Statistics · Probability Distributions
Normal Distribution Explained
Why the bell curve appears everywhere in nature, economics and social science — and how to use it to calculate probabilities.
Human heights, IQ scores, measurement errors, financial returns, and blood pressure readings all share a remarkable property: they cluster symmetrically around a central value, with fewer observations in the extremes. This pattern — the bell curve — is the normal distribution, and it is the single most important probability distribution in all of statistics.
📘 Key Term
The normal distribution is a continuous probability distribution that is symmetric, bell-shaped, and completely described by two parameters: its mean (μ) — which determines where the centre lies — and its standard deviation (σ) — which determines how wide or narrow the bell is. We write X ~ N(μ, σ²).
Properties of the Normal Distribution
✅ Symmetric about the mean (μ = median = mode)
✅ Bell-shaped curve that extends to ±∞
✅ Total area under the curve = 1 (represents 100% of probability)
✅ 68% of data within μ ± σ; 95% within μ ± 2σ; 99.7% within μ ± 3σ
✅ Defined by exactly two parameters: μ and σ²
The Standard Normal Distribution and Z-Scores
Any normal distribution can be converted to the standard normal distribution Z ~ N(0, 1) using the Z-score formula:
Z = (X − μ) / σ
The Z-score tells you how many standard deviations a value is above or below the mean. A Z-score of +2 means the value is 2 standard deviations above the mean; a Z-score of −1.5 means 1.5 standard deviations below it.
Worked Example: Adult Heights
Suppose adult male heights are normally distributed with μ = 175cm and σ = 8cm.
Q: What proportion of men are taller than 191cm?
Step 1: Calculate Z = (191 − 175) / 8 = 16/8 = 2.0
Step 2: From Z-table, P(Z < 2.0) = 0.9772
Step 3: P(X > 191) = 1 − 0.9772 = 2.28% of men are taller than 191cm
💡 Key Insight
The normal distribution’s ubiquity is explained by the Central Limit Theorem: regardless of how the underlying population is distributed, the distribution of sample means approaches normality as sample size increases. This is why the normal distribution applies to so many real-world averages and aggregated measurements.
⚠️ Common Error
The normal distribution assumes the variable is continuous and can theoretically take any value from −∞ to +∞. It is not appropriate for discrete variables (e.g. number of children), heavily skewed data (e.g. incomes), or data with natural boundaries (e.g. test scores capped at 100). Always check the appropriateness of normality before applying normal distribution methods.
Q1. Exam scores are normally distributed with a mean of 68 and a standard deviation of 10. What percentage of students scored between 58 and 88? [5 marks]
Answer: Z for 58: (58−68)/10 = −1.0. Z for 88: (88−68)/10 = +2.0. P(Z < 2.0) = 0.9772; P(Z < −1.0) = 0.1587. P(58 < X < 88) = 0.9772 − 0.1587 = 0.8185 = 81.85% of students scored between 58 and 88.
References
1. Wackerly, D., Mendenhall, W. and Scheaffer, R.L. (2008) Mathematical Statistics with Applications. Cengage.
2. Moore, D.S., McCabe, G.P. and Craig, B.A. (2021) Introduction to the Practice of Statistics. W.H. Freeman.
3. Hogg, R.V., McKean, J. and Craig, A.T. (2019) Introduction to Mathematical Statistics. Pearson.
4. Triola, M.F. (2022) Elementary Statistics. Pearson.
5. DeGroot, M.H. and Schervish, M.J. (2012) Probability and Statistics. Pearson.