Cogito
Probability and Statistics · Chapter 6 · Lesson 1
The Normal Distribution
The bell curve, and the rule that reads it.
12 problems · about 21 minutes · TEKS P.S.3.A, S-ID.A.4
What this lesson teaches
The student applies the 68-95-99.7 rule to normally distributed data.
- A normal distribution is symmetric and bell-shaped.
- Its mean sets the centre and its standard deviation sets the width.
- About 68%, 95% and 99.7% lie within one, two and three standard deviations.
Warm Up
Straightforward practice. Get the method working first.
5 problemsWhat percentage lies within one standard deviation of the mean?
Answer 68
Why 68%.
Why does 2.5% lie above two standard deviations rather than 5%?
Answer The 5% outside is split evenly between two tails.
Why Two tails share the 5%.
What percentage lies within two standard deviations?
Answer 95
Why The second number.
What percentage lies within three standard deviations?
Answer 99.7
Why The third number.
Mean 170, standard deviation 8. What value is two above the mean?
Answer 186
Why 170 + 16.
Build It Up
The same ideas with more to keep track of.
3 problemsMean 100, standard deviation 15. What value is one below the mean?
Answer 85
Why 100 − 15.
What percentage lies above the mean in a normal distribution?
Answer 50
Why It is symmetric.
What percentage lies outside one standard deviation?
Answer 32
Why 100 − 68.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each quantity by whether a normal model is reasonable.
Answer Roughly normal: Adult heights, Measurement errors · Not normal: Household income, Number of lottery wins per person
Why Strongly skewed quantities are not normal.
Mean 50, standard deviation 5. What value is three above the mean?
Answer 65
Why 50 + 15.
The Middle: What percentage lies within one standard deviation of the mean?
Answer 68 %
Why 68%.
The Tail: What percentage lies more than two standard deviations above the mean?
Answer 2.5 %
Why 2.5%.