Z-Score Calculator

About the Z-Score Calculator

A z-score tells you how far a value sits from the mean, measured in standard deviations. It turns scores from different tests, heights from different populations or measurements in different units into a single common scale, so you can compare them directly and look up how unusual they are.

This calculator performs the six z-score tasks that come up in statistics courses: computing $z$ from a raw value, recovering a raw value from $z$ , finding the area to the left or right of $z$ , finding the area between two z-scores and finding the z-score for a given percentile. Areas are computed from the standard normal distribution directly, so there is no table to read.

What is a z-score?

For a value $x$ from a population with mean $\mu$ and standard deviation $\sigma$ , the z-score is $z = \frac{x - \mu}{\sigma}$ . A z-score of 0 means the value equals the mean, a positive z-score means it lies above the mean and a negative one means it lies below.

If the data follow a normal distribution, the z-score also tells you the percentile. The area under the standard normal curve to the left of $z$ , written $\Phi(z)$ , is the proportion of values below $x$ .

How to use the Z-Score Calculator

Choose the calculation: z-score from x, raw value from z, area left, area right, area between or z-score for a percentile.
Enter the inputs the mode asks for: x, the mean and the standard deviation, or one or two z-scores, or a percentile.
Percentiles can be entered as a decimal such as 0.95 or a percentage such as 95%.
Press Calculate.
Read the z-score or area, and the percentile interpretation.

Formulas

Z-score

z = \frac{x - \mu}{\sigma}

Raw value from z

x = \mu + z\sigma

Area to the right

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

Area between

P(z_1 < Z < z_2) = \Phi(z_2) - \Phi(z_1)

Worked example: x = 85, μ = 70, σ = 10

The example takes a test score of $x = 85$ in a class with mean $\mu = 70$ and standard deviation $\sigma = 10$ . Subtracting the mean gives 15 and dividing by the standard deviation gives $z = 1.5$ : the score is one and a half standard deviations above average.

The area to the left of $z = 1.5$ is about 0.9332, so if the scores are normally distributed the student scored higher than roughly 93.32% of the class.

The solution below is generated by the same engine that powers the calculator above.

Final answer

z = 1.5

How we read your problem

Z-score from x, mean and SD

Step-by-step solution

Subtract the mean from the value
$x - \mu = 85 - 70 = 15$
Divide by the standard deviation
$z = \frac{15}{10} = 1.5$
Look up the area to the left (percentile)
$\Phi(1.5) \approx 0.933193$

Formulas used

Z-score
$z = \frac{x - \mu}{\sigma}$
Standard normal CDF
$\Phi(z) = \tfrac12\left[1 + \operatorname{erf}\!\left(\tfrac{z}{\sqrt2}\right)\right]$

85 lies 1.5 standard deviations above the mean; about 93.32% of a normal population lies below it.

Verification

Reverse: μ + zσ = 70 + (1.5)(10) = 85.

Detailed explanation

Reading areas under the normal curve

The total area under the standard normal curve is 1, and the curve is symmetric about 0. The area to the left of $z = 1.25$ is about 0.8944; the area to the right of $z = 1.96$ is about 0.025, which is why 1.96 appears in 95% confidence intervals. The area between $-1$ and $1$ is about 0.6827.

These areas give the empirical rule: about 68% of normally distributed values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three.

Going backwards: from percentile to z

The inverse mode solves $\Phi(z) = p$ . For the 95th percentile it returns $z \approx 1.645$ . Combine this with $x = \mu + z\sigma$ to find the cut-off value: with a mean of 70 and a standard deviation of 10, the 95th percentile is about $70 + 1.645 \times 10 \approx 86.45$ .

The raw-value mode performs that second step directly. Entering $z = 1.5$ with the same mean and standard deviation returns $x = 85$ , reversing the original example.

Comparing values on different scales

Suppose one student scores 85 on a test with mean 70 and standard deviation 10, and another scores 620 on a test with mean 500 and standard deviation 100. Their z-scores are 1.5 and 1.2, so the first performance is relatively stronger, even though 620 looks like the bigger number.

Z-scores also flag outliers: values with $|z| > 3$ are rare under a normal model and deserve a second look. Keep in mind that percentile interpretations depend on the data being approximately normal; for strongly skewed data the z-score still measures distance from the mean, but the area lookup may be misleading.

Common mistakes

Subtracting in the wrong order; it is the value minus the mean, not the mean minus the value.
Dividing by the variance instead of the standard deviation.
Reporting the left area when the question asks for the probability above a value.
Entering a percentile like 95 instead of 95% or 0.95.
Applying normal-curve percentiles to data that are clearly not normal.

Tips

Sketch the bell curve and shade the region you need before calculating.
Use symmetry: $P(Z < -z) = P(Z > z)$ .
Remember the landmark values 1.645, 1.96 and 2.576 for 90%, 95% and 99% two-sided intervals.
Use the standard deviation calculator first if you only have raw data.
Round z-scores to two decimal places only at the end.

Frequently asked questions

How do you calculate a z-score?

Subtract the mean from the value and divide by the standard deviation: z = (x − μ) ÷ σ. For x = 85, μ = 70 and σ = 10, z = 1.5.

What does a negative z-score mean?

The value is below the mean. A z-score of −2 means the value is two standard deviations below average.

How do I convert a z-score to a percentile?

Find the area under the standard normal curve to the left of z. For z = 1.5 the area is about 0.9332, the 93rd percentile.

How do I find the z-score for a percentile?

Choose the inverse mode and enter the percentile, for example 95%. The calculator returns z ≈ 1.645.

What is a good z-score?

It depends on context. Values between −1 and 1 are typical; values beyond ±2 are unusual and beyond ±3 are rare under a normal model.

Do I need a z-table?

No. The calculator computes the normal areas directly, so you get more precise values than a printed table provides.

Why does the calculator reject a standard deviation of 0?

Dividing by zero is undefined, and a standard deviation of 0 means every value equals the mean, so standardizing is meaningless.

Find your z-score

Choose a calculation above and enter your numbers to get the z-score, area or percentile.

Use the Z-Score Calculator Ask the AI Math Solver