Worked example: IQ scores
IQ scores are designed to be normally distributed with mean μ = 100 and standard deviation σ = 15.
- P(85 < X < 115): z = −1 and z = 1, so the probability is 0.8413 − 0.1587 = 0.6827. About 68% of people score within 15 points of 100.
- P(X > 130): z = (130 − 100) ÷ 15 = 2, so the probability is 1 − 0.9772 = 0.0228, about 2.3%.
- Top 5%: the value with 95% below it is 100 + 1.6449 × 15 ≈ 124.7.
The normal distribution in one paragraph
The normal distribution is the symmetric, bell-shaped curve that describes many natural measurements: heights, measurement errors, test scores and more. It is fully described by two numbers: the mean μ (its center) and the standard deviation σ (its width). The total area under the curve is 1, so the area over any range is the probability of a value falling in that range.
Check that your data is roughly normal
These probabilities are only accurate if the data really is close to normal. Draw a histogram with the normal curve switched on to check. If the data is clearly skewed, describe it with the median and IQR instead, using the five-number summary calculator.
The 68–95–99.7 rule
Another example: heights
Adult heights in a population are roughly normal with μ = 170 cm and σ = 9 cm. The share of adults under 160 cm is P(X < 160) = Φ((160 − 170) ÷ 9) = Φ(−1.11) ≈ 0.133, about 13%. The height that only 5% exceed is 170 + 1.6449 × 9 ≈ 184.8 cm.
Common mistakes to avoid
- Using it for clearly skewed data. Normal probabilities can be badly wrong for incomes, wait times and other skewed data.
- Mixing up “above” and “below”. P(X > a) = 1 − P(X < a).
- Using the variance instead of the standard deviation. σ is the square root of the variance.
- Expecting P(X = a) to be positive. For a continuous distribution, any exact single value has probability 0.
Do it in other software
| Software | How |
|---|---|
| Excel / Google Sheets | =NORM.DIST(a, mean, sd, TRUE) for P(X < a); =NORM.INV(p, mean, sd) for the inverse. |
| TI-84 | 2nd › DISTR › normalcdf(lower, upper, μ, σ); invNorm(p, μ, σ). |
| Python | scipy.stats.norm.cdf(a, loc=mean, scale=sd); norm.ppf(p, loc=mean, scale=sd). |
Related tools: Z-Score Calculator, Histogram Maker, Standard Deviation Calculator, Descriptive Statistics Calculator and Mean, Median & Mode Calculator.