The Normal Distribution and z-Scores

Key Terms

Normal distribution: A symmetric, bell-shaped distribution completely described by its mean (μ) and standard deviation (σ).
68-95-99.7 rule: About 68% of data falls within 1σ of the mean; 95% within 2σ; 99.7% within 3σ.
z-score: z = (x − μ) / σ — measures how many standard deviations x is from the mean; z > 0 is above the mean.
Standardising: Converting a raw score to a z-score allows comparison across different normal distributions.
Symmetry: The normal curve is symmetric about μ; P(X < μ) = P(X > μ) = 0.5.
CAS use: Use the Normal CDF (cumulative distribution function) on ClassPad to find probabilities for a given range of x values.

The Normal Distribution

68–95–99.7% Rule (empirical rule for a normal distribution):
• 68% of values lie within μ ± σ (1 standard deviation)
• 95% of values lie within μ ± 2σ (2 standard deviations)
• 99.7% of values lie within μ ± 3σ (3 standard deviations)

z-score formula: z = (x − μ) / σ (QCAA Formula Sheet)
A z-score measures how many standard deviations a value is above or below the mean.

Properties of the Normal Distribution

Property	Detail
Shape	Bell-shaped, symmetric about the mean
Mean = Median = Mode	All three are equal (at the centre of the bell)
z = 0	At the mean
Positive z	Value is above the mean
Negative z	Value is below the mean

Worked Example 1 — Using the 68–95–99.7% Rule

Exam scores: μ = 60, σ = 10. What percentage of students scored between 50 and 70?

Step 1: Identify the range in terms of standard deviations.
50 = 60 − 10 = μ − σ and 70 = 60 + 10 = μ + σ

Step 2: The range μ ± σ contains 68% of all values.

Answer: Approximately 68% of students scored between 50 and 70.

Worked Example 2 — Calculating and Using z-Scores

Physics test: μ = 65, σ = 8. Chemistry test: μ = 70, σ = 10.
Alex scored 77 on Physics and 84 on Chemistry. Which was the better relative performance?

Physics z-score: z = (77 − 65) / 8 = 12/8 = 1.5

Chemistry z-score: z = (84 − 70) / 10 = 14/10 = 1.4

Conclusion: Alex’s Physics result (z = 1.5) was slightly better relative to the class than Chemistry (z = 1.4), even though the raw Chemistry score was higher.

Hot Tip: The 68–95–99.7% rule only applies to normally distributed data. For one-tailed questions (e.g. “what % scored above 80?”), use symmetry: if 95% lie between μ − 2σ and μ + 2σ, then 5% lie outside this range, and by symmetry 2.5% lie above μ + 2σ.

Full Lesson: The Normal Distribution and z-Scores

What is the Normal Distribution?

The normal distribution (or “bell curve”) describes data that clusters symmetrically around a central mean. Many naturally occurring measurements follow this pattern: heights, birth weights, IQ scores, measurement errors, and exam results in large populations.

Key visual features of the bell curve:
• Highest point is at the mean
• Symmetrical — equal area on both sides of the mean
• Tails extend infinitely but approach zero
• The total area under the curve = 1 (or 100%)

Using the 68–95–99.7% Rule

The rule lets us answer percentage questions without tables or calculators:

Strategy for one-sided questions:
If 95% lie within μ ± 2σ, then 5% lie outside (by symmetry, 2.5% above and 2.5% below).
If 68% lie within μ ± σ, then 32% lie outside (16% above μ+σ, 16% below μ−σ).

Half-rule: 50% of values lie above the mean, 50% below (symmetry).
So: percentage above μ+σ = (100% − 68%) / 2 = 16%.

z-Scores for Comparing Data from Different Distributions

A z-score standardises a value by expressing it as a number of standard deviations from the mean. This allows fair comparison across different tests, groups, or measurements.

z = (x − μ) / σ → rearranged: x = μ + zσ

This rearrangement is useful when you know the z-score and want the original value.

The Statistical Investigation Process

The normal distribution is a key tool in statistical investigations. A complete investigation follows: formulate a question → collect/obtain data → analyse using appropriate statistics → interpret and communicate conclusions. When data is approximately normal, z-scores and the 68–95–99.7% rule provide powerful analytical tools.

Lesson Tip: When using the z-score formula, always identify μ and σ for the specific distribution you’re working with. A z-score by itself is meaningless without knowing which distribution it comes from. z = +2 means “2 standard deviations above the mean of that distribution.”

Mastery Practice

See Answers ➔

Fluency
IQ scores are normally distributed with μ = 100, σ = 15. What percentage of people have an IQ between 85 and 115?
Fluency
Heights of adult women: μ = 165 cm, σ = 6 cm. What percentage of women are taller than 177 cm?
Fluency
Calculate the z-score for x = 78 given μ = 70 and σ = 8.
Fluency
A z-score is −1.5. If μ = 50 and σ = 10, find the original value x.
Understanding
Test scores are normally distributed with μ = 65, σ = 10. Between what two values do the middle 95% of scores fall?
Understanding
Maya scored 82 on a Biology test (μ = 75, σ = 7) and 76 on a Chemistry test (μ = 70, σ = 8). Which was the better performance relative to the class? Use z-scores to justify your answer.
Understanding
Normal body temperature: μ = 36.8°C, σ = 0.4°C. A patient has a temperature of 38.0°C.
1. Calculate the z-score.
2. Using the 99.7% rule, is this temperature unusually high? Explain.
Understanding
A normally distributed dataset has μ = 200, σ = 25. Approximately what percentage of values lie: (a) above 250? (b) below 150? (c) between 175 and 225?
Problem Solving
A machine fills cereal boxes with mean fill μ = 500 g, σ = 8 g.
1. What percentage of boxes contain between 484 g and 516 g?
2. The machine is recalibrated so that μ = 510 g while σ stays at 8 g. What percentage of boxes now contain less than 494 g?
3. Under which setting is the machine performing better from a quality-control perspective? Explain.
Problem Solving
University entrance requires a minimum z-score of 0.5. Entrance exam results: μ = 68%, σ = 12%.
1. Calculate the minimum mark (raw score) needed to gain entry.
2. Jordan scored 74%. Will Jordan be admitted? Calculate their z-score.
3. Approximately what percentage of students achieve a z-score above 0.5? (Note: by symmetry, 50% are above the mean, and 34% are between the mean and μ + σ, so about 16% are above μ + σ. Use this reasoning to estimate.)