Density Curves
These are mathematical models for
distributions.
Density curves:
• display the shape of a distribution.
• are always on or above the x-axis.
• have an area under the curve of exactly 1.
This is a simple density curve. Can you verify that the graph
meets the requirements of a density curve? This distribution
is called a uniform distribution, the height of the distribution
is uniform for all values of x.
This is another density curve. Verify that the area under the
curve is 1.
1
0
2
The histogram represents some real data. The smooth curve is
an idealized model of the data. In order for this to be a
density curve the scale must be adjusted so that the total area
is one.
In statistics we will use density curves frequently. We will
often ask for the area corresponding to a given interval of x.
This figure shows the area corresponding to 7 < x < 8.
Area under the curve corresponds to the percent of observations
between two values of x. If we have simple geometric shapes, we
find areas by simple geometry. See that the area under the curve
corresponding to 0 < x < .5 is (.5)(.5)(.5) = .125, using the formula
for area of a triangle.
1
0
.5
2
The area under the curve corresponding to
1 < x < 2 is (1)(1/3)=1/3 for this uniform
distribution.
1/3
0
1
2
3
The area under this bell-shaped curve between two particular
values of x is a quantity we will often calculate in Statistics.
We will use tables or calculator functions to find estimates of
the areas of more complex shapes.
-3
-2
-1
0
1
2
3
We might be interested in finding the area under the curve
between 1 and 2.
The value of x that divides the area under the curve in half
defines the median of a density curve. In the case of a
symmetrical density curve, the median is also the mean.
The mean is the balance point of a density curve. Extreme
values have a greater impact than values closer to the center.
Imagine balancing on a seesaw, the further from the fulcrum
the greater the effect.
Skewed density curves have medians and means that are
different. The large values to the right “pull” the mean up.