Z-score Calculator - Standard Score, Percentile, and Raw Value

A z-score calculator answers a deceptively simple question: how unusual is this number, once you strip away the original units? Exam points, milligrams per liter, and reaction times cannot be compared until each is expressed in the same currency - standard deviations from a mean. The standard score z = (x − μ) / σ does that conversion. Positive z sits above the mean; negative z sits below; z = 0 is exactly μ.

The formula is only as honest as μ and σ. In a textbook problem those parameters are given as population values. In a lab course you often estimate them from a sample and should switch to a t procedure instead - see the p-value calculator when the test statistic is t, not z. This page assumes a known (or treated-as-known) σ > 0 and a normal model when it reports percentiles and tail probabilities.

Everything runs in the browser. The default example is the classroom classic: a raw score of 85 against μ = 70 and σ = 10 produces z = 1.5. Invert the same numbers with x = μ + zσ and you recover 85. Last inputs stay in localStorage on this device for up to thirty days; they are not uploaded.

The two panels split input from output. Drag the divider on a wide screen if you want more room for the numbers or for the result. Calculation updates as you type.

1. Choose compute z from x, μ, σ or compute x = μ + zσ.
2. Enter the known pieces. Commas are ignored. σ must be strictly greater than zero - a zero or negative spread is rejected rather than producing Infinity or a silent NaN.
3. Read z, the raw value, the percentile Φ(z)×100, one-tailed p = 1 − Φ(|z|), and two-tailed p ≈ 2(1 − Φ(|z|)).
4. Use Copy for a text block, or Clear All to wipe the fields. Fullscreen hides the article so you can screenshot just the score.

Φ is approximated with the Abramowitz–Stegun error-function polynomial (a1 = 0.254829592, …) via Φ(z) = ½(1 + erf(z/√2)). That is plenty for homework and most lab write-ups; it is not a 50-digit library CDF.

Subtract the mean so the origin sits at μ, then divide by the standard deviation so one unit on the new axis is one σ. The inverse is equally useful when a table or a software output already gave you z and you need the original scale: x = μ + zσ.

Example - default 85, 70, 10

• x − μ = 85 − 70 = 15
• z = 15 / 10 = 1.5
• Φ(1.5) ≈ 0.9332, so the score sits near the 93rd percentile
• One-tailed p ≈ 0.0668; two-tailed p ≈ 0.1336

Example - below the mean

A value of 55 with the same μ and σ gives z = (55 − 70)/10 = −1.5. The percentile drops to about 6.7%. Two-tailed p is unchanged versus +1.5 because that probability uses |z|.

Example - recover x from z = 1.96

With μ = 70 and σ = 10, x = 70 + 1.96×10 = 89.6. That is the raw cutoff people loosely call “two standard deviations up” when they round 1.96 to 2.

Once z is in hand, the standard normal CDF Φ(z) is the probability of drawing a value at or below that z if the population is N(μ, σ²). Multiply by 100 for a percentile rank. A student at the 93rd percentile outscored about 93% of the reference population - not 93% of the class sitting in the room unless that class is the reference.

One-tailed p on this page is the far-tail probability 1 − Φ(|z|): the chance of being at least this extreme in the direction of the observed sign. Two-tailed p doubles that, 2(1 − Φ(|z|)), which is the usual “as or more extreme in either direction” figure. Neither is a hypothesis test by itself; you still need a null, an alternative, and a decision rule. For a full p-value from z, t, or chi-square, open the p-value calculator.

Those critical values are also the building blocks of a mean confidence interval. After you have z, the confidence interval calculator attaches a margin of error to x̄.

Students sometimes treat a minus sign as an error to be discarded. It is information. z = −2.3 says the observation is 2.3 standard deviations below the mean - a low exam score, a short waiting time, a concentration under a specification, depending on context. The magnitude |z| still measures unusualness; the sign tells you which side.

Percentiles respect the sign: Φ(−1.96) ≈ 0.025, about the 2.5th percentile. Two-tailed p does not: the same 0.05 attaches to both −1.96 and +1.96. If your assignment asks “what fraction scored lower,” use the percentile. If it asks “two-sided unusualness versus the mean,” use two-tailed p.

Standardizing does not make a skewed raw variable normal. If the underlying measurements are badly skewed, Φ(z) is the wrong percentile engine. Compute μ and σ with the standard deviation calculator if you only have a list of numbers, then decide whether a normal z is even appropriate.

A z-score uses σ. A t-score uses s and carries degrees of freedom n − 1. When σ is truly known - a long-running production line with a settled process sigma, a problem statement that hands you σ - z is the right standard score. When σ is a sample estimate and n is small, the extra uncertainty in s fattens the tails; critical values grow, and you should not pretend s is σ.

The slogan “n ≥ 30, use z” is a classroom convenience, not a theorem. It leans on the central limit theorem for the mean and on s being close to σ in large samples. A sample of 30 from a wild mixture distribution can still misbehave. A sample of 12 from a near-Gaussian lab instrument may be fine with t. This calculator will still compute z = (x − μ) / σ for any positive σ you type; it will not silently switch to t.

If you are building an interval for a mean rather than scoring one observation, prefer the dedicated interval tool (z* or t* times s/√n). For a broader stats workspace, see the statistics calculator.

Percentiles and p-values assume X ~ N(μ, σ²), or at least that you are willing to use the normal as a reference distribution. The algebra for z itself does not require normality; it is just a linear rescaling. The Φ lookup does.

Extreme |z| (beyond about 8) will show a one-tailed p that underflows toward 0 in ordinary floating point; the Abramowitz–Stegun erf is a few-digit approximation, not a tail-asymptotic expansion. For journal-grade tiny p-values, use a statistics library. For homework, 1.5 and 1.96 are well inside the comfortable range.

This tool does not compute a sample mean or sample s from a paste of numbers. It does not apply a continuity correction, and it does not produce a t-score. Feed it μ, σ, and either x or z - nothing else.

For a normal population, about 68% of values fall within one standard deviation of the mean (|z| < 1), about 95% within two (|z| < 2, loosely; the exact two-tailed 95% cutoff is 1.96), and about 99.7% within three. That empirical rule is a sketch, not a replacement for Φ(z). A z of 1.5 is inside the middle 87% or so, which is why its two-tailed p is about 0.13 rather than “almost 0.05.”

The most common homework error is dividing by the standard error of a mean when the problem asked you to score a single observation. (x − μ) / (σ/√n) is a test statistic for x̄, not a z-score of one student’s exam. The second error is feeding a sample s into this page and then reading Φ as if σ were known. The arithmetic will run; the tail probability will be slightly too small because t tails are heavier. The third is treating percentile Φ(z)×100 as a grade: being at the 93rd percentile of a national reference is not the same as scoring 93% on the test.

A fourth trap is standardizing already-standardized data. If a spreadsheet column is already z, putting it through (x − 70)/10 again is nonsense. Check the units of μ and σ against x before you trust the 1.5.