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statistics ToolUpdated 2026-09-22

Percentile Calculator

Analyze data sets and statistical distributions with our free percentile calculator. Computes sample variance, population variance, interquartile ranges (IQR), and outliers instantly.

Educational Disclaimer: Just ask and get calculations and formulas are provided strictly for educational and informational purposes. Verify critical computations independently before executing legal, engineering, or binding financial transactions.

Formula & Mathematical Methodology

Sample Variance s² = ∑(x - x̄)² / (n - 1) | Population Variance σ² = ∑(x - μ)² / N

Calculates the arithmetic mean, sums squared deviations from the mean, and divides by degrees of freedom.

How to Use This Percentile Calculator

  1. Enter numbers separated by commas or spaces in the dataset box.
  2. Choose Sample (n-1) or Population (N) variance mode.
  3. View the mean, variance, standard deviation, quartiles, and detected outliers.

Step-by-Step Worked Example

Scenario: Dataset: 12, 18, 25, 29, 34, 38, 42, 55

1. Count n = 8, Sum = 253, Mean = 31.625
2. Sum of squared deviations ∑(x - x̄)² = 1,323.875
3. Sample variance: 1,323.875 ÷ (8 - 1) = 189.125
4. Sample standard deviation: √189.125 = 13.75
5. Q1 = 21.5, Median (Q2) = 31.5, Q3 = 40.0, IQR = 18.5
Final Calculated Result: Variance = 189.125, Std Dev = 13.75, IQR = 18.5

What Does Your Result Mean?

Measures dispersion and variability around the central tendency of the dataset.

Common Practical Applications:

  • Data science & statistical research
  • Hypothesis testing & quality control
  • Identifying anomaly and outlier values

Important Limitations & Assumptions:

  • Requires at least 2 distinct numerical values for sample variance calculation.

Frequently Asked Questions

What is the difference between sample and population variance?

Sample variance divides by (n - 1) (Bessel's correction) to provide an unbiased estimator of an unknown population variance.

How is an outlier mathematically defined?

Under Tukey's fence rule, any data point less than Q1 - 1.5×IQR or greater than Q3 + 1.5×IQR is considered a statistical outlier.

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