Z-Score Calculator
Free Z-Score Calculator. Instantly solve z-score problems with detailed solutions and interactive formula breakdowns Try it now — no signup required.
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The Z-score is the raw score minus the mean, divided by the standard deviation.
Last reviewed: December 2025
Worked Examples
Example 1: Test Score
Example 2: Below Average
Background & Theory
**Z-Score (Standard Score):** Describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. **Formula:** **Z = (x - μ) / σ** Where: * **Z:** Z-score * **x:** The raw score (data point) * **μ (mu):** The population mean * **σ (sigma):** The population standard deviation **Interpretation:** * **Z = 0:** The score is exactly the mean. * **Z > 0:** The score is above the mean. * **Z < 0:** The score is below the mean. * **|Z| > 2:** Usually considered "unusual" or an outlier.
History
The concept of the Z-score and standardizing data relates to the development of the normal distribution (bell curve) by Carl Friedrich Gauss in the early 19th century. Standard scores allow statisticians to compare data points from different normal distributions.
Key Features
- Compute Pearson and Spearman correlation coefficients and full covariance matrices from pasted data columns, highlighting strongly correlated feature pairs.
- Plan train, validation, and test splits and k-fold cross-validation schemes by entering dataset size and desired fold count, with stratification guidance for imbalanced classes.
- Apply min-max normalization and z-score standardization to feature columns, showing before-and-after distributions to confirm correct scaling.
- Calculate model accuracy, precision, recall, F1-score, and Matthews correlation coefficient from a 2x2 or multi-class confusion matrix with interpretive guidance.
- Estimate ROC-AUC from true positive rate and false positive rate pairs, plotting the curve and computing the area using the trapezoidal rule.
- Determine the minimum sample size per group for an A/B test given desired statistical power, significance level, and expected effect size using two-proportion z-test formulas.
- Apply Simpson's rule and the trapezoidal rule for numerical integration of discrete data points, with error bound estimation for smooth functions.
- Estimate dominant frequency components from a time-series data set using DFT approximations, helping identify periodicity and seasonal patterns.
Frequently Asked Questions
Sources & References
Formula
Z = (x - μ) / σ
The Z-score is the raw score minus the mean, divided by the standard deviation.
Frequently Asked Questions
What is a Z-Score?
A Z-score (standard score) describes the position of a raw score in terms of its distance from the mean, when measured in standard deviation units.
What does a Z-score of 0 mean?
A Z-score of 0 indicates that the data point\'s score is identical to the mean score.
What does a positive Z-score mean?
It means the value is above the average. For example, +1.0 means it is one standard deviation higher than the mean.
What does a negative Z-score mean?
It means the value is below the average. -2.0 means it is two standard deviations lower than the mean.