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Mathematics Standard · Year 12 · Module 5

The Empirical Rule

68–95–99.7: unlock the power of standard deviations in a normal distribution.

MS-S5 Lesson 10 ~35 min

Think First

In a class where exam marks are normally distributed with mean 70 and standard deviation 10, roughly what percentage of students would you expect to score between 60 and 80? How did you arrive at that estimate?

See key ideas

60 and 80 are each exactly one standard deviation from the mean (70 ± 10). The empirical rule tells us that approximately 68% of data lies within one standard deviation of the mean. So about 68% of students scored between 60 and 80.

Learning Intentions

Key Terms

Empirical rule
For normal distributions: 68% within 1σ, 95% within 2σ, 99.7% within 3σ of the mean.
68–95–99.7 rule
The three key percentage thresholds of the empirical rule, memorised in this order.
Within k standard deviations
The interval (μ − kσ, μ + kσ), symmetric about the mean.
Unusual value
A value more than 2 standard deviations from the mean (in the outer 5% of the distribution).
01

The Empirical Rule: 68–95–99.7

For any normally distributed dataset with mean μ and standard deviation σ:

Interval Approximate percentage of data
μ − σ to μ + σ (within 1 standard deviation) 68%
μ − 2σ to μ + 2σ (within 2 standard deviations) 95%
μ − 3σ to μ + 3σ (within 3 standard deviations) 99.7%
Memory trick: 68, 95, 99.7 — increasing by roughly double each time. The remaining percentages outside each band are 32%, 5%, and 0.3%.

Book Notes

Write the three rows of the empirical rule table from memory. Also note the "outside" percentages: 32%, 5%, 0.3%.

Quick check: What percentage of normally distributed data lies within 2 standard deviations of the mean?

02

Applying the Empirical Rule

To apply the empirical rule, first identify how many standard deviations each boundary is from the mean.

Worked example: IQ scores are normally distributed with μ = 100, σ = 15. What percentage of people have IQ between 70 and 130?

  1. Find the distance from the mean: 100 − 70 = 30 = 2σ and 130 − 100 = 30 = 2σ.
  2. So the interval (70, 130) is within 2 standard deviations of the mean.
  3. By the empirical rule: approximately 95% of people have IQ between 70 and 130.

Worked example 2: What percentage have IQ between 85 and 115?

  1. 100 − 85 = 15 = 1σ and 115 − 100 = 15 = 1σ.
  2. Interval (85, 115) is within 1 standard deviation.
  3. Approximately 68%.

Book Notes

Write the 3-step process: (1) subtract each boundary from μ, (2) divide by σ to find number of SDs, (3) apply 68/95/99.7 rule.

Quick check: A dataset has μ = 50, σ = 5. What percentage lies between 35 and 65?

03

One-Sided Intervals Using Symmetry

The empirical rule gives two-sided (symmetric) percentages. Use symmetry to find one-sided percentages.

Key facts derived from symmetry:

Worked example: μ = 100, σ = 15. What percentage have IQ above 130?

  1. 130 = μ + 2σ, so we want the percentage above μ + 2σ.
  2. 2.5% is above μ + 2σ (from 5% ÷ 2).
  3. Answer: approximately 2.5%.

Book Notes

Create a table with columns "beyond 1σ", "beyond 2σ", "beyond 3σ" and rows "total" and "one side". Fill in: 32%/16%, 5%/2.5%, 0.3%/0.15%.

Quick check: In a normal distribution with μ = 60, σ = 8, approximately what percentage of data is above 76?

04

Combining Intervals

More complex questions ask for intervals between different multiples of σ. Use the empirical rule plus subtraction.

Example: μ = 100, σ = 15. What percentage lies between 85 and 115 but NOT between 70 and 130?

This asks: what is in the 95% band but not in the 68% band?

The 13.5% rule: Between 1σ and 2σ on each side sits 13.5% of the data. This is very commonly tested.

Similarly, between 2σ and 3σ on each side: (99.7 − 95) ÷ 2 = 2.35%.

Book Notes

Draw a labelled normal curve showing: 34% + 34% (within 1σ), 13.5% + 13.5% (between 1σ and 2σ), 2.35% + 2.35% (between 2σ and 3σ). These sum to 99.7%.

Quick check: In a normal distribution, what percentage of data lies between μ + σ and μ + 2σ?

05

Identifying Unusual Values

A value is considered unusual if it falls more than 2 standard deviations from the mean. This is because only about 5% of data lies outside the 2σ band — making such values relatively rare.

Worked example: A factory produces bolts with length μ = 10 mm, σ = 0.2 mm. A bolt measures 10.5 mm. Is this unusual?

  1. Distance from mean: 10.5 − 10 = 0.5 mm
  2. In terms of σ: 0.5 ÷ 0.2 = 2.5 standard deviations above the mean
  3. Since 2.5 > 2, this value is more than 2σ from the mean — it is unusual.
Decision rule: Calculate (value − μ) ÷ σ. If the result is greater than 2 (in absolute value), the value is unusual.

Book Notes

Write the decision rule for unusual values: (x − μ) ÷ σ. If |result| > 2, the value is unusual.

Quick check: A dataset has μ = 200 and σ = 25. Is the value 260 unusual?

Activities

Activity 1 — Empirical Rule Calculations

Exam scores in a large test are normally distributed with μ = 65, σ = 12. Find:

  1. The percentage of students scoring between 53 and 77
  2. The percentage scoring between 41 and 89
  3. The percentage scoring above 89
  4. The percentage scoring below 53
See answers
  1. 53 = 65 − 12 = μ − σ and 77 = 65 + 12 = μ + σ. This is within 1σ → 68%.
  2. 41 = 65 − 24 = μ − 2σ and 89 = 65 + 24 = μ + 2σ. Within 2σ → 95%.
  3. Above μ + 2σ: 5% ÷ 2 = 2.5%.
  4. Below μ − σ: 32% ÷ 2 = 16%.

Activity 2 — Is It Unusual?

A machine fills juice bottles with μ = 500 mL, σ = 8 mL. Determine whether each bottle is unusual. Show working.

  1. A bottle containing 484 mL
  2. A bottle containing 520 mL
  3. A bottle containing 510 mL
See answers
  1. (484 − 500) ÷ 8 = −2.0. Exactly 2σ below. This is borderline — on the boundary. Some would say not unusual (rule: strictly more than 2σ).
  2. (520 − 500) ÷ 8 = 2.5. More than 2σ above the mean → unusual.
  3. (510 − 500) ÷ 8 = 1.25. Less than 2σ from mean → not unusual.

Multiple Choice

1. In a normal distribution with μ = 30 and σ = 4, what percentage of data lies between 22 and 38?

  1. 68%
  2. 95%
  3. 99.7%
  4. 50%
Answer

B. 22 = 30 − 8 = μ − 2σ and 38 = 30 + 8 = μ + 2σ. Within 2 standard deviations → 95%.

2. The heights of plants are normally distributed with μ = 45 cm, σ = 6 cm. Approximately what percentage of plants are taller than 57 cm?

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

C. 57 = 45 + 12 = μ + 2σ. The percentage above μ + 2σ is 5% ÷ 2 = 2.5%.

3. What percentage of normally distributed data lies between μ − 2σ and μ − σ?

  1. 27%
  2. 34%
  3. 13.5%
  4. 2.35%
Answer

C. Between 1σ and 2σ on one side: (95 − 68) ÷ 2 = 13.5%.

4. A dataset has μ = 150 and σ = 20. Which value would be considered unusual?

  1. 160
  2. 140
  3. 170
  4. 95
Answer

D. (95 − 150) ÷ 20 = −2.75. This is more than 2 standard deviations below the mean, so it is unusual. Options A, B, C are all within 2σ.

5. In a normal distribution, approximately what percentage of data lies more than 3 standard deviations from the mean?

  1. 3%
  2. 0.3%
  3. 5%
  4. 0.03%
Answer

B. 99.7% lies within 3σ, so 100 − 99.7 = 0.3% lies outside.

Short Answer

SAQ 1. Newborn baby weights are approximately normally distributed with μ = 3.4 kg, σ = 0.5 kg. (a) What percentage of babies weigh between 2.4 kg and 4.4 kg? (b) A baby weighs 2.2 kg. Is this an unusual weight? Show your working. (c) What percentage of babies weigh more than 3.9 kg?

See answer

(a) 2.4 = 3.4 − 1.0 = μ − 2σ and 4.4 = 3.4 + 1.0 = μ + 2σ. Within 2σ → approximately 95%.

(b) (2.2 − 3.4) ÷ 0.5 = −1.2 ÷ 0.5 = −2.4. Since |−2.4| > 2, this weight is more than 2 standard deviations below the mean → unusual.

(c) 3.9 = 3.4 + 0.5 = μ + σ. The percentage above μ + σ is 32% ÷ 2 = 16%.

SAQ 2. Draw a fully labelled normal distribution diagram for a dataset with μ = 20 and σ = 3. Mark the values at μ ± σ, μ ± 2σ, and μ ± 3σ on the x-axis. Label the percentage of data in each region between consecutive standard deviation marks.

See answer

x-axis values: 11, 14, 17, 20, 23, 26, 29. Regions from left to right: 0.15% | 2.35% | 13.5% | 34% | 34% | 13.5% | 2.35% | 0.15%. These sum to approximately 99.7% (with 0.3% in the extreme tails beyond 3σ).

Full Answers

MC 1: B  |  MC 2: C  |  MC 3: C  |  MC 4: D  |  MC 5: B

SAQ 1: (a) 95%; (b) −2.4 SDs → unusual; (c) 16%.

SAQ 2: Bell curve with x-values 11, 14, 17, 20, 23, 26, 29 and regions 0.15%, 2.35%, 13.5%, 34%, 34%, 13.5%, 2.35%, 0.15%.

Revisit

Can you recite the empirical rule from memory (68–95–99.7) and use it to find one-sided probabilities and identify unusual values? Can you draw the labelled bell curve with all seven σ-boundary regions?