AP Statistics
Describing Location in a Distribution
Density Curves
Name_________________________
1. What is a percentile?
It is the percent of data values at or below a given point. (Depending on which
definition we use.)
2. Is there a difference between the 80th percentile and the top 80%? Explain.
The 80th percentile is that has 80% of the data at or below it. That means we are
talking the bottom 80% not the top.
3. Is there a difference between the 80th percentile and the lower 80%? Explain.
Yes. The 80th percentile is an actual point that marks where the bottom 80% of data
falls. The lower 80% is a range of numbers.
4. What is a density curve?
A density curve is a curve whose area between the curve and the horizontal axis
underneath it is exactly 1.
5. What does the area under a density curve represent?
The proportion of observations that fall in a given interval.
6. Describe the shape of the normal density curve. Draw a picture to aid your
description.
A density curve where 68% of the area falls within plus or minus 1 standard
deviation of the mean, 95% within two standard deviations, and 99.7%
2 DESCRIBING LOCATION IN DISTRIBUTIONS
7. Where is the mean of a density curve located?
At its balancing point.
8. Where is the median of a density curve located?
At the point that divides the area into 2 equal halves.
9. What is a uniform distribution?
A distribution that looks the same, for example, a box with height 1. (See problem
11 for an example.
10. Below is a density curve made from segments.
a) Verify that the graph is a density curve.
If you split the shape into a triangle and a rectangle, all you have to do is find the
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area of each piece. The area of the triangle is 𝐴 = 2 (. 4 ∗ 1) = .2. The area of the
rectangle is 𝐴 = 1 ∗ .8 = .8. Add the two areas together and it adds to 1, so this
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
0.2
c) 0 ≤ X ≤ 0.4
0.6
d) 0 ≤ X ≤ 0.2
0.35
e) The median of the density curve is a point between X = 0.2 and X = 0.4.
Explain why.
Because the point that divides the area into two equal halves is some where in
that interval.
AP Statistics
Describing Location in a Distribution
Density Curves
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) Why is the total area equal to 1? The area of the square is A = 1*1 = 1.
b) What percent of the observations lie above 0.8? 0.2
c) What percent of the observations lie below 0.6? 0.6
d) What percent of the observations lie between 0.25 and 0.75? 0.5
e) What is the mean 𝜇 for this distribution? 0.5
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.
B = median C = mean
A=mean and median
B = median A = 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.
The 13.57 is the percentile for her times only. She is comparing her time to her
previous times, so if 13.57 is at the 10th percentile, only 10 percent of her times
were worse, so it was a bad race for her. However, her time was still better than
40% of the other athletes at the state meet.
4 DESCRIBING LOCATION IN DISTRIBUTIONS
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.
Area of the triangle is .4, and the area of the
rectangle is .6, so yes it is a density curve.
b) Determine the median. Mark the median and the approximate locations of
the quartiles Q1 and Q3 on your sketch.
Median = 0.5, Q1=.03, Q2=0.7
c) What percent of the observations lie below 0.3?
The first line segment has an equation of y = 0.6 +1.6x . Thus, the height of the
density curve at 0.3 is 0.6 + 1.6 × 0.3 = 1.08 . The total area under the density
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curve between 0 and 0.3 is 2 × 0.3𝑥0.48 + 0.3 × .48𝑥0.6. Thus, 25.2% of the
observations lie below 0.3.
d) What percent of the observations lie between 0.3 and 0.7?
Using symmetry of the density curve, the area between 0.3 and 0.7 is 1 − 2×0.252 =
0.496. Therefore, 49.6% of the observations lie between 0.3 and 0.7.