The normal curve
The normal distribution is a continuous distribution whose graph is the bell-shaped normal curve. It is described by its mean μ and standard deviation σ, written X ~ N(μ, σ²). Heights, measurement errors and many test scores are roughly normal.
Properties of the normal curve:
- It is symmetric about x = μ, and the mean, median and mode are equal.
- It is highest at the mean and falls off on both sides, approaching the x-axis but never touching it.
- The total area under it is 1; area between two values is the probability of X lying between them.
- Half the area lies on each side of the mean.
Standard normal variate
Any normal variable can be converted to the standard normal variate Z = (X − μ)/σ, which has mean 0 and standard deviation 1. Z tells you how many standard deviations X lies from the mean. Example: μ = 50, σ = 5, X = 60 gives Z = 2.
Areas under the standard normal curve are read from tables, usually as P(0 < Z < z). Using symmetry:
- P(Z > 0) = P(Z < 0) = 0.5
- P(Z < −a) = P(Z > a)
- P(Z > a) = 0.5 − P(0 < Z < a)
Empirical (68–95–99.7) rule
- About 68.27% of values lie within μ ± σ.
- About 95.45% lie within μ ± 2σ.
- About 99.73% lie within μ ± 3σ.
Example
Marks of 1000 students are normal with μ = 60, σ = 10. For X = 80, Z = 2. With P(0 < Z < 2) = 0.4772, P(Z > 2) = 0.5 − 0.4772 = 0.0228, so about 23 students score above 80.
Key points
- Z = (X − μ)/σ has mean 0 and SD 1.
- The curve is symmetric, so areas on the left mirror areas on the right.
- About 95% of values lie within 2 SD of the mean.
- Probabilities equal areas; the probability of a single exact value is 0.
Common mistakes
- Dividing by the variance instead of the standard deviation.
- Forgetting to subtract the table value from 0.5 for a tail area.