Calculate Z-scores with three modes: compute Z from a value and distribution, reverse-solve for the raw value from a Z-score, or calculate Z-scores for an entire dataset. Shows percentile, two-tailed probability, step-by-step formula, and a bell curve visual.
Mode 1: Calculate Z from a raw score, mean, and standard deviation (z = (x - μ) / σ). Mode 2: Reverse-solve — find the raw score x from a given Z, mean, and SD. Mode 3: Calculate Z-scores for an entire dataset (enter values, get Z-score for each).
For Mode 1, enter the raw score (x), population mean (μ), and standard deviation (σ). The result shows the Z-score, the percentile rank, one-tailed probability, and two-tailed probability, all with step-by-step formula working.
The interactive bell curve diagram shows your Z-score marked on the standard normal distribution, with the shaded area representing the probability. The Z-score is interpreted in plain language: how many standard deviations above or below average, and what proportion of a normal population would score lower.
A Z-score (also called standard score) measures how many standard deviations a value is from the mean of a distribution. The formula is z = (x - μ) / σ, where x is the raw value, μ is the population mean, and σ is the standard deviation. z = 0 means the value equals the mean. z = 1 means the value is exactly 1 standard deviation above the mean. z = -2 means the value is 2 standard deviations below the mean. Z-scores allow comparison across different distributions by standardising to a common scale.
A percentile tells you what percentage of values in a normal distribution fall below a given value. For a standard normal distribution (mean=0, SD=1), z=0 corresponds to the 50th percentile (half the values are below the mean). z=1 ≈ 84th percentile (about 84% of values fall below). z=2 ≈ 97.7th percentile. z=-1 ≈ 16th percentile. The conversion from Z-score to percentile uses the cumulative distribution function (CDF) of the standard normal distribution, calculated using the error function.
For any normal distribution: 68.27% of data falls within ±1σ (z between -1 and 1). 95.45% of data falls within ±2σ (z between -2 and 2). 99.73% of data falls within ±3σ (z between -3 and 3). Only 0.27% of data has |z| > 3, making values with |z| > 3 rare outliers. This is why z = ±1.96 is used as the critical value for 95% confidence intervals, and z = ±2.576 for 99% confidence intervals.
The p-value associated with a Z-score is the probability of observing a value at least as extreme as the given Z-score under the null hypothesis of a normal distribution. The one-tailed p-value is the probability of being in one tail (above or below the Z-score). The two-tailed p-value is the probability of being in either tail (|Z| ≥ |z_observed|). A two-tailed p-value below 0.05 is conventionally considered statistically significant (corresponding to |z| > 1.96).
Use Z-score when: (1) the population standard deviation (σ) is known, and (2) the sample size is large (n > 30). Use the T-distribution (t-score) when: (1) the population standard deviation is unknown and estimated from a sample, or (2) the sample size is small (n < 30). In practice, Z-scores are used for standardised tests, population studies, and quality control. T-scores are more common in experimental research with small samples.
Standardised testing: IQ scores (mean=100, SD=15), SAT scores (mean=500 per section, SD=100), and many academic assessments use Z-scores to compare performance across different test administrations. Healthcare: Growth charts for children use Z-scores to indicate whether a child's height or weight is within normal range. Finance: Z-scores identify outlier returns and are used in the Altman Z-score for bankruptcy prediction. Quality control: Values beyond ±3σ trigger investigation in statistical process control. Research: Meta-analysis combines Z-scores from different studies to assess overall effect size.
