BravoCalc

Standard Deviation Calculator

Calculate statistical measures including mean, median, mode, range, variance, and standard deviation.

Standard Deviation Calculator
Calculate statistical measures for your data set
Results
Statistical measures for your data set

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Sum

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Mean

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Median

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Mode

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Range

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Minimum

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Maximum

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Variance (sample)

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Standard Deviation (sample)

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Understanding Statistical Measures

Mean

The arithmetic average of a data set, calculated by summing all values and dividing by the number of values.

Formula: μ = (Σx) / n

Median

The middle value in a sorted data set. If there is an even number of values, the median is the average of the two middle values.

Mode

The value(s) that appear most frequently in a data set. A data set may have one mode, multiple modes, or no mode.

Range

The difference between the maximum and minimum values in a data set.

Formula: Range = Max - Min

Variance

A measure of how spread out the values are from the mean.

Population Formula: σ² = Σ(x - μ)² / N
Sample Formula: s² = Σ(x - x̄)² / (n-1)

Standard Deviation

The square root of the variance, representing the average distance from the mean.

Population Formula: σ = √σ²
Sample Formula: s = √s²

Population vs. Sample

When calculating variance and standard deviation, it's important to know whether your data represents an entire population or just a sample:

  • Population: Includes all members of a specified group. When calculating variance for a population, divide by N (the total number of values).
  • Sample: Includes only a subset of a population. When calculating variance for a sample, divide by (n-1) to get an unbiased estimator of the population variance.

Interpreting Standard Deviation

The standard deviation helps you understand how spread out your data is:

  • A low standard deviation indicates that the values tend to be close to the mean.
  • A high standard deviation indicates that the values are spread out over a wider range.
  • For normally distributed data, approximately 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Applications

Standard deviation and other statistical measures are used in various fields:

Finance

  • Measuring investment risk
  • Portfolio diversification
  • Option pricing models

Science & Research

  • Experimental error analysis
  • Quality control
  • Confidence intervals

Business & Economics

  • Market research
  • Process improvement
  • Forecasting models