The document defines percentiles and how they divide data into hundredths, with quartiles specifically dividing data into fourths. It describes how to compute quartiles and the interquartile range. Finally, it explains how to create and interpret box-and-whisker plots using the five number summary to visually depict the spread and symmetry of a data set.

1. 3.3 Percentiles and
Box-and-Whisker Plots
CHAPTER 3
AVERAGES AND VARIATION

2. Percentiles and Box-and-Whisker Plots


3. Percentiles
 For whole numbers P (where 1 ≤ P ≤ 99), the Pth
percentile of a distribution is a value such that P%
of the data fall at or below it and (100 – P)% of the
data fall at or above it.
 There are 99 percentiles, and in an ideal situation, the 99
percentiles divide the data set into 100 equal parts. However, if
the number of data elements is not exactly divisible by 100, the
percentiles will not divide the data into equal parts.

4. Finding Percentiles
 There are several widely used conventions for finding
percentiles. They lead to slightly different values for
different situations, but these values are close together.
 To find percentiles:
1. Order the values from smallest to largest.
2. A natural way to find the Pth percentile is to then find a value such
that P% of the data fall at or below it.
This will not always be possible, so we take the nearest value
satisfying the criterion. It is at this point that there is a variety of
processes to determine the exact value of the percentile.
We will not be very concerned about exact procedures for
evaluating percentiles in general.

5. Quartiles
 Quartiles are those percentiles that divide the data
into fourths. The first quartile Q1 is the 25th
percentile, the second quartile Q2 is the median and
the 50th percentile, and the third quartile Q3 is the
75th percentile.

6. Page 112
How to Compute Quartiles
1. Order the data from smallest to largest
2. Find the median. This is Q2.
3. The first quartile Q1 is then the median of the lower
half of the data; that is it is the median of the data
falling below the Q2 position (and not including
Q2).
4. The third quartile Q3 is then the median of the
upper half of the data; that is it is the median of the
data falling above the Q2 position (and not
including Q2).

7. Interquartile Range
 The interquartile range (IQR) is the difference
between the third and first quartiles.
 Tells the spread of the middle half of the data
 Resistant to change because extreme values would be found in
the lower and upper quartiles
IQR = Q3 – Q1

8. Page 112
Example 9 - Quartiles
In a hurry? On the run? Hungry as well? How about
an ice cream bar as a snack? Ice cream bars are
popular among all age groups. Consumer Reports
did a study of ice cream bars. Twenty-seven bars
with taste ratings of at least “fair” were listed, and
cost per bar was included in the report. Just how
much does an ice cream bar cost?
(a) Find the quartiles.
(b) Find the interquartile range.

9. Example 9 - Quartiles
Ordered Cost of Ice Cream Bars (in dollars)

10. Box-and-Whisker Plots
 Five Number Summary:
Lowest Value
Q1
Median
Q3
Highest Value
 The five number summary is used to create a
graphic sketch of the data called a box-and-
whisker plot.

11. Box-and-Whisker Plots
 A box-and-whisker plot is a visual representation
a five-number summary.
 provides a graphic display of the spread of the data about the
median and a quick overall summary of symmetry/skewness
 The box shows the middle half of the data. One quarter of the data
is located along each whisker.
 If the median line is near one end of the box, the data are
skewed toward the other end of the box.
 If the median is approximately centered in the box and the
whiskers are about the same length, the data is approximately
symmetric.

12. Outliers
 How far away is “far enough” to be considered an
outlier?
 Any number that is beyond 1.5(IQR) from either Q1 or Q3 is
considered an outlier

13. Page 114
How to Make a Box-and-Whisker Plot
1. Find the five number summary
2. Draw a vertical scale to include
the lowest and highest data values
3. To the right of the scale, draw a
box from Q1 to Q3
4. Include a solid line through the
box at the median level
5. Draw vertical lines, called Box-and-Whisker Plot
Figure 3-6
whiskers, from Q1 to the lowest
value and from Q3 to the highest
value.

14. Using the Calculator:
Find the Five Number Summary
 Enter the data in L1
 Hit STAT, tab over to CALC, choose 1: 1-VarStats
 Hit 2nd 1 ENTER
 Tab down to view the five number summary

15. Using the Calculator:
Box-and-Whisker Plot
1. Enter the data into L1 4. Highlight 1:Plot1, hit
2. Hit WINDOW ENTER, make the
Xmin=lowest value selections
Xmax=highest value
Xscl=1
Ymin=-5
Ymax=5
Yscl=1 5. Hit GRAPH
6. Press TRACE and tab
3. Hit 2nd y=
left/right to view values
in the Five Number
Summary

16. Assignment
 Page 117
 #1 – 5, 9, 11, 12