What is Z-Score Calculator?

Z-Score Calculator converts raw scores into standardized z-scores and vice versa. Enter an observed value along with the population mean and standard deviation to get the z-score, cumulative probability, and percentile rank. Useful for statistics coursework and data analysis.

The calculator handles seven modes from one panel. Forward converts a raw x value to a z-score; reverse goes the other way. Solve μ and Solve σ back-solve a missing distribution parameter when you already know the z-score. Between two Z computes the probability that a standard normal falls between two cut-offs, useful for hypothesis testing. Confidence → Z reads a confidence percentage (e.g. 95) and returns the two-sided critical z (≈ 1.96 for 95%). Batch takes a whole list of observed values and returns a z-score, cumulative probability, and percentile for each, with a CSV download. Each single-value result shows the formula substituted with your numbers, a plain-English interpretation when applicable, and a bell-curve diagram with the relevant region shaded.

How to use

1. Enter the observed value (x), population mean (μ), and standard deviation (σ).
2. It shows the z-score, cumulative probability, percentile, a plain-English interpretation, and the worked-out formula steps.
3. Switch modes with the tabs to back-solve the mean or standard deviation from a known z-score, compute the area between two z-scores, derive the critical z from a confidence level, or run a whole list of values at once in Batch mode (with CSV download) — the bell curve and steps update to match.

When to use

• Comparing an exam score against the class distribution to find a percentile rank.
• Finding p-values for a one- or two-tailed z-test in an intro statistics assignment.
• Working out which raw cutoff corresponds to the top 5% in a normally distributed dataset.

Result

A student scored 78 on an exam where the class mean was 70 with a standard deviation of 5. The z-score is 1.6, meaning they scored in the 94.5th percentile — better than 94.5% of the class.

FAQ

What does a z-score actually represent?

It's the number of standard deviations a value sits above or below the mean. A z of +1.5 means the value is one and a half SDs above average; a z of -2 means two SDs below. Negative z-scores aren't bad, they're just on the left side of the distribution.

Why is the cumulative probability different from the two-tailed value?

Cumulative probability is the area to the LEFT of your z-score, useful for percentiles. The two-tailed p-value sums the areas in both far tails (beyond |z|), which is what most hypothesis tests need when the direction of the effect isn't pre-specified.

Can I use this if my data isn't normally distributed?

The z-score itself still standardises any value, but the probability and percentile outputs assume normality. For skewed data, the percentile is misleading. Larger samples (n > 30) often work fine via the Central Limit Theorem when you're standardising sample means.

What's reverse mode for?

Use it when you know the z-score and want the raw value. Example: SAT scores have mean 1050 and SD 200, what raw score is the 95th percentile? Enter z = 1.645 in reverse mode with those parameters and you get 1379.

How accurate is the normal CDF used here?

The tool uses an Abramowitz-Stegun approximation accurate to about seven decimal places across the practical range (|z| up to ~6). For everyday coursework and applied stats, that's more than enough precision.

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