Standard Deviation Calculator
Enter your dataset to calculate variance and standard deviation instantly.
Please enter a valid list of numbers separated by commas or spaces.
Standard Deviation:
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Variance:
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Mean (Average):
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Count (n):
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Sum:
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What is Standard Deviation?
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the data points tend to be close to the mean (average), while a high standard deviation indicates that the data points are spread out over a wider range of values.
The Difference Between Sample and Population
When calculating standard deviation, it is crucial to know if you are working with a Population or a Sample:
- Population: Use this if your dataset includes every member of the group you are studying (e.g., the heights of every student in a specific classroom).
- Sample: Use this if your dataset is a subset of a larger population (e.g., the heights of 10 random students from a whole school). The calculation for a sample uses "n-1" (Bessel's correction) to account for potential bias.
How to Calculate Standard Deviation Manually
To find the standard deviation, follow these five steps:
- Find the Mean: Add all numbers together and divide by the count.
- Subtract the Mean: Subtract the mean from each individual data point.
- Square the Results: Square each of the differences found in step 2.
- Find the Variance: For population, find the average of those squared differences. For sample, divide the sum of squared differences by (n – 1).
- Square Root: Take the square root of the variance to get the standard deviation.
Example Calculation
Dataset: 2, 4, 4, 4, 5, 5, 7, 9
1. Sum = 40 | Count = 8
2. Mean = 40 / 8 = 5
3. Squared Differences: (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
4. Sum of Squares = 9+1+1+1+0+0+4+16 = 32
5. Sample Variance = 32 / (8-1) = 4.57
6. Sample SD = √4.57 = 2.138
2. Mean = 40 / 8 = 5
3. Squared Differences: (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
4. Sum of Squares = 9+1+1+1+0+0+4+16 = 32
5. Sample Variance = 32 / (8-1) = 4.57
6. Sample SD = √4.57 = 2.138