The normal distribution is a symmetric, bell-shaped distribution centered on its mean. Values produced by many small factors adding up, such as height, measurement error and many test scores, often look close to normal. In addition, the sample mean becomes approximately normal for large samples regardless of the original distribution (the central limit theorem), which is why many confidence intervals and tests are built on the normal distribution.
Two numbers define it
A normal distribution is fully determined by its mean μ and standard deviation σ. It is written N(μ, σ²) (the second slot is the variance).
- μ sets the center of the bell.
- σ sets its width. A larger σ gives a lower, wider bell; a smaller σ a taller, narrower one.
- The mean, median and mode are all equal to μ.
- The total area under the curve is 1, and the area over an interval is the probability of landing in it.
z-scores: putting different scales on one
A z-score tells how many standard deviations a value is from the mean.
z = (x − μ) / σ
Converting to z-scores turns any normal distribution into the standard normal distribution N(0, 1), with mean 0 and standard deviation 1. That is why one table, or one calculator, gives probabilities for every normal distribution. z-scores have no units, so they also let you compare results on different scales. A score of 85 on a test with mean 70 and SD 10 (z = 1.5) is relatively better than 80 on a test with mean 60 and SD 20 (z = 1.0).
The 68–95–99.7 rule
Intervals measured in standard deviations always contain the same share of a normal distribution.
| Interval | Share inside (exact) | Share outside |
|---|---|---|
| μ ± 1σ | about 68.27% | about 31.73% |
| μ ± 2σ | about 95.45% | about 4.55% |
| μ ± 3σ | about 99.73% | about 0.27% |
The rule is a mental shortcut. The interval that contains exactly 95% is ±1.96σ, not ±2σ, which is where the 1.96 in confidence intervals comes from. Likewise, 90% is ±1.645σ and 99% is ±2.576σ.
Reading cumulative probabilities
The basic quantity in normal calculations is the cumulative probability P(X ≤ x), the probability of being at or below x. For the standard normal it is written Φ(z). Every other probability is built from it.
- At or above x:
P(X ≥ x) = 1 − Φ(z) - Between a and b:
P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a) - Symmetry:
Φ(−z) = 1 − Φ(z)
In a continuous distribution a single point has probability 0, so ≤ versus < does not change the result.
Example 1: Probability below a score
Suppose test scores follow a normal distribution with mean 70 and standard deviation 10.
Probability of scoring 85 or less
- z-score:
z = (85 − 70) / 10 = 1.5 - Cumulative probability:
Φ(1.5) ≈ 0.9332 - Interpretation: about 93.32% of test takers score 85 or less, so 85 is in the top 6.68% or so.
Probability of scoring between 60 and 85
- z-score of 60:
(60 − 70) / 10 = −1.0→Φ(−1.0) ≈ 0.1587 - z-score of 85:
1.5→Φ(1.5) ≈ 0.9332 - Difference:
0.9332 − 0.1587 ≈ 0.7745 - Interpretation: about 77.45% of test takers score between 60 and 85.
Probability of scoring 55 or less
The z-score is −1.5, and by symmetry Φ(−1.5) = 1 − 0.9332 ≈ 0.0668, exactly the same as the probability of scoring 85 or more.
Example 2: Working back from a top-x% cutoff
Now go the other way, from a probability to a score. "What score puts you in the top 10%?" asks for the score with cumulative probability 0.90.
- Find the z-value with cumulative probability 0.90:
z = Φ⁻¹(0.90) ≈ 1.2816 - Convert back to the original scale:
x = μ + z × σ = 70 + 1.2816 × 10 ≈ 82.82 - Interpretation: a score of about 82.82 or higher is in the top 10%.
In the same way, the top 5% uses z ≈ 1.6449, giving 70 + 16.449 ≈ 86.45. For a bottom x% cutoff, use cumulative probability x directly.
| Cutoff | Cumulative probability | z | Score (μ=70, σ=10) |
|---|---|---|---|
| Top 25% | 0.75 | about 0.6745 | about 76.74 |
| Top 10% | 0.90 | about 1.2816 | about 82.82 |
| Top 5% | 0.95 | about 1.6449 | about 86.45 |
| Top 1% | 0.99 | about 2.3263 | about 93.26 |
Example 3: Applying it to a sample mean
Normal calculations also work for sample means, not just individual scores. On the same test (mean 70, SD 10), pick 25 students at random. What is the probability that their average is 74 or higher?
- Standard deviation of the sample mean (standard error):
σ / √n = 10 / √25 = 2 - z-score:
z = (74 − 70) / 2 = 2.0 - Probability:
1 − Φ(2.0) ≈ 0.0228
A single student scoring 74 or more is common, 1 − Φ(0.4) ≈ 0.3446, but a 25-student average of 74 or more happens only about 2.28% of the time. Averages vary far less than individual values, and this is the starting point for confidence intervals and hypothesis tests.
Cautions
- Check the normality assumption. Applying a normal model to right-skewed data such as income or waiting time gets tail probabilities badly wrong. Look at a histogram or Q-Q plot first.
- Bounded data (such as a 0 to 100 test) can depart from normality near the ends. Be especially careful when working back to extreme cutoffs like the top 1%.
- Separate parameters from estimates. If μ and σ were estimated from a sample, the results carry that estimation error too.
Enter a mean and standard deviation into this site's normal distribution calculator to get cumulative probabilities, interval probabilities and inverse values instantly.
Key takeaways
- A normal distribution is set by μ and σ, and
z = (x − μ) / σconverts it to the standard normal. - About 68.27%, 95.45% and 99.73% fall within μ ± 1σ, 2σ and 3σ; the exact 95% interval is ±1.96σ.
- With mean 70 and SD 10, P(score ≤ 85) ≈ 0.9332 and P(60 ≤ score ≤ 85) ≈ 0.7745.
- The standard error of a 25-student mean is
10 / √25 = 2, and P(mean ≥ 74) ≈ 0.0228. - A top-10% cutoff comes from the inverse cumulative probability:
70 + 1.2816 × 10 ≈ 82.82. - Do not apply the normal model blindly to skewed data.