Mathematics Standard • Year 12 • Module 5 • Lesson 10
The Empirical Rule, Master
Mastery Challenge for The Empirical Rule. Work through each self-contained problem, show your reasoning, then open the worked answers to check your method.
1. Core skill
For a normal variable with μ=72 and σ=8, calculate the bounds at μ−2σ, μ−σ, μ+σ and μ+2σ. Label the approximate percentage between each symmetric pair. 3 marks
2. Use the idea
Estimate the percentage between 56 and 80 (μ−2σ to μ+1σ). Show the partition used. 3 marks
3. Check your reasoning
A measurement is 88.1 when μ=72 and σ=8. Calculate its z-distance and explain why it falls just beyond the 2σ threshold, while acknowledging the empirical rule is approximate. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (3 marks)
μ−2σ=56, μ−σ=64, μ+σ=80, μ+2σ=88. About 68% lies from 64 to 80; about 95% lies from 56 to 88.
2. Use the idea (3 marks)
From −2σ to +2σ is about 95%. The band from +1σ to +2σ is (95%−68%)/2=13.5%. Therefore from −2σ to +1σ is 95%−13.5%=81.5%.
3. Check your reasoning (2 marks)
z=(88.1−72)/8=16.1/8=2.0125, just over 2 standard deviations above the mean. Values beyond +2σ are in the upper tail, about 2.5% in total; the rule gives an approximate proportion, not an exact count.