The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is also called the z-distribution, because every value in it is expressed as a z-score — the number of standard deviations a data point sits above or below the mean. A z-score of 0 falls exactly on the mean, a z-score of +0.5 is half a standard deviation above it, a z-score of +1.5 is one and a half standard deviations above it, and a z-score of +2 is two standard deviations above it.
You convert any raw value into a z-score with one formula:
z = (x − μ) ÷ σ
where x is the individual value, μ (mu) is the population mean, and σ (sigma) is the standard deviation.— The z-score (standard score) formula
Rearranging the same formula lets you go the other way and recover a raw value from a z-score: x = μ + (z)(σ). For example, if a normal distribution has a mean of 6 and a standard deviation of 3, then the value 12 lies two standard deviations above the mean:
x = μ + (z)(σ) = 6 + (2)(3) = 12
In the standard normal distribution the mean is 0 and the standard deviation is 1, so the formula simplifies neatly — a z-score and the value it represents become one and the same:
x = μ + (z)(σ) = 0 + (z)(1) = z
The standard normal (z) distribution
Difference Between a Standard Normal Distribution and a Normal Distribution
Like every normal distribution, the standard normal distribution is symmetrical about its mean and forms a smooth, bell-shaped curve. The difference is purely one of parameters:
- A normal distribution can have any mean and any positive standard deviation. Changing those two parameters shifts the curve left or right and stretches or squeezes it.
- The standard normal distribution is the single, fixed case where the mean is 0 and the standard deviation is 1.
Think of the standard normal curve as any normal curve that has been slid horizontally until its centre sits on 0 and rescaled until one standard deviation equals one unit on the x-axis. Because every normal distribution can be converted to this single reference shape, you only ever need one z-table to find probabilities for all of them — provided you are confident the data are genuinely normally distributed.
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Properties of the Standard Normal Distribution
The standard normal distribution shares the core shape of any normal distribution but has its own fixed features:
- The mean, median and mode are all equal to 0.
- The standard deviation is 1.
- The total area under the curve equals 1 (i.e. 100% of the probability).
- The curve is perfectly symmetrical about 0, so the area below 0 and the area above 0 are each 0.5.
- The curve is bell-shaped and extends indefinitely in both directions, getting ever closer to the x-axis without ever touching it (the tails are asymptotic).
The 68–95–99.7 (Empirical) Rule
The most useful fact about any normal distribution — and the standard normal distribution in particular — is the empirical rule, sometimes called the 68–95–99.7 rule. It describes how the data are spread around the mean in terms of standard deviations (z-scores):
- About 68% of values fall within ±1 standard deviation of the mean (z between −1 and +1).
- About 95% of values fall within ±2 standard deviations (z between −2 and +2).
- About 99.7% of values fall within ±3 standard deviations (z between −3 and +3).
This is why a z-score of 2 or more is often treated as unusual: roughly only 2.5% of values sit above z = +2, and another 2.5% sit below z = −2. A value at z = +2 lands at about the 97.7th percentile (more precisely, 97.72% of the distribution lies below it).
“For an approximately normal data set, almost all (99.7%) of the data fall within three standard deviations of the mean.”— The empirical (68–95–99.7) rule, a standard result in inferential statistics
How to Standardise a Normal Distribution
Standardising means converting an ordinary normal distribution into the standard normal distribution so that its mean becomes 0 and its standard deviation becomes 1. Every raw value x is replaced by its z-score. Standardising is worth doing because it lets you:
- Compare scores that come from different distributions with different means and standard deviations (for example, comparing a maths mark with a reading mark).
- Find the probability of an observation using a single z-table.
- Test whether a sample mean differs significantly from a known population mean.
Reading the Sign of a z-score
- A positive z-score means the value is greater than the mean.
- A negative z-score means the value is less than the mean.
- A z-score of zero means the value is equal to the mean.
The z-score formula again, with its parts labelled:
z = (x − μ) ÷ σ
x = the individual value | μ = the mean | σ = the standard deviation
How to Read a z-table
A z-table (standard normal table) tells you the probability — the area under the curve — that lies to the left of a given z-score. Because the total area is 1, this left-tail area is also the percentile of that z-score. To use a typical cumulative z-table:
- Find the row matching the z-score to one decimal place (e.g. the 1.5 row).
- Find the column for the second decimal place (e.g. .00 for exactly 1.50).
- Read the cell where they meet — that value is the cumulative probability P(Z ≤ z).
For negative z-scores, use the symmetry of the curve: P(Z ≤ −z) = 1 − P(Z ≤ +z). The table below lists the cumulative probabilities and percentiles for the most commonly searched z-scores.
| z-score | Area to the left (cumulative probability) | Percentile | Area to the right |
|---|---|---|---|
| −2.00 | 0.0228 | 2.3rd | 0.9772 |
| −1.50 | 0.0668 | 6.7th | 0.9332 |
| −0.50 | 0.3085 | 30.9th | 0.6915 |
| 0.00 | 0.5000 | 50th | 0.5000 |
| +0.50 | 0.6915 | 69.1st | 0.3085 |
| +1.00 | 0.8413 | 84.1st | 0.1587 |
| +1.50 | 0.9332 | 93.3rd | 0.0668 |
| +2.00 | 0.9772 | 97.7th | 0.0228 |
The Process of Standardisation — Step by Step
To turn a data set into z-scores, subtract the mean from every value (this centres the mean on 0), then divide every result by the standard deviation (this rescales the spread to 1). Worked through on a small set:
Take the normally shaped data set 2, 3, 3, 4, 4, 4, 5, 5, 6. Its mean is 4 and its standard deviation is approximately 1.15.
- Step 1 — subtract the mean (4): −2, −1, −1, 0, 0, 0, 1, 1, 2. The new mean is now 0.
- Step 2 — divide each value by the standard deviation (1.15): −1.73, −0.86, −0.86, 0, 0, 0, 0.86, 0.86, 1.73. The standard deviation of this new set is 1.
The data set now has a mean of 0 and a standard deviation of 1 — it has been standardised. Each transformed number is a z-score.
Worked Example: From z-score to Probability
Apply the formula z = (x − μ) ÷ σ:
z = (200 − 150) ÷ 25
z = 50 ÷ 25
z = 2.0
Your score lies 2 standard deviations above the mean.
Look up z = 2.00 in a z-table. The area to the left is 0.9772, so:
P(Z ≤ 2.0) = 0.9772 = 97.72%
About 97.7% of students scored below you, and only about 2.3% scored higher. This matches the 68–95–99.7 rule: roughly 95% of scores fall within ±2 standard deviations, leaving 2.5% in each tail.
The same two-step method — standardise to a z-score, then read the z-table — works for any normal distribution. If you needed the probability of scoring between two values, you would find the z-score for each and subtract their cumulative areas. For more on what these tail probabilities mean for hypothesis testing, see our guides to statistical significance and the p-value.
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