Z-score Calculator

Related Calculators

Raw value (x): The original score or data value on the problem's own scale, such as 85 points on a test.

Mean (mu): The average value, or center, of the distribution.

Standard deviation (sigma): A measure of spread that tells how far values usually are from the mean.

Z-score (z): How many standard deviations a value is above or below the mean; z = 0 is exactly at the mean [1].

Percentile: The percent of values at or below a point under the normal model.

Area to the left: The cumulative probability to the left of a value or z-score; in this calculator it matches the percentile.

Area to the right: The probability to the right of a value or z-score.

Area between mean and point: The probability between the mean and the entered point, ignoring whether the point is above or below the mean.

Area between bounds: The probability between two entered values or two entered z-scores.

Mistake: Entering 0 or a negative number for "Standard deviation (sigma)".
Fix: Use a value greater than 0, because "Z-score", "Percentile", and probability outputs cannot be calculated with zero spread.

Mistake: Typing a value into "Raw value (x)" when the mode is "Find raw value from z-score" and expecting that number to be used.
Fix: In that mode, enter "Z-score (z)" instead and read the result from "Raw value".

Mistake: Confusing "Percentile" with "Area to the right".
Fix: Read "Percentile" and "Area to the left" as the percent at or below the point, while "Area to the right" is the percent above it.

Mistake: Entering only one bound when "Probability type" is "Area between two values".
Fix: Fill both "Raw value (x)" and "Upper raw value (x2)" or both "Z-score (z)" and "Upper z-score (z2)", depending on "Probability input uses".

Mistake: Thinking a negative "Z-score" is wrong.
Fix: A negative "Z-score" simply means the point is below the "Mean (mu)"; check the "Interpretation" card for the plain-language meaning.

Mistake: Expecting "Area between mean and point" to be negative when the point is below the mean.
Fix: That output is always an unsigned area, so use "Interpretation" and the sign of "Z-score" to tell direction.

• This calculator assumes a normal distribution. If your data are strongly skewed or not bell-shaped, the results may not match real data well.
• "Standard deviation (sigma)" must be greater than 0. If it is 0 or negative, the calculation is invalid and should be corrected.
• Percentile, "Area to the left", and "Area to the right" are model-based probabilities from the normal curve, not guaranteed counts from a real sample.
• Very large positive or negative z-scores can round to 100% or 0%. That means the true value is extremely close to the limit, not exactly beyond it.
• In "Area between two values", the calculator uses the lower and upper bounds in order for the math, even if you type them in reverse.
• Rounding can make "Area to the left" plus "Area to the right" differ from exactly 100.0000% by a tiny amount.

Core idea

The calculator uses standard normal distribution formulas to convert between a raw value, a z-score, and probability areas under the normal curve [1].

z = (x - mu) / sigma

x = mu + z * sigma

P(Z <= z) = Phi(z)

P(Z >= z) = 1 - Phi(z)

P(a <= Z <= b) = Phi(b) - Phi(a)

area between mean and point = |Phi(z) - 0.5|

How each mode works

In "Find z-score from raw value," the calculator takes your "Raw value (x)", subtracts the "Mean (mu)", and divides by "Standard deviation (sigma)". It then uses that z-score to find "Percentile", "Area to the left", "Area to the right", and "Area between mean and point".

In "Find raw value from z-score," the calculator multiplies your "Z-score (z)" by the standard deviation and adds the mean. It also reports the same probability views for that z-score so you can interpret where the result sits on the curve.

In "Find probability or percentile," the calculator first decides whether your input is a raw value or a z-score. If you enter raw values, it converts them to z-scores before finding left-tail, right-tail, or between-bounds probability.

Worked mini-example

Suppose "Raw value (x)" is 85, "Mean (mu)" is 70, and "Standard deviation (sigma)" is 10.

z = (85 - 70) / 10 = 1.5

The left-tail probability for z = 1.5 is about 0.9332, so the calculator shows "Area to the left" and "Percentile" as about 93.32%. The right-tail area is about 6.68%, and the area between the mean and the point is about 43.32%.

Interpretation rules

A positive z-score means the point is above the mean. A negative z-score means it is below the mean. A z-score of 0 means the point is exactly at the mean [1].

The "Interpretation" output turns that into plain language, such as "1.5 standard deviations above the mean" or "1.2 standard deviations below the mean."

Calculation notes

The normal cumulative function Phi(z) is computed numerically and then converted to percent by multiplying by 100. Results are clamped to the valid probability range from 0 to 1 before percent formatting so extreme z-scores do not display NaN or Infinity.

For "Area between two values," equal bounds give 0%. If bounds are entered in reverse order, the calculator sorts them internally before applying the between-bounds formula.

Assumptions used

The outputs are based on a normal model, use your entered mean and standard deviation exactly, and show rounded display values. Real-world data may differ from the model, especially in the tails or when the data are not close to normal.