Z-Score Calculator (Standard Score & P-Value Probability)
Calculate standard Z-scores, normal distribution p-values, and percentile ranks.
TL;DR: Z-Score Calculator transforms raw scores into standard deviations from the mean using $Z = (X - \mu) / \sigma$ and computes standard normal cumulative probabilities (p-values).
What Is a Z-Score and How Do You Calculate It in Statistics?
A Z-score (standard score) measures how many standard deviations a raw data point (X) is above or below the population mean (μ). It is calculated using the formula: Z = (X - μ) / σ. A positive Z-score indicates a value above average, while a negative Z-score indicates below average.
How to Use the Z-Score Calculator
Our Z-Score Calculator performs high-precision mathematical operations directly in your browser with zero latency and complete client-side privacy.
- Enter the Raw Data Value ($X$).
- Enter the Population/Sample Mean ($\mu$ or $\bar{x}$).
- Enter the Standard Deviation ($\sigma$ or $s$).
- Click 'Calculate Z-Score' to compute the standard score.
- Review the two-tailed p-value, left-tail probability $P(Z < z)$, and percentile rank.
Mathematical Formula & Equations
Standardizes normal distribution data values into standard deviations from the mean and calculates left/right tail p-values.
Calculation Example
An exam score of $X = 85$ in a class with mean $\mu = 75$ and standard deviation $\sigma = 5$: $$Z = \frac{85 - 75}{5} = +2.00 \quad (\text{Top } 2.28\% \text{ percentile})$$
100% Client-Side Privacy & Data Security
All calculations, amortization schedules, variables, and sensitive numerical datasets execute 100% locally in your web browser memory. Your financial, medical, and personal values are never transmitted, logged, or uploaded to any external server.
Frequently Asked Questions
- A Z-score of 0 means the raw score is exactly equal to the mean of the dataset.
- The inverse formula is: `X = μ + Z × σ`.
- The standard normal distribution is a normal bell curve with a mean of `0` and a standard deviation of `1`.
- Under the empirical rule, approximately 95.45% of data in a normal distribution falls within 2 standard deviations of the mean.
- Z-scores are compared against critical values (e.g., Z = ±1.96 for 95% confidence) to determine whether to reject or fail to reject a null hypothesis.
- A Z-score of 0 indicates the value equals the population mean. A positive Z-score is above average, while a negative score is below the group average.
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