Descriptive statistics
Normal distribution
Let's make a histogram of everyone's heights.
There are lots of histograms that look like this. For example, if I take a die and roll it
1,000 times and take the average. If I repeat that 100 times, I will also get a normal
distribution.
Normalizing the normal
Normal distributions come in different sizes, depending on the mean and standard deviation.
However, if you require that the mean be 0 and the standard deviation be 1, it is unique.
Any distribution curve or histogram can be normalized by shifting and dividing by the
standard deviation.
Another way to think about this is in terms of standard units.
For example, with the HANES study, the average height for women (18-74) was 63.5 inches and
the SD was 2.5 inches.
- 66 inches in standard units would be +1,
- what about 58.5 inches?
- 64 inches?
- What is -1.2 standard units in inches?
This standard normal distribution has the following properties:
- the area between -1 and +1 is approx 68%
- the area between -2 and +2 is approx 95%
- the area between -3 and +3 is approx 99.7%
Using the table on p. A-105 in the book, find the following:
- between -1.2 and 1.2
- between 0 and 1
- between -2 and 0
- between -2 and 1
- to the right of 1
- to the right of 2
- to the left of 2
- between 1 and 2
Approximating a histogram with a normal curve
Example 1
In the HANES study, the average height for men (18-74) was 69 inches and the SD was 3 inches.
Using the normal curve, estimate the percentage of men between 63 and 72 inches.
- convert the numbers to standard units.
- sketch the normal curve
- add up the areas
Example 2
In the HANES study, the average height for women (18-74) was
63.5 inches and the SD was 2 inches.
Using the normal curve, estimate the percentage of women above 60 inches in height.
- convert the numbers to standard units.
- sketch the normal curve
- add up the areas
Example 3
Take the data from our class and estimate the number of people above 68 inches.
Compare it with the actual number.