🔔 AINMATH · Classroom Worksheet
Normal Distribution
High school · Probability & Statistics
⏱ 20–25 min
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Learning goal — Be able to apply the 68-95-99.7 rule for normal distributions, and calculate the standard score z=(x−μ)/σ to compare different sets of data.
About 68% falls within μ±1σ, about 95% within μ±2σ, and about 99.7% within μ±3σ.
Standard score z = (x − μ) / σ
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1Applying the 68-95-99.7 Rule
The heights of students at a certain school follow a normal distribution with a mean of 170 cm and a standard deviation of 5 cm.
① About what percent of students are between 165 cm and 175 cm tall?
② About what percent of students are between 160 cm and 180 cm tall?
2Calculating the Standard Score (z-score)
Q. On a certain exam, the mean is 70 points and the standard deviation is 8 points. Find the standard score z for a student who scored 86.
3Think Further — Comparing Different Exams
Q. Exam A has a mean of 75 and a standard deviation of 5, and Eunji scored 85. Exam B has a mean of 60 and a standard deviation of 15, and Minsu scored 90. Find each person's standard score, then decide who did relatively better (or whether they did the same), and explain your reasoning.
For teachers — cover before handing out to students
Answers & Solutions
Activity 1 — ① 165–175 is the μ±1σ range, so about 68% ② 160–180 is the μ±2σ range, so about 95%
Activity 2 — z = (86−70)/8 = 2.0
Activity 3 — Eunji: z=(85−75)/5=2.0, Minsu: z=(90−60)/15=2.0. Since both standard scores come out to exactly 2.0, even though their raw scores differ, they each did relatively equally well on their own exam.