- Define and explain The normal distribution in your own words
- Use key terms such as Normal distribution accurately
- Apply what you have learned to new examples and questions
- Avoid the common mistakes learners make with this topic
This lesson focuses on The normal distribution: the bell curve, standardisation with z-scores, and finding probabilities from tables or technology.
The bell curve, standardisation with z-scores, and finding probabilities from tables or technology.
Key ideas
Standardising the normal
Convert X ~ N(μ, σ²) to Z ~ N(0, 1) via z = (x − μ)/σ, then read probabilities from tables. The total area under the curve is 1, and the curve is symmetric about μ, so the probability that X falls below the mean μ is 0.5.
Finding probabilities from the curve
Because the total area under the curve is 1, areas are probabilities. Use symmetry first: P(X greater than μ) = 0.5, and P(μ − a less than X less than μ + a) is symmetric. For other values, standardise with z = (x − μ)/σ and read the probability from tables or a calculator. The empirical rule is a handy check: about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
Key term — Normal distribution: A continuous symmetric bell-shaped distribution, fully described by its mean μ and variance σ², written X ~ N(μ, σ²).
X ~ N(100, 15²). Find P(X less than 115).
Standardise: z = (115 − 100)/15 = 1. From tables, P(Z less than 1) ≈ 0.8413.
Answer: P(X less than 115) ≈ 0.8413.
- Using the variance where the standard deviation belongs The z-score formula needs σ, not σ²: for X ~ N(100, 15²), σ = 15, so z = (x − 100)/15.
- Answering P(X less than x) when the question asks P(X greater than x) Read carefully: P(X greater than x) = 1 − P(X less than x). Symmetry only gives you 0.5 exactly at the mean.
Practice
z = (130 − 100)/15 = 2.
z = (65 − 50)/10 = 1.5.
About 95%.
0.5 — half the area lies above the mean.
Quick check
Which of these best defines "Normal distribution"?
X ~ N(100, 15²). What is P(X less than 100)?
- The normal distribution: the bell curve, standardisation with z-scores, and finding probabilities from tables or technology.
- Standardising the normal: Convert X ~ N(μ, σ²) to Z ~ N(0, 1) via z = (x − μ)/σ, then read probabilities from tables.
- Finding probabilities from the curve: Areas are probabilities; use symmetry, z = (x − μ)/σ with tables, and the 68–95–99.7 rule as a check.
- Watch out for: using the variance where the standard deviation belongs