1. What We Know So Far… Data plots or graphs
Look for the overall pattern and identify deviations and outliers
Numerical summary to briefly describe center and spread

2. Density Curves
The key idea is that we can use a mathematical model (a density curve) as an approximation to the overall pattern of data.
If the pattern is sufficiently regular, approximate it with a smooth curve.

3. Density Curves
Software can replace the separate bars of a histogram with a smooth curve that represents the overall shape of a distribution.
The software doesn’t start from the histogram—it starts with the actual observations and cleverly draws a curve to describe their distribution.
In the figure, the software has caught the overall shape and shows the ripples in the long right tail more effectively than does the histogram. It struggles a bit with the peak. For example, it has extended the curve below zero in an attempt to smooth out the sharp peak. For the irregular distribution in the figure, we can’t do better.

4. Here is a histogram of vocabulary scores of 947 seventh graders.
Example
Here is a histogram of vocabulary scores of 947 seventh graders.
The smooth curve drawn over the histogram is a mathematical “idealization” for the distribution.

5. Example
The areas of the shaded bars in this histogram represent the proportion of
scores in the observed data that are less than or equal to This proportion is equal to

6. Example
Now the area under the smooth curve is shaded. Its proportion to the total area is
now equal to (not 0.303).
This is what the proportion on the previous slide would equal to if we had LOTS of data.

7. Density Curves: Definition
Always on or above the horizontal axis
Have area exactly 1 underneath curve
Area under the curve indicates the “theoretical” proportion of values in that range.
Remember the density is only an approximation, but it simplifies analysis and is generally accurate enough for practical use.
Come in a variety of shapes, bell-shaped density is commonly used.

8. Mean The mean of a density curve is the
balance point, at which the curve would
balance if made of solid material.

9. Median The median of a density curve is the
equal-areas point, the point that divides
the area under the curve in half.

10. The mode is the peak point of the curve.

11. Median
The median and the mean are always equal on a symmetric density curve
Mean > median for a right-skewed
distribution
Mean < median for a left-skewed distribution

12. Example
Determine which letter corresponds to the median, mode, and the mean of the following density curves.

13. Example Copy this picture into your notes.
Guess where you think the Median and the Mean are by drawing an arrow pointing to them
Guess what values they actually are.

14. Examples
Copy this histogram into your notes. Sketch a smooth curve that describes the distribution well. Mark your best guess with an arrow for the mean and median of the distribution.

15. Examples
Sketch a smooth curve that describes the distribution well. Mark your best guess with an arrow for the mean and median of the distribution.

16. Uniform Distribution
The figure below shows the density curve of a uniform distribution. The curve takes the constant value. Sketch the curve.

17. Example
The curve takes the constant value 1 over the interval from 0 to 1. This means that data described by this distribution take values that are uniformly spread between 0 and 1.
(a) Why is the total area under this curve equal to 1? The area under the curve is a rectangle with
height 1 and width 1. Thus the total area under the curve = 1 ´ 1 = 1
(b) What percent of the observations lies above 0.8? 20%. (The region is a rectangle with height 1
and base width 0.2; hence the area is 0.2.)
(c) What percent of the observations lie below 0.6? 60%.
(d) What percent of the observations lie between 0.25 and 0.75? 50%.
(e) What is the mean µ of this distribution? µ = 0.5, the “balance point” of the density curve

18. Questions for Previous Picture
(a) Why is the total area under this curve equal to 1? The area under the curve is a rectangle with height 1 and width 1. Thus the total area under the curve = 1 ´ 1 = 1
(b) What percent of the observations lies above 0.8? 20%. (The region is a rectangle with height 1 and base width 0.2; hence the area is 0.2.)
(c) What percent of the observations lie below 0.6? 60%.
(d) What percent of the observations lie between 0.25 and 0.75? 50%.
(e) What is the mean of this distribution? 0.5, the “balance point” of the density curve
(a) Why is the total area under this curve equal to 1? The area under the curve is a rectangle with
height 1 and width 1. Thus the total area under the curve = 1 ´ 1 = 1
(b) What percent of the observations lies above 0.8? 20%. (The region is a rectangle with height 1
and base width 0.2; hence the area is 0.2.)
(c) What percent of the observations lie below 0.6? 60%.
(d) What percent of the observations lie between 0.25 and 0.75? 50%.
(e) What is the mean µ of this distribution? µ = 0.5, the “balance point” of the density curve

19. Summary The mode is the peak point of the curve
The median of a density curve is the equal-areas point
The mean of a density curve is the balance point
The median and the mean are always equal on a symmetric density curve
Mean < median for a left-skewed distribution
Mean > median for a right-skewed distribution
Uniform distribution has a constant curve.

20. Density Curve Worksheet Due Wednesday
HOMEWORK
Density Curve Worksheet
Due Wednesday