Statistics Calculator - Sample & Population Descriptive Statistics

Statistics Calculator – Sample & Population Statistics Guide

Descriptive statistics summarize the central location, dispersion, and spread of a dataset. This calculator parses a numeric series and computes the arithmetic mean, sample variance, population variance, sample standard deviation, population standard deviation, sum of squares (SSD), geometric mean, and harmonic mean.

How Statistical Calculation Formulas Work

Arithmetic mean and sum of squares. Mean barx = (sum x_k / N), and the sum of squared deviations is SSD = sum (x_k - barx)^2.

Population vs. sample variance and standard deviation.

Population Variance = (SSD / N) qquad Sample Variance = (SSD / N - 1)

Population SD = √(Population Variance) qquad Sample SD = √(Sample Variance)

Geometric and harmonic means (when all values are positive).

Geometric Mean = exp≤ft((sum ln(x_k) / N)) qquad Harmonic Mean = (N / sum (1/x_k))

Worked Examples

Example 1: Dataset 2, 4, 8, 16

Count N = 4, sum = 30, mean = 30/4 = 7.5.

Squared deviations: (2-7.5)^2 = 30.25, (4-7.5)^2 = 12.25, (8-7.5)^2 = 0.25, (16-7.5)^2 = 72.25. SSD = 30.25 + 12.25 + 0.25 + 72.25 = 115.00.

Population variance = 115/4 = 28.7500; population SD = √(28.75) ≈ 5.3619. Sample variance = 115/3 ≈ 38.3333; sample SD = √(38.3333) ≈ 6.1914.

Geometric mean = (2 × 4 × 8 × 16)^1/4 = 1024^0.25 ≈ 5.6569. Harmonic mean = (4 / 1/2 + 1/4 + 1/8 + 1/16) = (4 / 15/16) ≈ 4.2667.

Example 2: Dataset 1, 3, 9, 27, 81

Count N = 5, sum = 121, mean = 121/5 = 24.2.

SSD = (1-24.2)^2 + (3-24.2)^2 + (9-24.2)^2 + (27-24.2)^2 + (81-24.2)^2 = 538.24 + 449.44 + 231.04 + 7.84 + 3226.24 = 4452.80.

Population variance = 4452.80/5 = 890.5600; population SD ≈ 29.8423. Sample variance = 4452.80/4 = 1113.2000; sample SD ≈ 33.3647.

Geometric mean = (1 × 3 × 9 × 27 × 81)^1/5 = 9.0000 exactly, because this dataset is consecutive powers of 3. Harmonic mean = (5 / 1/1 + 1/3 + 1/9 + 1/27 + 1/81) ≈ 3.3471.

What This Calculator Does Not Include

Frequently Asked Questions

Why do we divide by N - 1 in sample standard deviation?

Dividing by N minus 1 (Bessel's correction) compensates for the fact that a smaller sample tends to underestimate the overall variability of the parent population, providing an unbiased estimate of population variance.

When should I use population vs sample standard deviation?

Use Population SD when the dataset includes the entire group being studied. Use Sample SD when the dataset is a sample representing a larger population.

What are geometric mean and harmonic mean?

Geometric mean is the Nth root of the product of N values, useful for compound growth rates. Harmonic mean is the reciprocal of the average of reciprocals, useful for averaging rates like speed. Both require all values to be positive.

Why do geometric mean and harmonic mean disappear if any value is zero or negative?

Geometric mean requires taking the Nth root of a product, and harmonic mean sums reciprocals (1/x) — a zero value makes the reciprocal undefined, and a mix of negative and positive values makes the product's root ambiguous in the real numbers. The calculator only computes these two means when every value in the dataset is strictly positive.