Mathematics CalcyZone Calculator

Z-Score Calculator (with Probability)

Q: What is a good z-score?

It depends on context; |z| > 1.96 is outside the middle 95% of a normal distribution.

Convert a value to a z-score and find left-tail, right-tail and two-tailed normal probabilities.

z-score

1.5

P(Z < z) (left tail)

0.93319

P(Z > z) (right tail)

0.06681

Two-tailed p

0.13361

Step-by-step working

z = (x − μ) / σ = (85 − 70) / 10 = 1.5
P(Z < z) = ½[1 + erf(z/√2)] = 0.93319

Mathematical Formula

z = (x − μ) / σ. Φ(z) = ½[1 + erf(z / √2)], with erf from Abramowitz & Stegun 7.1.26 (|error| < 1.5 × 10⁻⁷).

Overview & Explanation

A z-score expresses how many standard deviations a value is from the mean; the normal CDF converts it to a probability.

How It Works

Standardise the value.
Evaluate the standard normal CDF.
Right tail = 1 − Φ(z); two-tailed = 2·min(Φ, 1 − Φ).

Practical Applications

Hypothesis testing.
Comparing exam scores across tests.

What the z-score formula means

z = (x - mean) / standard deviation. The numerator is how far the value sits from the average, and dividing by the standard deviation expresses that distance in units of spread. A z of 0 is exactly average, and a positive z is above average.

Example: a test score of 85 in a class with mean 70 and standard deviation 10 gives z = (85 - 70) / 10 = 1.5. On a normal distribution, about 93.3% of values fall below z = 1.5. A z of -1 sits at about the 15.9th percentile.

The 68-95-99.7 rule

For roughly normal data, about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. So |z| above 2 is uncommon and above 3 is rare. These percentiles only apply when the data really is close to bell-shaped.

Example 1: x = 85, μ = 70, σ = 10

z = 1.5.

Inputs

Value x: 85 · Mean μ: 70 · Standard deviation σ: 10

= entered

Step 1

z = (x − μ) / σ = (85 − 70) / 10 = 1.5

Step 2

P(Z < z) = ½[1 + erf(z/√2)] = 0.93319

z-score

Result

= 1.5

Example 2: z = 1.96

The 95% two-tailed critical value.

Inputs

Value x: 1.96 · Mean μ: 0 · Standard deviation σ: 1

= entered

Step 1

z = (x − μ) / σ = (1.96 − 0) / 1 = 1.96

Step 2

P(Z < z) = ½[1 + erf(z/√2)] = 0.975

z-score

Result

= 1.96

How it's calculated & sources

z = (x − μ) / σ. Φ(z) = ½[1 + erf(z / √2)], with erf from Abramowitz & Stegun 7.1.26 (|error| < 1.5 × 10⁻⁷).

Frequently Asked Questions

What is a good z-score?

It depends on context; |z| > 1.96 is outside the middle 95% of a normal distribution.

Does this assume normality?

Yes, the probabilities assume a normal distribution.

z to percentile?

The left-tail probability × 100 is the percentile.

Can I compare scores from different tests with z-scores?

Yes, that is a common use. A 1.5 on one test and 0.8 on another show the first result was further above its own class average, even if the raw marks differ.

Do I use population or sample standard deviation?

Use the population value when you have every member of the group, and the sample value (dividing by n - 1) when your data is a sample from a larger group.