Measuring Variation 2
Lecture 17
Sec. 5.3.3
Mon, Oct 3, 2005
The Five-Number Summary

Five-number summary – A summary of a
sample or population consisting of the five
numbers
Minimum
 First quartile Q1
 Median
 Third quartile Q3
 Maximum


This does a better job of indicating the spread
than any single number can do.
TI-83 – Five-Number Summary

Use the TI-83 to find a five-number summary
of the data in Exercise 4.29, p. 252.
Min = 75.4
 Q1 = 77.1
 Median = 81.15
 Q3 = 84.9
 Max = 90.3

Excel – Five-Number Summary

Use Excel to find a five-number summary of the
same data.

OnTimeArrivals.xls.
The Five-Number Summary

From the 5-number summary of these data, can
we detect skewness in the distribution?

Answer: Maybe.
Boxplots

Boxplot – A graphical display of a five-number
summary.
Draw and label a scale representing the variable.
 Draw a box over the scale with its left and right ends
at Q1 and Q3.
 Draw a vertical line through the box at the median.
 Draw a left tail (whisker) from the box to the
minimum.
 Draw a right tail from the box to the maximum.

Example

Draw a boxplot of the data in the previous
example.
Boxplots and Shape



What would a boxplot for a uniform distribution
look like?
What would a boxplot for a symmetric
distribution look like?
What would a boxplot for a left-skewed
distribution look like?
TI-83 – Boxplots

Press STAT PLOT.

Select Plot1
Turn Plot 1 On.
 Select the Boxplot Type.
 Specify list L1.


Press WINDOW.



Set minX and maxX appropriately.
Press GRAPH.
See the instructions on p. 316.
TI-83 – Boxplots

Press TRACE.

Use the arrow keys to see the values of the
minimum, Q1, the median, Q3, and the maximum.
Modified Boxplots

Modified boxplot – A boxplot in which the
outliers are indicated.
Modified Boxplots





Draw the box part of the boxplot as usual.
Compute STEP = 1.5  IQR.
The inner fences are at Q1 – STEP and Q3 +
STEP.
Extend the whiskers from the box to the
smallest and largest values that are within the
inner fences.
Draw as individual dots any values that are
outside the inner fences. These dots represent
outliers.
Example: DePaul University

For an example of modified boxplots, see
DePaul University’s web page on retention.
Example


Draw a modified boxplot of the data from the
earlier example.
Are there any outliers?
TI-83 – Modified Boxplots

Follow the same steps as for a regular boxplot,
but for the Type, select the modified-boxplot
icon, the first icon in the second row.


It looks like a boxplot with a couple of extra dots.
Use the TI-83 to find a modified boxplot of the
data from the previous example.
Let’s Do It!



Let’s do it! 5.9, p. 320 – Five-number Summary
and Outliers.
Let’s do it! 5.10, p. 320 – Cost of Running
Shoes.
Let’s do it! 5.11, p. 321 – Comparing Ages–
Antibiotic Study.