AP Statistics
Describing Location in a Distribution
Density Curves
Name_________________________
1. What is a percentile?
2. Is there a difference between the 80th percentile and the top 80%? Explain.
3. Is there a difference between the 80th percentile and the lower 80%?
Explain.
4. What is a density curve?
5. What does the area under a density curve represent?
6. Describe the shape of the normal density curve. Draw a picture to aid your
description.
7. Where is the mean of a density curve located?
8. Where is the median of a density curve located?
9. What is a uniform distribution?
2 DESCRIBING LOCATION IN DISTRIBUTIONS
10. Below is a density curve made from segments.
a) Verify that the graph is a density curve.
For each of the following, use areas under this density curve to find the
proportion of observations within the given interval:
b) 0.6 ≤ X ≤ 0.8
c) 0 ≤ X ≤ 0.4
d) 0 ≤ X ≤ 0.2
e) The median of the density curve is a point between X = 0.2 and X = 0.4.
Explain why.
11. The figure below shows the density curve for the uniform distribution. The
curve takes the constant value 1 over the interval from 0 to 1.
a)
b)
c)
d)
e)
Why is the total area equal to 1?
What percent of the observations lie above 0.8?
What percent of the observations lie below 0.6?
What percent of the observations lie between 0.25 and 0.75?
What is the mean 𝜇 for this distribution?
AP Statistics
Describing Location in a Distribution
Density Curves
12. Refer to problem 11. Can you construct a modified boxplot for the uniform
distribution? If so, do it. If not, explain why not.
13. For each of the images below, identify the mean and the median.
14. McKenzie is a star runner on the track team. In the 100m dash, her time of
13.57 would fall at the 10th percentile of all her race times that season.
However, her performance places her at the 4oth percentile in the state
championship meet. Explain how this is possible.
15. A certain density curve looks like an inverted letter V. The first segment goes
from the point (0,0.6) to the point (0.5, 1.4). The second segment goes from
the point (0.5, 1.4) to (1, 0.6).
a) Sketch the curve. Verify that the area under the curve is 1, so that it is a
valid density curve.
b) Determine the median. Mark the median and the approximate locations
of the quartiles Q1 and Q3 on your sketch.
c) What percent of the observations lie below 0.3?
d) What percent of the observations lie between 0.3 and 0.7?