šŸ“Š

Standard Deviation Calculator

Calculate standard deviation, variance, mean, and range for sample or population data. Measure data dispersion and variability with step-by-step statistical calculations.

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Enter Your Data

Enter at least 2 numbers separated by commas, spaces, or line breaks
Use sample SD when your data represents a subset of a larger group (most common)

Standard Deviation Results

Sample Standard Deviation (s)
5.2372
Sample Variance (sĀ²)
27.4286

šŸ“Š Interpretation

Sample standard deviation: 5.2372. This estimates the spread of the population based on your sample data with mean 18.00.

Descriptive Statistics

Mean (Average)
18
Count
8
Minimum
10
Maximum
23
Range
13
Sum
144
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Both Standard Deviations

For reference, here are both calculations:

Sample Standard Deviation (s)
5.2372
Population Standard Deviation (Ļƒ)
4.899

šŸ“ Formulas Used

Sample SD: s = āˆš[Ī£(x - xĢ„)Ā² / (n-1)]

Population SD: Ļƒ = āˆš[Ī£(x - Ī¼)Ā² / n]

Where x is each value, xĢ„/Ī¼ is the mean, n is count, and Ī£ means sum

68-95-99.7 Rule

For normally distributed data, here's what your standard deviation tells you:

68% of data falls within:
12.76 to 23.24
(Mean Ā± 1 SD)
95% of data falls within:
7.53 to 28.47
(Mean Ā± 2 SD)
99.7% of data falls within:
2.29 to 33.71
(Mean Ā± 3 SD)

Understanding Standard Deviation

Standard deviation is a measure of how spread out numbers are from their average (mean). It tells you whether your data points are clustered close to the mean or scattered far from it.

How Standard Deviation is Calculated

  1. Calculate the mean (average) of all numbers
  2. Subtract the mean from each number (these are deviations)
  3. Square each deviation
  4. Calculate the average of squared deviations (this is variance)
  5. Take the square root of variance (this is standard deviation)

Sample vs Population

  • Population SD (Ļƒ): Used when you have data for EVERYONE in a group. Divide by N. Example: grades for all students in your class.
  • Sample SD (s): Used when you have data for SOME people from a larger group. Divide by N-1 (Bessel's correction). Example: heights of 30 people representing a city.
  • Why N-1? Bessel's correction gives a better estimate when inferring about a larger population from a sample.

Interpreting Standard Deviation

  • Low SD: Data points are close to the mean (consistent, less variability)
  • High SD: Data points are spread out from the mean (more variability, less consistent)
  • SD = 0: All values are identical
  • Units: SD is in the same units as your data (unlike variance which is squared)

The 68-95-99.7 Rule (Empirical Rule)

For normally distributed data:

  • ~68% of data falls within 1 standard deviation of the mean
  • ~95% of data falls within 2 standard deviations of the mean
  • ~99.7% of data falls within 3 standard deviations of the mean

Practical Applications

  • Finance: Measure investment risk and volatility
  • Quality Control: Monitor manufacturing consistency
  • Education: Analyze test score distributions
  • Healthcare: Evaluate treatment effectiveness
  • Research: Report data variability in studies
  • Weather: Describe temperature variation

Example

Consider two classes with the same mean test score of 75:

  • Class A scores: 73, 74, 75, 76, 77 (SD ā‰ˆ 1.6) - Very consistent
  • Class B scores: 50, 65, 75, 85, 100 (SD ā‰ˆ 18.7) - Highly variable

Both classes have the same average, but Class A is much more consistent, while Class B has students with widely varying abilities.

Frequently Asked Questions

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