Statistics Tools

Standard Deviation Calculator

Comprehensive statistical analysis with visualizations

Enter Your Data

Input numbers separated by commas, spaces, or new lines

Accepts comma-separated, space-separated, or line-separated values

Use "Sample" for a subset of data, "Population" for complete data

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Enter data in the Calculator tab and click Calculate to see results
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Enter data in the Calculator tab to view charts
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Enter data in the Calculator tab to view data analysis

Formulas & Definitions

Statistical formulas used in this calculator

Mean (Average)
Formula
x̄ = Σx / n

The arithmetic mean is the sum of all values divided by the count of values. It represents the central point of the data.

Standard Deviation
Sample Standard Deviation (s)
s = √[ Σ(x - x̄)² / (n - 1) ]
Population Standard Deviation (σ)
σ = √[ Σ(x - μ)² / n ]

Standard deviation measures the spread of data around the mean. A low standard deviation means data points are close to the mean; a high standard deviation means they are spread out.

When to use Sample (n-1)

Use when your data is a sample from a larger population. The (n-1) correction (Bessel's correction) provides an unbiased estimate.

When to use Population (n)

Use when your data represents the entire population, not a sample. This gives the exact standard deviation.

Variance
Variance
Variance = (Standard Deviation)²

Variance is the square of the standard deviation. It represents the average of the squared differences from the mean.

Z-Score
Z-Score Formula
z = (x - x̄) / s

A Z-score indicates how many standard deviations a value is from the mean. Positive Z-scores are above the mean; negative Z-scores are below.

  • Z = 0: Value equals the mean
  • |Z| ≤ 1: Within 1 standard deviation (68% of data)
  • |Z| ≤ 2: Within 2 standard deviations (95% of data)
  • |Z| > 2: Potential outlier
Other Statistics
Coefficient of Variation (CV)

CV = (s / x̄) × 100%

Measures relative variability as a percentage.

Standard Error of Mean (SEM)

SEM = s / √n

Estimates how far the sample mean is from the population mean.

Interquartile Range (IQR)

IQR = Q3 - Q1

The range of the middle 50% of data.

Skewness

Measures asymmetry of distribution.

  • Skew = 0: Symmetric
  • Skew > 0: Right-skewed (tail on right)
  • Skew < 0: Left-skewed (tail on left)

📊 The Empirical Rule (68-95-99.7)

68% of Data

Falls within ±1 standard deviation from the mean (μ ± 1σ)

95% of Data

Falls within ±2 standard deviations from the mean (μ ± 2σ)

99.7% of Data

Falls within ±3 standard deviations from the mean (μ ± 3σ)

This rule applies to normally distributed (bell-curve) data.