What is a normal distribution?
A normal distribution is a symmetrical, bell-shaped curve set by two numbers: the mean, where the peak is, and the standard deviation, how spread out it is. The area under the curve is 1, so the area over a stretch of values is the chance a value lands there (“less than 65” and “65 or less” are the same).
Source OpenStax · Introductory Statistics 2e, 6.1 The Standard Normal Distribution ↗OpenStax · Introductory Statistics 2e, 6.2 Using the Normal Distribution ↗
How to calculate normal distribution probability
Here it is with your numbers:
- z = (65 − 63) ÷ 5
- z = 0.4
- P(X < 65) is the area left of z.
- P(X < 65) = 0.6554
It means a value picked at random has a 65.54% chance of being below 65.
In Excel or Google Sheets, =NORM.DIST(x, mean, standard deviation, TRUE) gives the probability below x, and =NORM.INV(percent, mean, standard deviation) the value at a percentile (0.9 for 90%).
Source Microsoft Support · NORM.DIST function ↗Microsoft Support · NORM.INV function ↗
The 68 – 95 – 99.7 rule for your numbers
About 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. For the numbers above:
| Standard deviations | Range | Share of values |
|---|---|---|
| Within 1 | 58 – 68 | 68.27% |
| Within 2 | 53 – 73 | 95.45% |
| Within 3 | 48 – 78 | 99.73% |
FAQs
What is the formula for calculating the normal probability distribution?
Turn the value into a z-score, z = (x − mean) ÷ standard deviation; the probability of a value below x is the area under the bell curve left of z. With a mean of 63 and a standard deviation of 5, 65 has a z-score of 0.4 and 0.6554 of the curve is below it.
Source OpenStax · Introductory Statistics 2e, 6.2 Using the Normal Distribution ↗
How to find probability from a normal distribution?
Below x is the area left of x; for above x, take that from 1: P(X > x) = 1 − P(X < x). With a mean of 63 and a standard deviation of 5, the chance of more than 65 is 0.3446.
What is the 68 %- 95 %- 99.7 rule?
The empirical rule: about 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3 (exactly 68.27%, 95.45% and 99.73%). With a mean of 63 and a standard deviation of 5, 99.7% of values are between 48 and 78.
Source OpenStax · Introductory Statistics 2e, 6.1 The Standard Normal Distribution ↗
How do you know when to use PDF or CDF?
Use the CDF (cumulative distribution function) for probabilities: the area below x. The PDF (probability density function) is the height of the curve at x, not a probability. In Excel, NORM.DIST(42, 40, 1.5, TRUE) is the CDF, 0.9087888.
Sources
- OpenStax: Introductory Statistics 2e, 6.1 The Standard Normal Distribution
- OpenStax: Introductory Statistics 2e, 6.2 Using the Normal Distribution
- Microsoft Support: NORM.DIST function
- Microsoft Support: NORM.INV function