The Normal Distribution — Practice
Some measurements cluster in a bell-shaped pattern that a normal distribution can approximate.
What these questions cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Percent — Parts per hundred. 37% = 37100 = 0.37.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Standard deviation — A measure of typical distance from the mean.
- z-score — How many standard deviations a value sits from the mean: z = x - μσ.
- Empirical rule — About 68%, 95% and 99.7% lie within one, two and three deviations.
- Percentile — The percentage of the population at or below a value.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
Using the empirical rule, approximately: what percent of a normal population lies within one standard deviation of the mean?
Worked answer
68
About 68%
Problem 2
Using the empirical rule, approximately: what percent lies within two standard deviations?
Worked answer
95
About 95%
Problem 3
Using the empirical rule, approximately: what percent lies within three standard deviations?
Worked answer
99.7
About 99.7%
Problem 4
A value is 2 deviations above the mean. What is its z-score?
Worked answer
2
z=2
Problem 5
Mean 50, deviation 10. Find the z-score of 70.
Worked answer
2
70-5010=2
Problem 6
Mean 100, deviation 15. Find the z-score of 85.
Worked answer
-1
85-10015=-1
Problem 7
Using the empirical rule, approximately: heights are normal, mean 170 cm, deviation 8 cm. What percent lie between 162 and 178?
Worked answer
68
One deviation either side
About 68%
Problem 8
Using the empirical rule, approximately: same heights. What percent lie between 154 and 186?
Worked answer
95
Two deviations either side
Problem 9
Using the empirical rule, approximately: same heights. What percent are taller than 186 cm?
Worked answer
2.5
5% lies outside two deviations
Half of it is above
Problem 10
Using the empirical rule, approximately: same heights. What percent are shorter than 162 cm?
Worked answer
16
32% lies outside one deviation
Half of it is below
Problem 11
Mean 500, deviation 100. What score is 2 deviations above the mean?
Worked answer
700
500+2(100)=700
Problem 12
Using the empirical rule, approximately: mean 500, deviation 100. What score is at the 84th percentile?
Worked answer
600
That is one deviation above
500+100=600
Problem 13
Using the empirical rule, approximately: mean 500, deviation 100. What score is at the 16th percentile?
Worked answer
400
500-100=400
Problem 14
Using the empirical rule, approximately: mean 20, deviation 3. What percent lie between 20 and 26?
Worked answer
47.5
Half of 95% is 47.5%
Problem 15
Which of these is least likely to be normally distributed?
- Household income
- Adult heights
- Measurement errors
- Shoe sizes
Worked answer
Household income
Income has a few very large values pulling the tail right
The others are roughly symmetric
Problem 16
In a group of 200 with normal scores, about how many lie within one deviation of the mean?
Worked answer
136
0.68 × 200 = 136