Statistics › 🔔 Random Variables & Distributions

Z-Scores & the 68-95-99.7 Rule

How to turn any value into a universal 'how far from average?' number — and read its percentile straight off the bell curve.

Intro StatisticsAP Statistics levelCollege Stats 101

💡

The big idea: A raw number on its own — a height of 189 cm, a time of 24 minutes — tells you almost nothing until you know what is typical and how spread out things are. The z-score fixes this: it rewrites any value as the number of standard deviations it sits from the mean. Once values speak that common language, you can compare across completely different distributions and, for normal data, read off exactly what fraction of the population falls above or below.

🎯 By the end, you'll be able to

Compute a z-score with z = (x − μ)/σ and read it as a number of standard deviations from the mean
Use z-scores to compare values that come from different normal distributions
Apply the 68-95-99.7 (empirical) rule to estimate how common a value is
Connect a z-score to a percentile using the tail area under the normal curve

📎 You should already know

Mean and standard deviation
The normal distribution and its bell shape

A number by itself doesn't tell you much

Suppose someone is 189 cm tall. Is that unusual? You cannot answer until you know two more things: what is the average height, and how much do heights typically vary? Among adult men (average about 175 cm, typical spread about 7 cm) 189 cm is quite tall. Among professional basketball players it is fairly ordinary.

So a raw value only becomes meaningful once you measure it against its own group's mean and standard deviation. That is exactly what standardizing does, and the result is called a z-score.

🔑 The z-score, in one sentence

A z-score is how many standard deviations a value sits above or below its mean. Subtract the mean to recenter, then divide by the standard deviation to rescale. A positive z is above average, a negative z is below, and \(z = 0\) is exactly average. Because the units cancel, a z-score is a pure number you can compare against any other z-score.

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

The z-score of a value x: distance from the mean μ, measured in standard deviations σ.

See standardizing in action

In the tool below, slide the raw value x, the mean μ, and the standard deviation σ. The readout shows \(z = (x-\mu)/\sigma\) updating live.

The picture is drawn on the standard normal scale, where the axis is marked in standard deviations (\(-3\sigma, -2\sigma, \mu, +2\sigma, \dots\)). The red line sits at your z, and the shaded left tail is the area \(P(Z < z)\) — the fraction of the population below x, i.e. its percentile. Notice how moving x or changing σ slides the line and grows or shrinks that shaded area.

🎮 Interactive: standardize a value and read its percentile LIVE

Slide x, μ, and σ. The z-score is how many standard deviations x is from the mean; the shaded left tail is P(Z < z), the percentile of x. Try holding z fixed while changing μ and σ — the percentile stays the same, because z alone determines the tail area.

✨ Why z-scores are the great equalizer

Two values from different normal distributions cannot be compared directly — a raw 24-minute time and a raw 23-minute time mean different things if the two races had different averages and spreads. But their z-scores can be compared, because every z-score is measured in the same universal unit: standard deviations from its own mean. Convert both to z, and whichever is further in the favorable direction is the stronger result. This is also why a z-score pins down a percentile on the standard normal: z is all the tail area depends on.

The 68-95-99.7 rule

For data that follows a normal (bell) shape, the z-score comes with a famous shortcut called the empirical rule. It says that almost all of the data sits within three standard deviations of the mean, split up in a very predictable way:

About 68% of values fall within \(\pm 1\sigma\) of the mean.
About 95% fall within \(\pm 2\sigma\).
About 99.7% fall within \(\pm 3\sigma\).

Because the bell is symmetric, the leftover tail splits evenly. Outside \(\pm 2\sigma\) lies \(5\%\), so just \(2.5\%\) sits above \(+2\sigma\) and \(2.5\%\) below \(-2\sigma\). That single fact turns a z-score into a quick estimate of how rare a value is.

\[ \begin{aligned} \mu \pm 1\sigma &\;\approx\; 68\% \\ \mu \pm 2\sigma &\;\approx\; 95\% \\ \mu \pm 3\sigma &\;\approx\; 99.7\% \end{aligned} \]

The empirical rule: the share of a normal distribution captured within 1, 2, and 3 standard deviations of the mean.

📝 Worked example: Adult male heights are approximately normal with mean μ = 175 cm and standard deviation σ = 7 cm. How unusual is a height of 189 cm?

Standardize: \(z = (189 - 175)/7 = 14/7 = 2.0\). So 189 cm is exactly two standard deviations above the mean.
Apply the empirical rule: about 95% of values lie within \(\pm 2\sigma\), leaving 5% in the two tails combined — so roughly 2.5% sit above \(+2\sigma\).
Read the percentile: being above all but ~2.5% puts 189 cm near the 97.5th percentile.

✓ z = 2.0, so about 2.5% of the population is taller — 189 cm sits around the 97.5th percentile: genuinely tall, but not off-the-charts extreme.

📝 Worked example: Finishing times in Race A are normal with μ = 30 min, σ = 4 min; Maria finishes in 24 min. In a different Race B, times are normal with μ = 27 min, σ = 2 min; Ben finishes in 23 min. Faster (lower) times are better. Relative to their own races, who ran better?

Maria's z: \(z = (24 - 30)/4 = -6/4 = -1.5\) — she is 1.5 standard deviations faster than her race's average.
Ben's z: \(z = (23 - 27)/2 = -4/2 = -2.0\) — he is 2.0 standard deviations faster than his race's average.
Compare on the common scale: faster is better, so the more negative z wins. You cannot compare the raw 24 min and 23 min directly, because the two races had different means and spreads.

✓ Ben's z = −2.0 beats Maria's z = −1.5: relative to his own race he was further ahead of the pack. Standardizing is what lets you compare across two different normal distributions.

Check your understanding

1. A value has a z-score of −1.5. This tells you the value is…

A z-score is measured in standard deviations, and a negative sign means below the mean: z = −1.5 is 1.5 standard deviations below average.

2. Measurements are normal with μ = 80 and σ = 6. A measurement of 68 has a z-score of…

z = (x − μ)/σ = (68 − 80)/6 = −12/6 = −2. The value is two standard deviations below the mean.

3. In a normal distribution, about what percent of values fall within 2 standard deviations of the mean?

The empirical rule: roughly 68% lie within ±1σ, 95% within ±2σ, and 99.7% within ±3σ.

4. Plant A grew to z = +1.2 in its field; Plant B grew to z = +0.9 in a different field (different mean and spread). Relative to its own field, which plant is more above average?

Z-scores share a common scale — standard deviations from each field's own mean — so they can be compared directly. z = +1.2 is further above average than z = +0.9, so Plant A.

✅ Key takeaways

A z-score, z = (x − μ)/σ, is how many standard deviations a value sits from its mean — positive above, negative below.
Because units cancel, z-scores let you compare values from different normal distributions on one common scale.
The empirical rule for normal data: about 68% within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean.
The area under the normal curve to the left of a value is its percentile; that tail area depends only on the z-score.
The tails are symmetric, so e.g. only about 2.5% of a normal population lies beyond +2σ.