ganak.appStandard Deviation Calculator
Calculate population and sample standard deviation, variance, mean, median, quartiles, and Z-scores for any dataset. Enter numbers separated by commas or spaces. Includes step-by-step formulas, distribution histogram, and coefficient of variation. Free online statistics calculator.
Supports commas, spaces, tabs, or new lines. Paste from spreadsheets works too.
Statistical Summary
Population and sample statistics for your dataset
Step-by-Step Calculation
How the population standard deviation is computed
Formula
σ = √( Σ(xᵢ − μ)² / N )
Step 1: Calculate the mean (μ)
μ = (2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) / 8 = 5.5000
Step 2: Find each deviation from the mean and square it
(2 − 5.50)² = (-3.50)² = 12.2500
(3 − 5.50)² = (-2.50)² = 6.2500
(4 − 5.50)² = (-1.50)² = 2.2500
(5 − 5.50)² = (-0.50)² = 0.2500
(6 − 5.50)² = (0.50)² = 0.2500
(7 − 5.50)² = (1.50)² = 2.2500
... 2 more values
Step 3: Sum the squared deviations and divide by N (8)
σ² = 5.2500
Step 4: Take the square root
σ = √5.2500 = 2.2913
Sample std dev: Divide by N−1 = 7 instead of N → s = 2.4495
Value Distribution
Frequency distribution of your dataset
Z-Scores
How many standard deviations each value is from the mean
| Value | Deviation | Z-Score |
|---|---|---|
| 4 | -1.5000 | -0.6547 |
| 8 | 2.5000 | +1.0911 |
| 6 | 0.5000 | +0.2182 |
| 5 | -0.5000 | -0.2182 |
| 3 | -2.5000 | -1.0911 |
| 7 | 1.5000 | +0.6547 |
| 2 | -3.5000 | -1.5275 |
| 9 | 3.5000 | +1.5275 |
Values with |Z| > 2 are highlighted as potential outliers.
What Is Standard Deviation?
Understanding data spread and variability
Standard deviation measures how spread out values are from the mean (average) of a dataset. A low standard deviation means data points cluster close to the mean, while a high standard deviation means data points are spread over a wider range. It is the most commonly used measure of variability in statistics.
Population Standard Deviation Formula
σ = √( Σ(xᵢ − μ)² / N )
Where σ (sigma) is the standard deviation, xᵢ represents each value, μ (mu) is the population mean, and N is the number of values. The formula squares each deviation from the mean, averages them, and takes the square root.
Population vs Sample Standard Deviation
When to use N vs N−1 (Bessel's correction)
Population Standard Deviation (σ)
Divides by N (the total number of values). Use this when your data represents the entire population — for example, test scores of every student in a class, or all daily temperatures in a month.
σ = √( Σ(xᵢ − μ)² / N )
Sample Standard Deviation (s)
Divides by N−1 (Bessel's correction). Use this when your data is a sample from a larger population — for example, a survey of 100 people from a city, or a batch of products from a production line. Dividing by N−1 corrects for the bias in estimating the population variance from a sample.
s = √( Σ(xᵢ − x̄)² / (N − 1) )
Related Statistical Measures
Variance, Z-scores, coefficient of variation, and quartiles
Variance (σ²)
The square of the standard deviation. Variance measures the average squared deviation from the mean. Standard deviation is its square root, returning the measure to the original unit of the data.
Z-Score
Z = (x − μ) / σ. Tells you how many standard deviations a value is from the mean. A Z-score of +2 means the value is 2 standard deviations above the mean. Values with |Z| > 2 are often considered potential outliers.
Coefficient of Variation
CV = (σ / |μ|) × 100%. A relative measure of variability that allows comparison between datasets with different units or scales. A CV of 20% means the standard deviation is 20% of the mean.
Quartiles & IQR
Q1 (25th percentile) and Q3 (75th percentile) divide the sorted data. The Interquartile Range (IQR = Q3 − Q1) measures the spread of the middle 50% and is robust to outliers, unlike standard deviation.
Real-World Example
Worked example: Test scores 85, 90, 78, 92, 88
A teacher wants to understand the spread of test scores: 85, 90, 78, 92, 88.
Step-by-Step
- Mean: (85+90+78+92+88) / 5 = 86.6
- Deviations: −1.6, 3.4, −8.6, 5.4, 1.4
- Squared: 2.56, 11.56, 73.96, 29.16, 1.96
- Sum of squares: 119.2
- Pop. variance: 119.2 / 5 = 23.84
Results
- Population σ: √23.84 ≈ 4.8827
- Sample s: √(119.2/4) ≈ 5.4589
- Mean: 86.6
- Range: 92 − 78 = 14
- CV: 4.88 / 86.6 × 100 ≈ 5.64%
The low CV (5.64%) tells us these scores are fairly consistent — most students scored within about 5 points of the average. If you were sampling from a larger class, you would use the sample standard deviation (s ≈ 5.46).
Frequently Asked Questions
Common questions about standard deviation, variance, and statistical measures