📊Z-Score Calculator
Calculate Z-scores from raw scores (Z = (x − μ) / σ), convert Z-scores to probabilities (one-tail and two-tail), find the probability between two Z-scores, or reverse-calculate a Z-score from a percentile. Shows the full standard normal distribution table values.
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75 is 1 standard deviations above the mean
Probability Distribution Breakdown
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Z-Score Calculator: Standard Scores, Probabilities, and Percentiles
A Z-score (standard score) measures how many standard deviations a value is from the mean: Z = (x − μ) / σ. A Z-score of 0 means the value equals the mean; Z = 2 means 2 standard deviations above; Z = −1.5 means 1.5 SDs below. The Z-score allows comparison across different normal distributions.
Key values: Z = ±1.96 → middle 95% of data; Z = ±2.576 → middle 99% of data
| Z-score | Left tail P(Z ≤ z) | Percentile |
|---|---|---|
| −1.645 | 0.0500 | 5th |
| 0 | 0.5000 | 50th |
| 1.960 | 0.9750 | 97.5th |
Z-scores are fundamental to statistics, standardized testing, quality control, and medical diagnostics. They appear whenever we need to ask "how unusual is this value?" or "what fraction of the population scores above this level?"
How to Calculate a Z-Score
The Z-score formula is Z = (x − μ) / σ where x is the data value, μ is the population mean, and σ is the population standard deviation. Example: a student scores 85 on a test where the mean is 75 and SD is 10. Z = (85 − 75) / 10 = 1.0. This means the student scored 1 standard deviation above the mean, placing them at approximately the 84th percentile.
Z-Score to Probability
The standard normal distribution table (Z-table) gives Φ(z) = P(Z ≤ z), the probability that a standard normal random variable is less than or equal to z. The right-tail probability is 1 − Φ(z). The two-tail probability (used in hypothesis tests) is 2 × min(Φ(z), 1 − Φ(z)). For Z = 1.96: left tail = 0.975, right tail = 0.025, two-tail = 0.05.
Empirical Rule (68-95-99.7)
In any normal distribution: approximately 68% of values fall within 1 standard deviation of the mean (Z between −1 and +1); 95% within 2 SDs (Z between −1.96 and +1.96 for exactly 95%); 99.7% within 3 SDs (Z between −3 and +3). These are the basis for common significance thresholds in statistics and the "3-sigma rule" in manufacturing quality control.
Frequently Asked Questions
What is a Z-score and what does it tell you?
A Z-score tells you how many standard deviations a specific value is above or below the mean of a data set. Z = (x − μ) / σ. A Z-score of +2 means the value is 2 standard deviations above the mean; −1.5 means 1.5 SDs below. Z-scores are useful because they put different measurements on a common scale. A student scoring Z = 1.5 on a math test performed better relative to their class than a student scoring Z = 1.0 on a history test, even if the raw scores differ.
How do you convert a Z-score to a probability or percentile?
To find the probability, look up the Z-score in a standard normal table (Z-table) to find Φ(z) = P(Z ≤ z). This value is also the percentile (multiply by 100 for the percentage). For Z = 1.5: Φ(1.5) ≈ 0.9332, so approximately 93.32% of values fall below Z = 1.5. For Z = −0.5: Φ(−0.5) ≈ 0.3085, so about 30.85% of values fall below. For a two-tailed probability (used in hypothesis testing), compute 2 × min(Φ(z), 1 − Φ(z)). For Z = 1.96: two-tail p ≈ 0.05.
What is a good or bad Z-score?
There is no universally good or bad Z-score — it depends on context. In academic testing, Z = +1.5 to +2.0 is typically considered strong performance (top 7–15%). In manufacturing quality control (Six Sigma), a Z-score ≥ 6 means fewer than 3.4 defects per million. In medical reference ranges, values with |Z| > 2 are often flagged as abnormal (outside the 95% reference interval). In hypothesis testing, |Z| > 1.96 indicates significance at α = 0.05. In most statistical contexts, |Z| < 1 is ordinary, |Z| = 2–3 is notable, and |Z| > 3 is rare (less than 0.3% of normal distribution).
What is the difference between a Z-score and a T-score?
Both measure standard deviations from the mean, but they differ in when you use them. A Z-score uses the population standard deviation σ and is appropriate when σ is known and/or the sample is large (n > 30, by the Central Limit Theorem). A T-score uses the sample standard deviation s with n−1 degrees of freedom, appropriate when σ is unknown and the sample is small. As sample size increases, the t-distribution approaches the normal distribution: at n = 30, t* ≈ 2.042 vs Z* ≈ 1.960 for 95% CI. For most practical large-sample statistics (polls, A/B tests), Z-scores are used.