Z-Score Calculator
Compute the Z-score (standard score) and cumulative probability instantly. Enter \u03bc and \u03c3 manually, or paste a dataset and let the calculator do the math.
Z-Score calculator
Compute the standard score and cumulative probability.
84.1% of the distribution falls below this value.
How Z-scores work
The Z-score formula standardizes any value relative to its distribution:
Where:
- x = the observed value
- \u03bc = the population mean
- \u03c3 = the population standard deviation
A Z-score of 0 means the value equals the mean. A Z-score of +2 means the value is 2 standard deviations above the mean (approximately the top 2.3% of a normal distribution).
The calculator also shows the cumulative probability \u2014 the percentage of the normal distribution that falls below your value. This is equivalent to looking up the Z-score in a standard normal table.
Common use cases
Hypothesis testing
Compare test statistics to critical Z-values to determine statistical significance.
Standardized test scores
Convert SAT, GRE, or IQ scores to percentiles using the Z-score and normal CDF.
Quality control
Identify defects or process deviations that fall outside ±3 sigma (Six Sigma methodology).
Finance & risk
Calculate Value-at-Risk (VaR) and identify unusual market movements or returns.
Grading on a curve
Normalize student scores to a standard distribution for fair grading.
Data science
Standardize features before machine learning to ensure equal weighting across variables.
Frequently asked questions
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. The formula is z = (x − μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. A Z-score of 0 means the value equals the mean; positive values are above the mean, negative values below.
How do I interpret the Z-score?
A Z-score of +1.0 means the value is 1 standard deviation above the mean. About 68% of data falls within ±1 SD, 95% within ±2 SD, and 99.7% within ±3 SD (the empirical rule). Values beyond ±3 are considered unusual outliers.
What is the cumulative probability shown?
P(X < x) is the percentage of a standard normal distribution that falls below your Z-score. For example, a Z-score of +1.0 gives P(X < x) ≈ 84.13%, meaning about 84% of the population scores lower. This is often called the "percentile."
What is the difference between manual and dataset mode?
In manual mode, you enter the mean and standard deviation directly. In dataset mode, you paste a list of numbers and the calculator computes the mean and population standard deviation automatically, then finds the Z-score for your target value.
Does this use population or sample standard deviation?
Dataset mode uses population standard deviation (divides by n, not n−1). If you’re working with a sample and want the sample standard deviation, use manual mode and enter your pre-calculated sample SD.
Can Z-scores be used for non-normal distributions?
The Z-score formula works for any distribution, but the probability percentages assume a normal (Gaussian) distribution. For non-normal data, the Z-score still measures distance from the mean in SD units, but the probability interpretation may not apply.
What are common Z-score thresholds?
Z = ±1.645 corresponds to the 90% confidence interval. Z = ±1.96 corresponds to 95% (most common in hypothesis testing). Z = ±2.576 corresponds to 99%. In quality control, Z = ±3 defines the "3-sigma" rule.
How accurate is the probability calculation?
This calculator uses the Abramowitz & Stegun approximation for the normal CDF, which is accurate to within 7.5 × 10⁻⁸ (essentially 7 decimal places). This is more than sufficient for educational, business, and most scientific purposes.