Z Score Calculator

The calculator has three modes:

**1. Raw score → z score and percentile**

1. Enter the raw score (x), the mean (μ) and the standard deviation (σ). 2. Read the z score, the percentile (area to the left) and the area to the right.

**2. z score → probability**

1. Enter a z value. 2. Read the left-tail, right-tail and two-tailed probabilities. This is the same as looking up a standard normal (z) table, but exact.

**3. Batch z scores for a column**

1. Paste a column of numbers (from Excel, Google Sheets or a CSV). 2. The calculator uses the column's own mean and SD (or values you enter) and returns a z score for every row, which you can copy back into your spreadsheet. Values with |z| > 3 are easy to spot as potential outliers.

No data are uploaded: all computation happens locally.

A z score (standard score) tells you **how many standard deviations a value lies above or below the mean**. A z of 0 is exactly average, +1 is one SD above, −2 is two SDs below.

Typical uses:

• **Comparing scores on different scales**, for example a student's maths and reading results when the tests have different means and SDs.
• **Finding percentiles** for normally distributed measures such as standardised test scores, height or IQ.
• **Flagging outliers** in a dataset (values with |z| above 3 are often checked).
• **Standardising variables** before combining them into an index or comparing regression coefficients.
• **Hypothesis tests** that produce a z statistic, such as large-sample tests of a proportion. To get the p value of any z, t, χ² or F statistic, you can also use the [p value calculator](/tools/p-value-calculator).

For a single value:

`z = (x − μ) / σ`

where x is the raw score, μ the mean and σ the standard deviation. When you only have sample data, use the sample mean and sample SD: `z = (x − x̄) / s`.

To go back from z to a raw score: `x = μ + z × σ`.

The percentile is the standard normal cumulative probability `Φ(z)`: the share of the distribution below z. The right tail is `1 − Φ(z)`, and the two-tailed probability is `2 × (1 − Φ(|z|))`.

**Assumptions.** Computing a z score itself requires nothing beyond a mean and SD. But **converting z to a percentile assumes the variable is approximately normally distributed**. For strongly skewed data (income, reaction times, counts), the normal-based percentile can be misleading, and an empirical percentile from the data is better.

A student scores **82** on an exam where the class mean is **70** and the standard deviation is **8**.

1. z = (82 − 70) / 8 = 12 / 8 = **1.50** 2. Area to the left, Φ(1.50) = **0.9332**, so the score is at about the **93rd percentile**. 3. Area to the right = 1 − 0.9332 = **0.0668**: about 6.7% of students would be expected to score higher, if scores are normal. 4. Two-tailed probability of |z| ≥ 1.50 = **0.134**.

Another student scores 60: z = (60 − 70) / 8 = **−1.25**, percentile Φ(−1.25) = 0.106, about the **11th percentile**.

Useful reference points: z = ±1.96 leaves 2.5% in each tail (the basis of 95% confidence intervals), and about 68%, 95% and 99.7% of a normal distribution lie within 1, 2 and 3 SDs of the mean.

• **Sign:** positive means above the mean, negative means below.
• **Size:** |z| around 1 is common, |z| above 2 is unusual (about 5% of a normal distribution lies beyond ±1.96), and |z| above 3 is rare.
• **Percentile:** the percentage of the reference population scoring below that value, if the distribution is normal.

APA 7 italicises *z* and reports values to two decimals. For a descriptive standard score:

> The participant's score was 1.50 standard deviations above the sample mean (*z* = 1.50), corresponding to the 93rd percentile.

For a z test statistic, give the exact p:

> The proportion differed significantly from .50, *z* = 2.17, *p* = .030.

If you are analysing a full dataset and want the complete write-up, Data2Paper's [Generate Paper](/generate-paper) can produce a full paper draft with descriptive statistics, tests, tables and figures, and lets you preview the first part free.

• **Using the variance instead of the SD** in the denominator.
• **Mixing population and sample values**, e.g. a raw score from one group with the mean and SD of another.
• **Reading percentiles off a normal curve for skewed data.**
• **Confusing left- and right-tail areas.** The percentile is the left tail; "what proportion scored higher" is the right tail.
• **Using z when σ is unknown and n is small** for a test of means; a t test is appropriate there. See the [t test calculator](/tools/t-test-calculator).
• **Deleting every |z| > 3 value automatically.** Check whether it is a data error or a genuine extreme case first.