What the normal curve describes
The normal distribution is a symmetric curve, tallest at the middle and falling fast to both sides. The NIST statistics handbook writes its density with two parameters: μ, the location parameter, which says where the center is, and σ, the scale parameter, which says how wide the curve is. The case μ = 0 and σ = 1 is called the standard normal. Changing μ slides the curve sideways without altering its shape; increasing σ flattens and widens it, because the same total area (which is always 1) has to spread out further.
f(x) = exp( −(x − μ)² ÷ (2σ²) ) ÷ ( σ × √(2π) )- μ
- the mean, the center of the curve.
- σ
- the standard deviation, which controls the width.
- x
- the value whose curve height you want.
View the data
| x | σ = 1 | σ = 2 |
|---|---|---|
| -4 | 0 | 0.03 |
| -3 | 0 | 0.06 |
| -2 | 0.05 | 0.12 |
| -1 | 0.24 | 0.18 |
| 0 | 0.4 | 0.2 |
| 1 | 0.24 | 0.18 |
| 2 | 0.05 | 0.12 |
| 3 | 0 | 0.06 |
| 4 | 0 | 0.03 |
Z-score: measuring everything in standard deviations
To use a standard normal table, every value has to be converted. The NIST table page says exactly that: standardize your value by subtracting the mean and dividing by the standard deviation, then look up the result. That result is the z-score, and it has no unit: it is "how many standard deviations above (positive) or below (negative) the mean".
z = (x − μ) ÷ σ| Value x | z | Percentile (P(X ≤ x)) | Above x |
|---|---|---|---|
| 70 | −2 | 2.28% | 97.72% |
| 85 | −1 | 15.87% | 84.13% |
| 130 | 2 | 97.72% | 2.28% |
The real power of z shows when comparing different scales. Suppose 82 points on an exam with mean 70 and standard deviation 8, and 640 points on another with mean 500 and standard deviation 100. In raw numbers you cannot say which was better. In z terms: the first result has z = (82 − 70) ÷ 8 = 1.5 (93.32nd percentile) and the second has z = (640 − 500) ÷ 100 = 1.4 (91.92nd percentile). The first score was relatively better, even though it looks "smaller".
The 68-95-99.7 rule
The empirical rule is the normal’s mental shortcut: about 68% of values fall within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. The exact values, which the calculator returns, are 68.27%, 95.45% and 99.73%. Applied to the scale with mean 100 and standard deviation 15, the rule becomes three concrete ranges.
| Range | Interval | Rule | Exact |
|---|---|---|---|
| μ ± 1σ | 85 to 115 | 68% | 68.27% |
| μ ± 2σ | 70 to 130 | 95% | 95.45% |
| μ ± 3σ | 55 to 145 | 99.7% | 99.73% |
Areas, percentiles and computed examples
Every probability question on the normal follows the same path: standardize, choose the region (below, above, between or outside) and read the area. The inverse exists too: given a percentile, find the value. The examples use exam scores with mean 70 and standard deviation 10, and the scale with mean 100 and standard deviation 15.
- Who scores above 85?z = (85 − 70) ÷ 10 = 1.5. The area to the right of 1.5 is 6.68%. About 1 in 15 students.
- Who scores above 60?z = (60 − 70) ÷ 10 = −1. The area to the right of −1 is 84.13%, the complement of the 15.87th percentile.
- What is the cutoff for the top 10%?Find the z that leaves 10% on the right: 1.2816. Then x = 70 + 1.2816 × 10 = 82.82.
- Between which values do the central 95% fall (μ = 100, σ = 15)?z = ±1.96, so from 100 − 1.96 × 15 = 70.60 to 100 + 1.96 × 15 = 129.40.
- What is the chance of landing between 90 and 110 (μ = 100, σ = 15)?z runs from −0.667 to 0.667. The area between them is 49.50%, well below the 68% for ±1σ, because the interval is narrower than 1 standard deviation.
Try your own values in the tool below: it draws the shaded area, shows the z of each limit and also solves the inverse direction (from percentile to value).
P-values and critical values
In a z test, the p-value is the area in the tails beyond the observed statistic. In a two-sided test both tails count; in a one-sided test only one does. A z of 1.96 leaves 2.5% in each tail, hence the two-sided p of 0.05. A z of 2.5 gives p = 0.0124; a z of 1.8 in a right-tailed one-sided test gives p = 0.0359. The critical values below are the ones that come up most.
| Confidence | α | Two-sided z | One-sided z |
|---|---|---|---|
| 90% | 0.10 | 1.645 | 1.282 |
| 95% | 0.05 | 1.960 | 1.645 |
| 99% | 0.01 | 2.576 | 2.326 |
| 99.9% | 0.001 | 3.291 | 3.090 |
When the population standard deviation is unknown and the sample is small, the reference becomes Student’s t distribution, with heavier tails. With a statistic of 2.1 and 9 degrees of freedom, the two-sided p is 0.0651 and the test does not reject at 5%. The same statistic read as a z would give 0.0357 and would reject. Choosing the wrong distribution changes the conclusion. The hypothesis test calculator helps set up the full test.
Common mistakes
Assuming the 68-95-99.7 rule works for any data
It describes the normal curve, not data in general. Check the shape with a histogram. On skewed data, the share within 2 standard deviations can be quite different from 95%.
Mixing up two-sided and one-sided
The critical z at 5% is 1.96 two-sided and 1.645 one-sided. Using the wrong one doubles or halves your real significance level. Decide the direction of the test before looking at the data.
Reading the curve’s height as a probability
Density can exceed 1 on narrow curves and is not a probability. Probability is always area. The chance of an exact value is zero; only intervals have a chance.
Using the wrong standard deviation
The z of one observation uses the standard deviation of the data. The z of a sample mean uses the standard error, σ ÷ √n. Swapping one for the other makes a result look far more (or far less) extreme than it is.
Treating a "large z" as certainty
A z of 3 leaves 0.27% in the two tails combined. Rare is not impossible: among thousands of observations, a few z values above 3 are expected. A low p-value does not prove the effect is large or that the hypothesis is false.
Frequently asked questions
What is a z-score?
How does the 68-95-99.7 rule work?
What is the difference between z and t?
Why does 1.96 appear so often?
Can I use the normal with any data set?
The normal distribution boils down to a mean, a standard deviation and areas under a curve. The z-score converts any value into standard deviations and lets you compare different scales; the 68-95-99.7 rule gives the intuition (with exact values of 68.27%, 95.45% and 99.73%); and p-values and critical values are just tail areas. The essential care is to check that the data really are approximately normal, to choose one or two tails before looking at the result, and not to confuse standard deviation with standard error. With that, probability questions stop being a recipe and become a calculation you can explain.