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VCE General Mathematics (Year 12) · Data Analysis · 14m read

The Normal Distribution and z-Scores

Was 72 in English better than 80 in Maths? Raw marks cannot tell you — the tests had different averages and spreads. The normal distribution and z-scores let you put any value on a common ruler, and the 68–95–99.7% rule turns a picture of a bell curve into instant percentages. Expect at least one question on this in every exam.

Learning objective: Apply the 68–95–99.7% rule and standardised scores to compare and interpret values from normal distributions.

By the end of this lesson you can:

  • I use the 68–95–99.7% rule to find percentages.
  • I calculate and interpret z-scores.
  • I compare values from different distributions using z-scores.

The normal curve is a bell. Most values ring near the middle; fewer at the edges. The 68–95–99.7 rule tells you how much of the bell lies within 1, 2, 3 SDs.

Bell curves and standard scores

NORMAL DISTRIBUTION:
• Symmetric, bell-shaped, single peak.
• Mean = median = mode, all at the centre.
• Described completely by its mean (x̄) and standard deviation (s).

THE EMPIRICAL RULE (68-95-99.7):
• About 68% of values lie within 1 SD of the mean.
• About 95% lie within 2 SD.
• About 99.7% lie within 3 SD.

SPLITTING THE PERCENTAGES (by symmetry):
• Mean to +1 SD: 34%.
• +1 SD to +2 SD: 13.5%.
• +2 SD to +3 SD: 2.35%.
• Beyond +3 SD: 0.15%.
• Above the mean: 50%.

STANDARDISED (z) SCORES:
• z tells you how many standard deviations a value is from the mean.
• Positive z → above the mean; negative z → below.
• z lets you compare values from DIFFERENT distributions.

Standard scores

  • z-score: z = \dfrac{x - \bar{x}}{s}
  • Rearranged: x = \bar{x} + z \times s

Quick reference

| Concept | Formula/rule |
|---|---|
| z-score | z = (x − mean) ÷ SD |
| Within 1 SD | 68% |
| Within 2 SD | 95% |
| Within 3 SD | 99.7% |
| Above +2 SD | 2.5% |
| Compare scores | Higher z = relatively better |

Key ideas at a glance

  • Use z to compare different scales: Standardise.
  • Symmetric bell: Halve tail percentages.
  • Negative z: Below mean.
  • Check normality: Before using 68–95–99.7.

Worked example (Medium)

Heights of adult men are approximately normal with mean 170 cm and SD 8 cm. (a) What percentage are taller than 186 cm? (b) What percentage are between 162 cm and 186 cm?

  1. Standardise 186. z = (186 − 170) ÷ 8 = 2.
  2. (a). Beyond +2 SD: (100% − 95%) ÷ 2 = 2.5%.
  3. Standardise 162. z = (162 − 170) ÷ 8 = −1.
  4. (b). −1 to 0 is 34%; 0 to +2 is 47.5%. Total 34% + 47.5% = 81.5%.

Answer: (a) 2.5% (b) 81.5%

Worked example (Easy)

Mia scored 72 in English (mean 65, SD 5) and 80 in Maths (mean 75, SD 10). In which subject did she perform better relative to her class?

  1. English z. (72 − 65) ÷ 5 = 1.4
  2. Maths z. (80 − 75) ÷ 10 = 0.5
  3. Compare. 1.4 > 0.5, so English is relatively better despite the lower raw mark.

Answer: English (z = 1.4 vs z = 0.5).

Common mistakes

  • Using 95% for "above 2 SD" 95% is the middle band; only the tails are outside it. Fix: Above +2 SD = (100 − 95) ÷ 2 = 2.5%.
  • Comparing raw scores across tests Different means and SDs make raw marks incomparable. Fix: Convert both to z-scores first.
  • Applying the rule to skewed data The percentages only hold for (approximately) normal distributions. Fix: Check the question says the data are normal or bell-shaped.

Standardising exam scores

VCE study scores are standardised so a score of 30 represents roughly the same relative position in every subject (mean ≈ 30, SD ≈ 7).

z-scores make it fair to compare results measured on different scales.

Practice questions

  1. Battery life is normal with mean 20 hours and SD 2 hours. What percentage last less than 16 hours?

    1. 2.5%
    2. 5%
    3. 16%
    4. 0.15%

    (1 marks)

  2. A student has z = −1.5 on a test with mean 60 and SD 6. What was their mark?

    1. 51
    2. 54
    3. 57
    4. 69

    (1 marks)

  3. In a group of 400 normally distributed results, how many would you expect between 1 SD below and 2 SD above the mean?

    (2 marks)

Summary

  • 68–95–99.7% of values lie within 1, 2, 3 SD of the mean.
  • Split the rule symmetrically: 34, 13.5, 2.35, 0.15.
  • z = (x − mean) ÷ SD.
  • Use z-scores to compare across distributions.

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