Standard Deviation Calculator

Standard Deviation is evaluated from Data Values and Calculation Type. The calculation reports Mean, Variance and Standard Deviation.

Results

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About the Standard Deviation Calculator

Standard Deviation is treated here as a quantitative relation between Data Values and Calculation Type and Mean, Variance, Standard Deviation and Coefficient of Variation.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
sigma = sqrt(Sigma(xᵢ - mu)^2 / N)
s = sqrt(Sigma(xᵢ - x̄)^2 / (n - 1))
CV = (s / x̄) x 100%

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

sigma = sqrt(Sigma(xᵢ - mu)^2 / N)
s = sqrt(Sigma(xᵢ - x̄)^2 / (n - 1))
CV = (s / x̄) x 100%

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Classic Textbook Dataset

Inputs

data: 2, 4, 4, 4, 5, 5, 7, 9 type: population
Mean: 5. Variance: 4. Standard Deviation: 2. Coefficient of Variation: 40%. Count: 8

With Data Values = 2, 4, 4, 4, 5, 5, 7, 9 and Calculation Type = population as the stated inputs, the result is Mean = 5, Variance = 4 and Standard Deviation = 2. Each value corresponds to the declared output fields.

Example 2: SAT Scores — Sample SD

Inputs

data: 1080, 1150, 1200, 1220, 1250, 1280, 1300, 1350, 1400, 1450 type: sample
Mean: 1,268. Variance: 12,773.333333. Standard Deviation: 113.019172. Coefficient of Variation: 8.9132%. Count: 10

With Data Values = 1080, 1150, 1200, 1220, 1250, 1280, 1300, 1350, 1400, 1450 and Calculation Type = sample as the stated inputs, the result is Mean = 1,268, Variance = 12,773.333333 and Standard Deviation = 113.019172. Each value corresponds to the declared output fields.

Example 3: Investment Returns — Risk Measurement

Inputs

data: 12.5, -3.2, 8.7, 22.1, 4.3, -8.5, 15.6, 6.8 type: sample
Mean: 7.2875. Variance: 98.009821. Standard Deviation: 9.899991. Coefficient of Variation: 135.8489%. Count: 8

With Data Values = 12.5, -3.2, 8.7, 22.1, 4.3, -8.5, 15.6, 6.8 and Calculation Type = sample as the stated inputs, the result is Mean = 7.2875, Variance = 98.009821 and Standard Deviation = 9.899991. Each value corresponds to the declared output fields.

Example 4: Manufacturing Quality Control

Inputs

data: 10.01, 9.98, 10.02, 10.00, 9.99, 10.03, 9.97, 10.01, 10.00, 9.99 type: population
Mean: 10. Variance: 0.0003. Standard Deviation: 0.017321. Coefficient of Variation: 0.1732%. Count: 10

With Data Values = 10.01, 9.98, 10.02, 10.00, 9.99, 10.03, 9.97, 10.01, 10.00, 9.99 and Calculation Type = population as the stated inputs, the result is Mean = 10, Variance = 0.0003 and Standard Deviation = 0.017321. Each value corresponds to the declared output fields.

Common Use Cases

  • Measure data spread in a test score dataset
  • Calculate process variability in manufacturing (Six Sigma)
  • Assess investment risk via return standard deviation
  • Compute stock price volatility