Standard Deviation Calculator
Calculate mean, variance, and standard deviation for any data set.
What is Standard Deviation?
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (average) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.
Sample vs. Population Standard Deviation
Choosing the correct calculation method is critical for accuracy:
- Population Standard Deviation: Used when you have data for every member of the group you are studying (e.g., test scores for every student in a specific class). It divides the sum of squares by N.
- Sample Standard Deviation: Used when the data is a random subset of a larger population (e.g., surveying 100 people to estimate the behavior of a whole city). It uses Bessel's correction, dividing by n-1, to provide an unbiased estimate.
How to Calculate Standard Deviation Manually
Follow these five steps to find the standard deviation of any data set:
- Find the Mean: Add all the numbers together and divide by the count.
- Calculate Deviations: Subtract the mean from each individual number.
- Square the Deviations: Square each result from Step 2 (to make all numbers positive).
- Find the Variance: Sum all the squared values. If using Sample data, divide by (n-1). If using Population data, divide by (n).
- Take the Square Root: The square root of the variance is your standard deviation.
Real-World Example
Suppose a coffee shop tracks the number of lattes sold over 5 days: 45, 50, 55, 60, 40.
1. Mean = (45+50+55+60+40) / 5 = 50
2. Squared differences: (45-50)²=25, (50-50)²=0, (55-50)²=25, (60-50)²=100, (40-50)²=100
3. Sum of Squares = 25+0+25+100+100 = 250
4. Population Variance = 250 / 5 = 50
5. Population Standard Deviation = √50 ≈ 7.07