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Standard Deviation Calculator

Calculate mean, variance, standard deviation and more for any data set — with full step-by-step working. Perfect for students, researchers and data analysts!

📊 Enter Your Data Set

Enter numbers separated by commas, spaces or new lines

Separate numbers with commas, spaces or new lines. Up to 1000 values supported.
Try a sample data set:
Classic Example
Linear Spread
No Variation
1 to 10
Exam Scores

📊 Your Statistics

📊 Sample Standard Deviation
2.0
Standard Deviation (s)
Mean
5.000
Median
4.500
Variance
4.000
Count (n)
8
Min
2
Max
9
Range
7
Sum
40
📊 Data Spread Analysis
SD = 2.0. About 68% of your data falls within 3.0 to 7.0 (mean ± 1 SD).
📋 Step-by-Step Working
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Standard Deviation Calculator — Complete Statistics Guide

Standard deviation is the most important measure of data spread in statistics. It tells you how much variation or dispersion exists in a data set. A low standard deviation means data points cluster closely around the mean. A high standard deviation means data is spread widely. Understanding standard deviation is essential for anyone working with data — from GCSE and A-Level students to university researchers, finance professionals and data scientists. Use our Average Calculator to find the mean of your data first, then use this tool for the full statistical breakdown!

How to Calculate Standard Deviation — Step by Step

The standard deviation formula has five clear steps. Our calculator shows every step so you can learn the process, check your working and understand exactly how the result is derived.

Standard Deviation Formula: Population SD: σ = √[ Σ(x - μ)² / N ] Sample SD: s = √[ Σ(x - x̄)² / (n-1) ] Where: x = each data value μ or x̄ = mean of all values N = population size n = sample size n-1 = Bessel's correction for sample data Step-by-step example: data set {2, 4, 4, 4, 5, 5, 7, 9} Step 1 - Find mean: (2+4+4+4+5+5+7+9) / 8 = 40/8 = 5 Step 2 - Subtract mean and square each: (2-5)² = 9 (4-5)² = 1 (4-5)² = 1 (4-5)² = 1 (5-5)² = 0 (5-5)² = 0 (7-5)² = 4 (9-5)² = 16 Step 3 - Sum of squared differences: 9+1+1+1+0+0+4+16 = 32 Step 4 - Variance (population): 32 / 8 = 4.0 Variance (sample): 32 / 7 = 4.571 Step 5 - Standard Deviation: √4 = 2.0 (population) √4.571 = 2.138 (sample)

Population vs Sample Standard Deviation — Which to Use?

This is the most common question in statistics. The key is whether your data represents the entire population or just a sample from it. For most research, surveys and classroom data sets you should use sample standard deviation. The sample formula uses n-1 in the denominator (called Bessel's correction) which corrects for the bias that occurs when estimating population variance from a sample. Use our Percentage Calculator to express variance as a percentage of the mean for easy comparison!

Situation Use Formula Example
Survey of 100 people from a citySample SD (s)n-1 denominatorSample of population
Exam scores for the whole classPopulation SD (σ)n denominatorAll data included
5 product samples from productionSample SD (s)n-1 denominatorSample of output
Height of all players on a teamPopulation SD (σ)n denominatorComplete group data

The Empirical Rule — 68-95-99.7

In a normal distribution the empirical rule describes exactly how data distributes around the mean. This rule is used in exam grading, quality control, finance and scientific research to identify typical and unusual values. It is also known as the three-sigma rule.

Range Data Included Example (Mean=70, SD=10) Interpretation
Mean ± 1σ68.27%60 to 80Typical range
Mean ± 2σ95.45%50 to 90Almost all values
Mean ± 3σ99.73%40 to 100Virtually all values

Real-World Applications of Standard Deviation

Statistics You Get From This Calculator

Beyond standard deviation our calculator computes a complete statistical summary of your data including mean, median, variance, range, sum, minimum, maximum and count. This gives you a full picture of your data distribution. The mean combined with standard deviation tells you the centre and spread. The median compared to the mean reveals skewness. The range shows absolute extremes. Together these statistics tell the complete story of any data set. For further statistical analysis try our Fraction Calculator for converting fractional data values!

📚 Note: This calculator handles up to 1000 data values. Results assume your data is numerical and correctly entered. For very large data sets with millions of values use dedicated statistical software such as R, Python (NumPy) or Excel. Standard deviation results are correct to 6 decimal places.

Frequently Asked Questions

What is standard deviation? +
Standard deviation measures how spread out numbers are in a data set. A low standard deviation means values cluster closely around the mean (average). A high standard deviation means values are spread widely. It equals the square root of the variance and is expressed in the same units as the original data making it easy to interpret. It is the most commonly used measure of statistical dispersion in science, finance and everyday data analysis.
What is the difference between population and sample standard deviation? +
Population SD uses N (total count) in the denominator and represents the true spread of an entire population. Sample SD uses n-1 (Bessel's correction) and is used when your data is a sample from a larger population. The n-1 correction prevents underestimating the true population variance. In practice use sample SD for surveys, experiments and research data. Use population SD only when you have data for every member of the group being studied.
How do you calculate standard deviation step by step? +
Step 1 — Find the mean: add all values and divide by count. Step 2 — For each value subtract the mean and square the result. Step 3 — Add all the squared differences together. Step 4 — Divide by n (population) or n-1 (sample) to get the variance. Step 5 — Take the square root of the variance. Our calculator shows every single step with your actual data making it a powerful learning tool for statistics students.
What is variance and how is it related to standard deviation? +
Variance is the average of the squared differences from the mean. Standard deviation is the square root of variance. Variance is harder to interpret because it is in squared units (e.g. kg² if your data is in kg). Standard deviation converts back to the original units making it more practical. For example if heights in cm have variance of 100 the standard deviation is 10 cm — a much more intuitive measure of spread.
What does the empirical rule (68-95-99.7) mean? +
In a normal distribution 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations and 99.7% within 3 standard deviations. This allows you to identify outliers: any value more than 2 SDs from the mean is unusual (outside the top 5%) and any value more than 3 SDs away is extremely rare (outside 99.7% of data). This rule underlies IQ scoring, exam bell curves and financial risk models.
What is a good or bad standard deviation? +
There is no universal good or bad standard deviation — it depends entirely on context. In manufacturing a low SD indicates consistent product quality. In finance a higher SD means higher volatility (more risk and potential reward). In test scores a very low SD means uniform performance while high SD shows diverse ability levels. The coefficient of variation (SD divided by mean multiplied by 100%) allows comparison across data sets with different scales.
What is the difference between standard deviation and standard error? +
Standard deviation describes spread within a data set. Standard error describes how precisely the sample mean estimates the true population mean. Standard error equals standard deviation divided by the square root of n. A smaller sample has larger standard error (less precise estimate). Standard error is used in confidence intervals and hypothesis testing to quantify uncertainty in statistical estimates rather than describing the raw data spread.

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