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Calculate z-scores (standard scores) to determine how many standard deviations a value is from the mean. Supports both forward and reverse calculations with p-value analysis.
Enter a value, mean, and standard deviation to find the z-score
Enter a z-score, mean, and standard deviation to find the original value
Please enter a value (x)
Please enter the mean (μ)
Please enter the standard deviation (σ)
This calculator provides z-score calculations for educational and statistical analysis purposes. Results assume a normal distribution.
P-values are calculated using the cumulative distribution function for the standard normal distribution. For critical statistical decisions, consult a statistician.
A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a distribution. A z-score of 0 indicates the value is exactly at the mean, while positive z-scores indicate values above the mean and negative z-scores indicate values below the mean. Z-scores are fundamental in statistics for comparing values from different distributions, calculating probabilities, and determining statistical significance.
Z-scores are used across many fields for standardization and comparison:
Academic Testing – Standardized test scores (SAT, ACT, IQ tests) use z-scores to compare performance across different test versions and populations.
Quality Control – Six Sigma methodology uses z-scores to measure process performance and defect rates in manufacturing.
Medical Research – Z-scores help compare patient measurements (BMI, bone density) to reference populations for diagnosis.
Finance – Z-scores are used to identify outliers in financial data and assess credit risk (Altman Z-score).
z = (x - μ) / σWhere z is the z-score, x is the value, μ (mu) is the mean, and σ (sigma) is the standard deviation
x = μ + z × σWhere x is the value, μ is the mean, z is the z-score, and σ is the standard deviation
There's no universal 'good' or 'bad' z-score—it depends entirely on context. In academic testing, a z-score of +1 to +2 typically indicates above-average performance. In quality control (Six Sigma), z-scores of ±3 or better are desirable. The interpretation depends on whether higher or lower values are preferred in your specific application.
Yes, z-scores can be negative. A negative z-score simply means the value is below the mean. For example, a z-score of -1.5 means the value is 1.5 standard deviations below the mean. Negative z-scores are perfectly normal and common in any distribution.
A z-score of 0 means the value is exactly equal to the mean. It's at the center of the distribution, corresponding to the 50th percentile. Half of all values in the distribution are below this point, and half are above.
Z-scores can be converted to percentiles using the cumulative distribution function. A z-score of 0 = 50th percentile, z = 1 ≈ 84th percentile, z = 2 ≈ 98th percentile, z = -1 ≈ 16th percentile, z = -2 ≈ 2nd percentile. Our calculator automatically provides this conversion.
Z-scores are used when the population standard deviation is known or for large samples. T-scores are used when the population standard deviation is unknown and must be estimated from a small sample. T-distributions have heavier tails than the standard normal distribution, especially for small sample sizes.
P-values indicate the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true. In hypothesis testing, p-values below a threshold (commonly 0.05) suggest statistical significance. Our calculator provides left-tailed, right-tailed, and two-tailed p-values for comprehensive analysis.
Standard deviation measures the spread of data around the mean. A standard deviation of zero would mean all values are identical (no variation), making z-score calculation mathematically undefined (division by zero) and meaningless. Real-world data always has some variation.
While z-scores are most meaningful for normal distributions, they can still be calculated for any distribution. However, the percentile and p-value interpretations assume normality. For highly skewed or non-normal data, other standardization methods or transformations may be more appropriate.