ÷ 7 ≈ 4.5714, so s ≈ 2.1381. Population: σ² = 32 ÷ 8 = 4, so σ = 2.
Example 2: a frequency table
Values 1, 2, 3, 4, 5 appear 3, 5, 8, 4 and 2 times. Then n = 22 and Σfx = 63, so the mean is 63 ÷ 22 ≈ 2.8636. Multiply each squared deviation by its frequency and add: Σf(x − x̄)² ≈ 28.5909. The sample variance is 28.5909 ÷ 21 ≈ 1.3615, so s ≈ 1.1668.
Example 3: grouped data
Classes 0–10, 10–20, 20–30, 30–40 and 40–50 have frequencies 4, 9, 15, 8 and 4. Use the midpoints 5, 15, 25, 35 and 45. Then n = 40, Σfm = 990 and the mean is 24.75. The sum of f(m − x̄)² is 4,897.5, so s = √(4,897.5 ÷ 39) ≈ 11.2061. This is an estimate, because the exact values inside each class are unknown.
The empirical rule and z-scores
For bell-shaped data, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. If test scores have mean 70 and standard deviation 8, roughly 68% fall between 62 and 78, and 95% between 54 and 86.
A z-score says how many standard deviations a value is from the mean: z = (x − mean) ÷ standard deviation. A score of 86 in that test has z = (86 − 70) ÷ 8 = 2, so it is 2 standard deviations above average. For one value at a time, use the z-score calculator.
Standard error, coefficient of variation and outliers
The standard error of the mean, s ÷ √n, describes how much a sample mean would vary from sample to sample. For Example 1 it is 2.1381 ÷ √8 ≈ 0.7559. It shrinks as the sample grows; the standard deviation does not.
The coefficient of variation (CV) is the standard deviation as a percent of the mean: 2.1381 ÷ 5 ≈ 42.76%. It lets you compare spread between data sets with different units or sizes.
Outliers inflate the standard deviation. In 12, 14, 13, 15, 14, 13, 12, 41, the quartiles are Q1 = 12.5 and Q3 = 14.5, so IQR = 2 and the upper fence is 14.5 + 1.5 × 2 = 17.5: 41 is an outlier. Removing it changes s from about 9.85 to about 1.11. The calculator also flags any value more than 3 standard deviations from the mean. Always check whether an outlier is a typing error before you remove it.
Common mistakes with standard deviation
| Mistake | Fix |
|---|---|
| Using N for a sample | Divide by n − 1 unless you have the whole population. |
| Forgetting the square root | Variance is in squared units; the standard deviation is its square root. |
| Adding deviations without squaring | Deviations always add to 0; square them first. |
| Rounding the mean early | Keep full precision until the last step. |
| Ignoring frequencies | Multiply each squared deviation by how often the value appears. |
Checking homework with the calculator
Estimate first: the standard deviation can never be negative, and it is usually less than half the range (for 2 to 9, less than 3.5). Work the problem by hand using the table method above, then compare each column with the deviation table in the steps. The first row that differs shows where the mistake is. On tests where calculators are allowed, a TI-84 gives the same numbers: Sx is the sample value and σx is the population value.
Standard deviation practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Mean 4; squared deviations 4, 0, 4 add to 8; σ² = 8 ÷ 3 ≈ 2.67; σ = √2.67 ≈ 1.63. The value 2 is the sample s (8 ÷ 2 = 4, √4 = 2).
Mean 6; squared deviations 9, 1, 1, 9 add to 20; s² = 20 ÷ 3 ≈ 6.67; s ≈ 2.58. Dividing by 4 instead gives the population value 2.24.
No value differs from the mean of 12, so every deviation is 0 and the standard deviation is 0.
The 40 students are part of the larger group you want to describe, so this is a sample: divide by n − 1.
65 and 85 are 2 standard deviations below and above the mean (75 ± 2 × 5). The empirical rule puts about 95% of values there.
z = (88 − 70) ÷ 6 = 18 ÷ 6 = 3, so the value is 3 standard deviations above the mean.
The data are 2, 3, 3, 4. Mean 3; squared deviations 1, 0, 0, 1 add to 2; σ² = 2 ÷ 4 = 0.5; σ ≈ 0.71. Ignoring frequencies (2, 3, 4) gives 0.82.
Quiz complete
Frequently asked questions
Is standard deviation calculated with the mean?
Yes. Every deviation is measured from the mean, so you find the mean first. Two data sets can share a mean and still have very different spreads: 4, 5, 6 and 1, 5, 9 both have mean 5, but their sample standard deviations are 1 and 4.
How do you find standard deviation on a TI-84?
Enter the data in L1 (STAT, Edit), then choose STAT, CALC, 1-Var Stats. Sx is the sample standard deviation and σx is the population standard deviation. For a frequency table, put frequencies in L2 and use 1-Var Stats L1, L2.
What is the difference between standard deviation and variance?
Variance is the average squared distance from the mean; standard deviation is its square root. The standard deviation is in the same units as the data, which makes it easier to interpret.
What is the difference between standard deviation and standard error?
Standard deviation describes the spread of individual values. Standard error describes how precisely a sample mean estimates the population mean. It equals s ÷ √n, so it gets smaller with more data.
Can the standard deviation be 0?
Yes, but only when every value is the same. Then nothing varies from the mean. It can never be negative. For the mean, median and mode themselves, use the mean, median and mode calculator.
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