Calculate mean, variance, standard deviation, and other descriptive statistics for your sample data. Learn the mathematical foundations behind each measure.
Understanding the mathematical foundations behind descriptive statistics
x̄ = (Σ xᵢ) / n
The arithmetic average of all sample values. Sum all values and divide by the number of observations.
Most common measure of central tendency, sensitive to outliers.
s² = Σ(xᵢ - x̄)² / (n - 1)
Measures the average squared deviation from the mean. Uses n-1 (Bessel's correction) for unbiased estimation.
Unbiased estimator of population variance, fundamental for statistical inference.
σ² = Σ(xᵢ - x̄)² / n
Measures variability using all n observations in the denominator. Used when treating data as the entire population.
Biased estimator when used on samples, but appropriate for population data.
s = √s² or σ = √σ²
Square root of variance, measures spread in the same units as the original data.
Easier to interpret than variance as it's in the same scale as the data.
Middle value when data is ordered
For odd n: middle value. For even n: average of two middle values.
Robust measure of central tendency, not affected by outliers.
Key concepts and applications in descriptive statistics
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Mathematical Statistics