Z-Score Calculator
Calculate standardized Z-score ($Z = \frac{X - \mu}{\sigma}$), cumulative normal distribution probabilities, percentiles, and two-tailed p-values.
Calculate Standardized Z-Score & Normal Distribution Probability
Enter raw score (X), mean (μ or x̄), and standard deviation (σ or s).
Calculation Results
Calculated using Standard Normal Distribution $N(0,1)$ CDF standards: Z = \frac{X - \mu}{\sigma}
Quick Summary
The Z-Score Calculator evaluates standardized Z-scores ($Z = \frac{X - \mu}{\sigma}$), cumulative left-tail probability $P(Z < z)$, percentile rank, and two-tailed p-values for normal distributions.
Formula Explanation
Z = \frac{X - \mu}{\sigma}
\text{Percentile Rank \%} = P(Z < z) \times 100
\text{Two-Tailed p-value} = 2 \times P(Z > |z|)How It Works
A Z-score (standard score) normalizes data from different normal distributions into a standardized $N(0,1)$ distribution with mean $0$ and standard deviation $1$. The Z-Score Calculator computes exact Z-values, percentiles, and cumulative probabilities under the normal bell curve.
Step-by-Step Worked Example
Practical Problem: An individual scores $X = 115$ on an IQ test with mean $\mu = 100$ and standard deviation $\sigma = 15$. Calculate Z-score, percentile rank, and cumulative probability.
- Step 1: Calculate numerator (deviation from mean): $115 - 100 = \mathbf{15.00}$.
- Step 2: Divide by standard deviation ($\sigma = 15$): $15 / 15 = \mathbf{+1.0000\text{ Z-Score}}$.
- Step 3: Look up cumulative normal distribution $P(Z < 1.00)$: $P(Z < 1.00) = \mathbf{0.8413}$.
- Step 4: Calculate Percentile Rank: $0.8413 \times 100 = \mathbf{84.13\text{th percentile}}$.
- Step 5: Interpretation: A score of 115 is $1.0$ standard deviation above the mean, outscoring $84.13\%$ of the population.
Real-World Calculation Examples
Scenario 1: IQ Test Score (X = 115, μ = 100, σ = 15)
Parameters: X = 115, mean = 100, SD = 15
Result: Z = +1.0000 (84.13th percentile, 1 SD above mean).
Scenario 2: High SAT Score (X = 1400, μ = 1050, σ = 200)
Parameters: X = 1400, mean = 1050, SD = 200
Result: Z = +1.7500 (95.99th percentile, top 4% score).
Scenario 3: Below Average Score (X = 85, μ = 100, σ = 15)
Parameters: X = 85, mean = 100, SD = 15
Result: Z = -1.0000 (15.87th percentile, 1 SD below mean).
Scenario 4: Extreme Mensa IQ Level (X = 130, μ = 100, σ = 15)
Parameters: X = 130, mean = 100, SD = 15
Result: Z = +2.0000 (97.72nd percentile, top 2% Mensa qualifier).
Key Benefits of Using This Calculator
Standardized Z-Score Formula
Applies standard $Z = (X - \mu) / \sigma$ normalization for cross-test comparisons.
Percentile Rank & CDF Probability
Calculates cumulative left-tail $P(Z < z)$ and right-tail $P(Z > z)$ normal curve probabilities.
Outlier & p-value Identification
Evaluates two-tailed p-values to identify statistically significant outliers ($|Z| > 2.0$).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a Z-score in statistics?
A Z-score (standard score) indicates how many standard deviations an individual raw score $X$ lies above or below the population mean $\mu$.
What does a positive vs negative Z-score mean?
A positive Z-score means the score is above average; a negative Z-score means the score is below average; a Z-score of 0 equals the mean.
What is the 68-95-99.7 Empirical Rule?
In a normal bell curve: 68% of scores fall between $Z = -1$ and $+1$; 95% fall between $Z = -2$ and $+2$; 99.7% fall between $Z = -3$ and $+3$.
How do I convert a Z-score to a percentile?
Look up the Z-score in a standard normal distribution table or use the CDF probability ($P(Z < z) \times 100$). A Z-score of $+1.0$ corresponds to the 84.13th percentile.
What Z-score threshold defines an outlier?
In statistics, scores with $|Z| > 2.0$ (top/bottom 2.5%) are considered unusual, and $|Z| > 3.0$ (top/bottom 0.13%) are extreme outliers.
How is Z-score used in financial Altman Z-score analysis?
Altman Z-score combines corporate financial ratios to predict company bankruptcy risk (Z > 2.99 = Safe Zone; Z < 1.81 = Distress Zone).
What is the difference between Z-score and T-score?
Z-score has mean $0$ and SD $1$; T-score transforms Z-score to mean $50$ and SD $10$ ($\text{T} = 50 + 10Z$), commonly used in bone density DEXA scans.
How do I convert raw score X back from a Z-score?
Use the inverse formula: $X = \mu + (Z \times \sigma)$. For example: $100 + (1.96 \times 15) = 129.4$.
What is a Standard Normal Distribution N(0,1)?
A Standard Normal Distribution is a bell curve with mean $\mu = 0$ and standard deviation $\sigma = 1$.
Can Z-score be calculated for non-normal distributions?
Z-scores can be computed for any dataset to measure distance in standard deviations, but percentile probabilities are only accurate if the underlying distribution is normal.