Z-Score Calculator — Standard Score, Percentile & Probability

A z-score answers a simple question: how far from the average is this particular value, measured in standard deviations? That framing transforms a raw number into something meaningful regardless of what's being measured. A test score of 78 doesn't tell you much — but a z-score of +1.4 tells you that score is well above average. The CalcHub Z-Score Calculator handles both the score calculation and the probability lookup.

The Z-Score Formula

z = (x − μ) / σ

Where:

• x = the raw value you're standardizing


• μ (mu) = population mean


• σ (sigma) = population standard deviation

Example: Test scores with mean = 70, standard deviation = 10. A student scores 85.

z = (85 − 70) / 10 = 1.5

That score is 1.5 standard deviations above the mean.

How to Use the Calculator

1. Enter the raw score (x), mean (μ), and standard deviation (σ).
2. Click Calculate.
3. The z-score appears, along with the percentile rank and the probability values (left tail, right tail, between).

You can also work in reverse: enter a z-score and get the raw value back, useful when you know "what score corresponds to the 90th percentile?"

Z-Score to Percentile Reference

Z-Score	Percentile	Meaning
−3.0	0.13%	Extremely below average
−2.0	2.28%	Well below average
−1.0	15.87%	Below average
0.0	50.00%	Exactly average
+1.0	84.13%	Above average
+2.0	97.72%	Well above average
+3.0	99.87%	Extremely above average

The empirical rule: about 68% of data falls within z = ±1, 95% within z = ±2, and 99.7% within z = ±3.

Probability Calculations

Given a z-score, you can find:

• P(Z < z) — probability that a random value is less than your score (left tail)
• P(Z > z) — probability that a random value exceeds your score (right tail)
• P(z₁ < Z < z₂) — probability of falling between two z-scores

For z = 1.5:

• P(Z < 1.5) ≈ 93.32% → a score of 85 beats about 93% of test takers
• P(Z > 1.5) ≈ 6.68% → only about 7% scored higher

Real Applications

Medical reference ranges: Many lab tests report results as "within 2 standard deviations of normal." A z-score outside ±2 is flagged as abnormal. Finance (Value at Risk): A portfolio with z = −1.65 at the 5th percentile means there's a 5% chance of losing that much or more in a given period. Manufacturing quality control: Measuring whether product dimensions fall within spec — parts outside ±3σ are defects. Education: Standardized tests like the SAT/ACT are designed around z-scores. An SAT score of 1200 with mean 1010 and SD 210 gives z = 0.90, or about the 82nd percentile.

Sample vs. Population

If your data is a sample (not the full population), the formula uses the sample mean (x̄) and sample standard deviation (s). The z-score formula itself looks the same, but the underlying SD is calculated differently (dividing by n−1 instead of n). The calculator has a toggle for this.

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When would a z-score be negative?

Whenever the raw value is below the mean. If the mean temperature in a city is 25°C and today's high is 20°C with SD=4, the z-score is (20−25)/4 = −1.25. Negative z-scores are completely normal and just mean "below average."

What's the difference between a z-score and a t-score?

Both standardize data, but a t-score is used with smaller samples (typically n < 30) when the population standard deviation is unknown. The t-distribution has heavier tails than the normal distribution, which accounts for the extra uncertainty in small samples.

Can I use z-scores for non-normal distributions?

You can calculate a z-score for any dataset — it's just arithmetic. But interpreting it as a percentile (using the normal distribution table) only makes sense if the data is approximately normally distributed. For heavily skewed data, percentile ranks should be determined directly from the data.

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Related calculators: Standard Deviation Calculator · Confidence Interval Calculator · Probability Calculator