Z-score Calculator - Бесплатный онлайн-калькулятор

The z-score (also called standard score or normal score) is one of the most powerful tools in statistics - it standardises any value to a common scale so that observations from different distributions become directly comparable. A student's CAT score of 99.5 percentile, an investment with a 1.8 Sharpe ratio, an IQ test score of 130, and a baby's growth chart percentile all use z-scores under the hood. The formula is simple: z = (x − μ) / σ. The result tells you exactly how many standard deviations above (positive z) or below (negative z) the mean your value is.

For Indian students preparing for entrance exams and competitive tests, z-score appears in two contexts: directly in Class 11-12 statistics chapters and JEE/NEET probability questions, and indirectly in normalised score reporting where exam committees convert raw marks to percentile/scaled scores using z-statistics. The empirical rule (68-95-99.7) for normal distributions translates directly to z-score: ~68% of data has |z| ≤ 1, ~95% has |z| ≤ 2, ~99.7% has |z| ≤ 3. This calculator gives both the z-score and the cumulative probability so you can interpret 'how unusual' a value is.

Formula: z = (x − μ) / σ. Subtract the mean to centre the value around zero; divide by standard deviation to express the offset in 'SD units.' Once standardised, all distributions become comparable on the same scale. The standard normal distribution has μ = 0, σ = 1; any value's position in any normal distribution maps to a unique z-score, then to a unique cumulative probability via the standard normal table (or its modern computer-based equivalents).

Probability interpretation: P(Z ≤ z) is the area under the standard normal curve to the left of z. For z = 1.96, P ≈ 0.975, meaning 97.5% of data lies below this value - the boundary for the upper 2.5% tail. For z = -1.96, P ≈ 0.025. For z = 0, P = 0.50 (the mean is the median in a symmetric distribution). These probabilities are the foundation of hypothesis testing, confidence intervals, and statistical decision-making.

When to use z vs t: use z-score when you know the population standard deviation σ. Use t-score when you only have a sample SD (most common in research) and your sample size n is small (< 30). For large samples, z and t converge to nearly identical values. Always check which is appropriate before quoting a probability from a table.