Calculate sample and population standard deviation, variance, mean, geometric mean, harmonic mean, quartiles, interquartile range, and full step-by-step deviation tables.
• Method 1 (Keypad): Type numbers and press ADD to push them into the running dataset.
• Method 2 (Batch Input): Paste or type comma-separated numbers in the textarea below.
• Both input methods stay automatically synchronized in real time.
Understanding central tendency, dispersion, Bessel's degrees of freedom, and Pythagorean means.
Descriptive statistics summarize and characterize the essential features of a collection of data. They are split into two primary dimensions:
Pinpoint the central or typical value around which data clusters. Includes the Arithmetic Mean (average), Median (middle score), and Mode (most recurring value).
Quantify how far data points scatter from the central location. Includes the Variance, Standard Deviation, Range, and Interquartile Range (IQR).
When computing variance from a sample to estimate population variance, using the standard divisor n causes a systematic underestimation (negative bias) because sample data points naturally lie closer to the sample mean x̄ than to the unknown true population mean μ.
Bessel's Correction replaces the divisor n with degrees of freedom n - 1 [s² = Σ(xᵢ - x̄)² / (n - 1)], making the sample variance an unbiased estimator of the true population parameter σ².
While the arithmetic mean is suitable for additive quantities, the Geometric Mean (GM) is mathematically required when dealing with multiplicative processes, ratios, percentages, and investment growth rates:
For example, if an investment portfolio gains 100% in year 1 (+2x) and loses 50% in year 2 (0.5x), the arithmetic mean return is +25%, but the true annualized return is 0% (GM = √(2 × 0.5) = 1.0).
The median is robust against extreme outliers and skewed distributions (such as household incomes or house prices), where a few extreme values would heavily distort the arithmetic mean.
Variance is expressed in squared units (e.g. dollars² or kg²), which can be difficult to interpret intuitively. Standard deviation is in the exact same original units as your data.
The IQR is the distance between the 75th percentile (Q₃) and the 25th percentile (Q₁): IQR = Q₃ - Q₁. It represents the spread of the middle 50% of your data and is unaffected by extreme outliers.
The geometric mean requires taking the nth root of the product of all values. The logarithm ln(x) is undefined for x ≤ 0 in real arithmetic, so GM is only defined for strictly positive real numbers (x > 0).