Free Z-Score Calculator with Animated Normal Distribution
- What this is
- A free, embeddable, animated normal distribution (z-score) diagram for statistics education on any website.
- Who uses it
- Statistics educators, psychology researchers, public health writers, data science bloggers, and anyone explaining normal distributions and probability to a general audience.
- How to embed
- Copy one line of HTML. No signup. No watermark. Works in WordPress, Webflow, Ghost, Substack, plain HTML.
- File size
- ~25 KB per diagram.
- License
- Free forever. Editorial explainer text included; updated centrally over time.
- URL
- scrollchart.com/z-score-calculator
Animated standard normal distribution
The diagram below shows the standard normal distribution with z-score bands highlighted. Scroll down to trigger the animation. This is the same component available for embedding.
The z-score formula
The z-score (or standard score) transforms a raw value x from a distribution with mean mu and standard deviation sigma into a standardized unit:
Where:
- x: the observed value
- mu: the population mean
- sigma: the population standard deviation
- z: the number of standard deviations x is from the mean
A positive z-score means x is above the mean. A negative z-score means x is below the mean. A z-score of zero means x equals the mean exactly.
The 68-95-99.7 rule
For any normal distribution, the empirical rule gives the percentage of values within each standard deviation band:
- 68% of values fall between z = -1 and z = +1 (within 1 standard deviation)
- 95% of values fall between z = -2 and z = +2 (within 2 standard deviations)
- 99.7% of values fall between z = -3 and z = +3 (within 3 standard deviations)
This rule is used to quickly estimate the probability of a value without a z-table. If a test score has z = +1.5, approximately 86.6% of the population scored below that value (area to the left of z = 1.5 in a standard normal table).
Common z-score probability values
These z-score cutoffs appear frequently in statistics and research:
- z = 1.645: 95th percentile (one-tailed, 5% significance level)
- z = 1.960: 97.5th percentile (two-tailed, 5% significance level)
- z = 2.326: 99th percentile (one-tailed, 1% significance level)
- z = 2.576: 99.5th percentile (two-tailed, 1% significance level)
- z = 3.090: 99.9th percentile (two-tailed, 0.1% significance level)
Common applications of z-scores
Z-scores appear across many fields because they enable comparison across distributions with different units:
- Clinical medicine: bone density T-scores and Z-scores (DEXA scan results), growth chart percentiles, lab reference ranges expressed as standard deviations from the mean
- Psychological testing: IQ scores (mean 100, SD 15 in most modern scales), SAT and standardized test scoring, personality inventory scales
- Finance: Altman Z-score for bankruptcy risk, portfolio returns expressed in standard deviation units (Sharpe ratio uses a related concept)
- Quality control: Six Sigma defect rates are defined by how many standard deviations the process mean is from the nearest specification limit
- Research: effect size calculation (Cohen's d is the difference between means divided by pooled standard deviation, equivalent to a z-score of the difference)
Z-score versus percentile
A z-score tells you how many standard deviations a value is from the mean. A percentile tells you what percentage of the population falls at or below that value. The two are related by the cumulative distribution function of the normal distribution. A z-score of +1.0 corresponds to approximately the 84th percentile; a z-score of +2.0 corresponds to approximately the 97.7th percentile.
Free forever
The diagram and explainer text are served from scrollchart.com so the content can be updated over time to reflect new statistics pedagogy or to link to relevant sources within the explainer. No subscription, no ads, no tracking pixels served to your readers.
Frequently asked
How do I calculate a z-score?
The z-score formula is z = (x - mu) / sigma, where x is the observed value, mu is the population mean, and sigma is the population standard deviation. The result tells you how many standard deviations x is above or below the mean. A z-score of +1.0 means the value is one standard deviation above the mean; a z-score of -2.0 means two standard deviations below the mean.
What probability corresponds to a z-score of 1.96?
A z-score of 1.96 corresponds to the 97.5th percentile of the standard normal distribution. The area to the left (below z = 1.96) is 97.5%. This is the boundary used in two-tailed hypothesis testing at the 5% significance level: values beyond +1.96 or -1.96 fall in the 5% rejection region (2.5% in each tail).
What is the 68-95-99.7 rule?
The empirical rule (also called the 68-95-99.7 rule) states that for a normal distribution: approximately 68% of values fall within 1 standard deviation of the mean (between z = -1 and z = +1), 95% fall within 2 standard deviations (z = -2 to z = +2), and 99.7% fall within 3 standard deviations (z = -3 to z = +3). The animated normal distribution diagram above shows these bands visually.
How do I interpret a z-score in a clinical or research context?
In clinical contexts (bone density, growth charts, lab values), a z-score expresses how far a patient's value is from the reference population mean in standard deviation units. A z-score below -2.0 is typically flagged as low; above +2.0 is flagged as high. In research, z-scores are used in hypothesis testing to determine whether an observed effect is statistically significant under the null hypothesis.
What is the difference between a z-score and a t-score?
A z-score assumes the population standard deviation is known. A t-score (from the t-distribution) is used when the population standard deviation is unknown and must be estimated from the sample. For large samples (n > 30), the t-distribution approximates the normal distribution and the two scores converge. For small samples, the t-distribution has heavier tails than the normal, reflecting greater uncertainty.
How do I embed an animated normal distribution in my website?
Scrollchart provides a free animated standard normal Z diagram that embeds with one line of HTML. The diagram shows the bell curve with z-score bands (68%, 95%, 99.7%) and animates the highlighted area as the reader scrolls to it. Copy the embed snippet from this page and paste it into any CMS or HTML file.
Does the z-score calculator work for non-English content?
Yes. The percentile labels, axis titles, and the 68-95-99.7 annotations render in any major language — English, French, Spanish, German, Mandarin, Hindi, Arabic, Japanese, Portuguese, and more. The numerical computation is the same; the rendering is localised to the language you ask for.
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