Standard Deviation Calculator
Calculate standard deviation, variance, mean, median, and more statistics
Enter Your Data
Parsed Data (8 values)
10, 12, 16, 16, 21, 23, 23, 23
Results
📊 Quick Answer
For your 8 data points, the sample standard deviation is 5.2372. The mean is 18, variance is 27.4286, and the data ranges from 10 to 23.
Dr. Snezana Lawrence
Mathematical Historian
15+ years experience
PhD from Yale University. Published mathematical historian ensuring precision in all calculations.
Education
PhD in Mathematical History - Yale University
📑 Table of Contents
🤔 What is Standard Deviation?
Standard deviation (σ) is a measure of how spread out numbers are in a dataset. It tells you, on average, how far each data point is from the mean (average) of the data set.
A low standard deviation means that data points tend to be close to the mean, while a high standard deviation means the data is spread out over a wider range.
Low Standard Deviation
Data: 48, 49, 50, 51, 52
Mean: 50, σ ≈ 1.58
Data points are clustered together
High Standard Deviation
Data: 10, 30, 50, 70, 90
Mean: 50, σ ≈ 31.62
Data points are spread out widely
📐 Standard Deviation Formula
Population Standard Deviation
σ = √(Σ(xᵢ - μ)² / N)
Sample Standard Deviation
s = √(Σ(xᵢ - x̄)² / (n-1))
Where:
- σ or s = Standard deviation
- xᵢ = Each value in the dataset
- μ or x̄ = Mean (average) of the values
- N or n = Number of values
- Σ = Sum of all values
📊 Sample vs. Population Standard Deviation
Population (σ)
Use when you have data for the entire population.
- • Divides by N (total count)
- • Example: Test scores for ALL students in a class
- • Symbol: σ (sigma)
Sample (s)
Use when you have a sample from a larger population.
- • Divides by n-1 (Bessel's correction)
- • Example: Survey of 100 customers from 10,000
- • Symbol: s
Why n-1? The sample standard deviation uses n-1 (Bessel's correction) because a sample tends to underestimate the true population variability. Using n-1 gives an unbiased estimate.
📝 Step-by-Step Calculation
Here's how to calculate standard deviation manually:
Calculate the Mean
Add all numbers and divide by the count. For [2, 4, 4, 4, 5, 5, 7, 9]: Mean = 40/8 = 5
Find the Deviations
Subtract the mean from each value: (2-5), (4-5), (4-5), (4-5), (5-5), (5-5), (7-5), (9-5) = -3, -1, -1, -1, 0, 0, 2, 4
Square Each Deviation
9, 1, 1, 1, 0, 0, 4, 16
Calculate Variance
Sum of squares = 32. For population: 32/8 = 4. For sample: 32/7 ≈ 4.57
Take the Square Root
Population σ = √4 = 2. Sample s = √4.57 ≈ 2.14
📈 Interpreting Standard Deviation
The Empirical Rule (68-95-99.7)
For normally distributed data:
Within ±1 standard deviation
Within ±2 standard deviations
Within ±3 standard deviations
Practical Applications
- Quality Control: Identifying manufacturing defects (Six Sigma)
- Finance: Measuring investment risk and volatility
- Education: Grading on a curve, test score analysis
- Science: Experimental error reporting, data reliability
❓ Frequently Asked Questions
What is standard deviation?
Standard deviation is a measure of how spread out numbers are from the mean (average). A low standard deviation means data points are close to the mean, while a high standard deviation indicates data is spread out over a wider range.
What's the difference between sample and population standard deviation?
Sample standard deviation uses n-1 in the denominator (Bessel's correction) to provide an unbiased estimate when working with a sample from a larger population. Population standard deviation uses n and is used when you have data for the entire population.
How do I calculate standard deviation?
To calculate standard deviation: 1) Find the mean of your data, 2) Subtract the mean from each data point and square the result, 3) Find the average of these squared differences (variance), 4) Take the square root of the variance.
What is variance?
Variance is the average of squared differences from the mean. It's the square of the standard deviation (σ²). Variance is useful in statistical calculations but is harder to interpret because it's in squared units.
When should I use standard deviation vs. range?
Range (max - min) is simpler but only considers two values. Standard deviation uses all data points, making it more reliable and less affected by outliers. Use standard deviation for more accurate analysis of data spread.
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Dr. Snezana Lawrence
Mathematical Historian | PhD from Yale
Dr. Lawrence is a published mathematical historian with a PhD from Yale University. She ensures mathematical precision and accuracy in all our calculations, conversions, and academic score calculators. Her expertise spans computational mathematics and educational assessment.
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