Z-Pontszám Kalkulátor – Standard Pontszám, Percentilis és Valószínűség
Calculate the z-score for any data point, convert z-scores to percentiles, and find the probability of values occurring in a normal distribution. Our z-score calculator handles both population and sample statistics with detailed explanations.
Hogyan használja ezt a számológépet
- Adja meg: Calculate
- Adja meg: Data Point (x)
- Adja meg: Mean (μ)
- Adja meg: Standard Deviation (σ)
- Adja meg: Z-Score
- Kattintson a Számít gombra
- Olvassa el a számológép alatt megjelenő eredményt
What Is a Z-Score?
A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a dataset. The formula is: z = (x − μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
Z-scores allow comparison of values from different distributions. For example, you can compare a test score of 80 out of 100 (mean 70, SD 10) to a score of 55 out of 75 (mean 50, SD 8) by converting both to z-scores.
A z-score of 0 means the value equals the mean. A z-score of +1 means one standard deviation above the mean. A z-score of −2 means two standard deviations below the mean.
The Standard Normal Distribution
The standard normal distribution (Z distribution) has mean = 0 and standard deviation = 1. Any normal distribution can be converted to the standard normal by calculating z-scores.
Key percentages of the empirical rule:
- 68% of data falls within 1 standard deviation (z between −1 and +1)
- 95% of data falls within 2 standard deviations (z between −2 and +2)
- 99.7% of data falls within 3 standard deviations (z between −3 and +3)
Values beyond z = ±3 are rare (0.3% probability) — these are statistical outliers by conventional definitions.
Z-Score to Percentile Conversion
| Z-Score | Percentile | Interpretation |
|---|---|---|
| −3.0 | 0.13% | Extremely low |
| −2.0 | 2.28% | Very low (bottom 2.3%) |
| −1.0 | 15.87% | Below average |
| 0.0 | 50.00% | Average (median) |
| +1.0 | 84.13% | Above average |
| +1.645 | 95.00% | Top 5% |
| +2.0 | 97.72% | Top 2.3% |
| +2.576 | 99.50% | Top 0.5% |
| +3.0 | 99.87% | Extremely high |
Applications of Z-Scores
- Education: Standardized test scores (SAT, GRE, IQ) are often expressed as z-scores or scaled equivalents
- Medical: BMI percentiles, bone density scores (T-scores, Z-scores in DEXA scans), growth charts
- Finance: Altman Z-score predicts bankruptcy probability; stock return analysis
- Quality control: Six Sigma methodology targets processes with 6 sigma (z = 6) quality
- Sports analytics: Comparing player performance across different eras, leagues, or positions
- Research: Hypothesis testing, p-values, confidence intervals all use the standard normal distribution
Hypothesis Testing with Z-Scores
In statistics, z-scores are central to hypothesis testing. A z-test compares a sample mean to a known population mean:
z = (x̄ − μ) / (σ / √n), where x̄ is the sample mean, μ is the population mean, σ is population standard deviation, and n is sample size.
The resulting z-score is compared to critical values for the chosen significance level (α):
- α = 0.05 (two-tailed): reject H₀ if |z| > 1.96
- α = 0.01 (two-tailed): reject H₀ if |z| > 2.576
- α = 0.001 (two-tailed): reject H₀ if |z| > 3.29
Z-Score vs T-Score
When should you use a z-score vs. a t-score?
- Z-score: Use when population standard deviation (σ) is known AND sample size is large (n > 30). Distribution is the standard normal.
- T-score: Use when population SD is unknown (using sample SD instead) or sample size is small (n ≤ 30). Distribution is t-distribution with n−1 degrees of freedom.
In practice, for large samples (n > 100), z and t distributions are nearly identical. The t-distribution has heavier tails, accounting for additional uncertainty from not knowing σ exactly.
Interpreting Extreme Z-Scores
Extreme z-scores have important practical implications:
- Z > 3 or Z < −3: Outlier. May indicate data entry error, measurement error, or genuinely unusual observation. Investigate before including in analysis.
- Z = 4 (p ≈ 0.003%): Extremely rare event. In particle physics, a "4 sigma" result is noteworthy but not conclusive evidence of new phenomena.
- Z = 5 (p ≈ 0.00003%): The "five sigma" standard used in physics to claim discovery of new particles (e.g., Higgs boson confirmation in 2012).
Utolsó frissítés: March 2026
Frequently Asked Questions
What is a good z-score?
There's no universally "good" z-score — it depends on context. In academic testing, a z-score of +1 to +2 (above average) is generally positive. In quality control, you want processes as close to z = 0 as possible. In finance, a high z-score (Altman) indicates financial stability.
How do you calculate a z-score?
z = (x − μ) / σ. Subtract the mean from your value, then divide by the standard deviation. Example: value = 75, mean = 70, SD = 10 → z = (75−70)/10 = 0.5. This means 75 is half a standard deviation above the mean.
What does a z-score of 1.5 mean?
A z-score of 1.5 means the value is 1.5 standard deviations above the mean. This corresponds to the 93.3rd percentile — the value is higher than 93.3% of the distribution.
Can z-scores be negative?
Yes. A negative z-score means the value is below the mean. A z-score of −2 means the value is 2 standard deviations below the mean, at approximately the 2.3rd percentile.
What is the z-score for 95th percentile?
The z-score for the 95th percentile is 1.645 (one-tailed). This means 95% of the distribution falls below this z-score. For a two-tailed 95% confidence interval, the critical z-values are ±1.96.
How do you find the area to the left of a z-score?
The area to the left of a z-score = the cumulative distribution function (CDF) of the standard normal distribution at that z. This equals the probability that a randomly selected value is less than z. Use a z-table or the erfc function: P(Z ≤ z) = 0.5 × erfc(−z/√2).
What is the difference between z-score and percentile?
Z-score measures distance from the mean in standard deviation units. Percentile ranks a value within the distribution (0–100%). They're related: a z-score of 0 = 50th percentile; z = 1 = ~84th percentile. Percentiles are more intuitive; z-scores are more useful mathematically.
What z-score corresponds to the 99th percentile?
A z-score of approximately 2.326 corresponds to the 99th percentile. This means only 1% of the distribution falls above this value.
Is z-score the same as standard deviation?
No. Standard deviation (σ) measures the spread of an entire dataset. Z-score measures where a single value sits relative to the distribution, expressed in standard deviation units. A z-score uses the standard deviation in its calculation.
What is a z-score in healthcare?
In healthcare, z-scores are used for bone density measurements (DXA/DEXA scans compare to age-matched peers as Z-score, and to young adults as T-score), growth charts (height/weight z-scores vs age-sex norms), and clinical lab reference ranges.