📈 Statistics Calculators

Measures of Central Tendency

The mean, median, and mode each summarise a dataset with a single representative value, but they behave very differently. The mean (arithmetic average) is sensitive to outliers — a single very large or small value pulls it significantly. The median (middle value when sorted) is resistant to outliers and better represents "typical" values in skewed distributions like income or house prices, where a few very high values inflate the mean. The mode (most frequent value) is most useful for categorical data and for identifying the most common outcome. In a perfectly symmetric, bell-shaped distribution, all three measures coincide. In right-skewed distributions (long tail to the right), mean > median > mode; in left-skewed, mean < median < mode.

Standard Deviation and Variance

Variance measures the average squared deviation from the mean: σ² = Σ(xᵢ − μ)² ÷ N for a population, or s² = Σ(xᵢ − x̄)² ÷ (N−1) for a sample. The (N−1) denominator in the sample formula is Bessel's correction — it compensates for the tendency of sample variance to underestimate population variance. Standard deviation is the square root of variance, returning to the original units of measurement. A low standard deviation means data points cluster near the mean; a high one means they are spread out. The coefficient of variation (CV = standard deviation ÷ mean × 100%) allows meaningful comparison of spread between datasets with different units or scales.

Probability and Z-Scores

The z-score measures how many standard deviations a value lies from the mean: z = (x − μ) ÷ σ. A z-score of 0 means the value equals the mean; z = 2 means the value is 2 standard deviations above the mean. Z-scores allow comparison across different normal distributions with different means and standard deviations. In a standard normal distribution, 68.3% of data falls within ±1 SD, 95.4% within ±2 SD, and 99.7% within ±3 SD (the empirical rule). Converting a raw score to a z-score and looking up a z-table gives the probability of observing that value or more extreme — the foundation of hypothesis testing and confidence intervals.

Permutations and Combinations

Counting methods are fundamental to probability calculations. A permutation is an ordered arrangement: the number of ways to arrange r items chosen from n is P(n,r) = n! ÷ (n−r)!. A combination is an unordered selection: C(n,r) = n! ÷ (r! × (n−r)!). The distinction matters because order often does not matter in real problems — a poker hand of A-K-Q-J-10 is the same regardless of the order the cards were dealt (combination), but the finishing order in a race matters (permutation). Combinations appear in binomial probability: the binomial coefficient C(n,k) gives the number of ways k successes can occur in n independent trials, making it central to calculating exact probabilities for coin flips, quality control, and clinical trials.