The Normal Distribution — Challenge
Some measurements cluster in a bell-shaped pattern that a normal distribution can approximate.
What these problems 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
What shape does a normal distribution have?
- A symmetric bell
- A straight line
- A rectangle
- A sawtooth
- A U shape
Worked answer
A symmetric bell
A normal distribution is highest at its mean and tails off evenly on both sides.
That gives the familiar symmetric bell shape.
Symmetry is the key property: the left half is a mirror image of the right.
Problem 2
In a normal distribution, roughly what percentage of values lie within one standard deviation of the mean?
- 50%
- 68%
- 95%
- 99.7%
- 34%
Worked answer
68%
This is the first part of the empirical rule for normal data.
About 68% of values lie within one standard deviation either side of the mean.
Each side holds about 34%, which is why that number also appears among the options.
Problem 3
In a normal distribution, roughly what percentage of values lie within two standard deviations of the mean?
- 68%
- 90%
- 95%
- 99.7%
- 50%
Worked answer
95%
The empirical rule gives about 68% within one standard deviation, 95% within two and 99.7% within three.
So the answer is about 95%.
This is why two standard deviations is the usual rough cut-off for an unusual value.
Problem 4
A distribution has mean 100 and standard deviation 15. What is the z-score of the value 115?
- 0
- 1
- 1.5
- 15
- 115
Worked answer
1
A z-score counts how many standard deviations a value sits above the mean: z = x-μσ.
115-10015 = 1515 = 1.
So 115 is exactly one standard deviation above the mean.
Problem 5
A distribution has mean 100 and standard deviation 15. What is the z-score of the value 70?
- -2
- 2
- -30
- 30
- -1
Worked answer
-2
z = 70-10015 = -3015.
That gives -2.
The sign matters: a negative z-score means the value is below the mean.
Problem 6
What is the mean of the standard normal distribution?
- 0
- 1
- 50
- 100
- -1
Worked answer
0
The standard normal is the normal distribution written in terms of z-scores.
A z-score of 0 means a value at the mean, so the mean of the standard normal is 0.
Its standard deviation is 1, which is the other half of the definition.
Problem 7
What is the standard deviation of the standard normal distribution?
- 0
- 1
- 10
- 100
- 0.5
Worked answer
1
A z-score is measured in units of one standard deviation.
So in the standard normal the standard deviation is exactly 1, and the mean is 0.
Problem 8
In a normal distribution, how do the mean, median and mode compare?
- They are all equal
- They are all different
- The mean is largest
- The median is largest
- The mode is largest
Worked answer
They are all equal
A normal distribution is symmetric, so the mean sits at the centre and so does the median.
It is also highest at that centre, so the mode is there too.
All three therefore coincide. In a skewed distribution they separate.
Problem 9
A distribution has mean 50 and standard deviation 5. Which value has a z-score of 2?
- 52
- 55
- 58
- 60
- 65
Worked answer
60
Rearranging z = x-μσ gives x = μ+zσ.
50+2· 5 = 60.
Adding only the 2 would give 52, which ignores the size of the standard deviation.