1. Displaying
Quantitative Data

2. How do we examine the distribution of a
quantitative variable?
Dotplot
 Stemplot
 Histogram
 Ogive

3. To interpret graphs of
quantitative data:
• Look for an overall pattern and for notable
departures from the pattern.

4. Keep these features in mind:
 Shape – mode?, skewed left/right, symmetric
 Outlier - unusual features
 Center – middle value
 Spread - smallest to largest values

5. Shape
• Is the distribution
approximately bell
shape?
• Skewed left?
• Skewed right?
• Is there a mode?

6. Outlier
• Are there any values that fall outside the
overall pattern?

7. Center
• Find a value that divides the observations so
that about half take larger values and about
half take smaller values.
• Median

8. Spread
• Give the smallest and largest values.
• Range

9. Dotplot
Things to remember:
 You only need a properly labeled horizontal axis
 You need to be neat
 Title the graph
 Each dot represents a count of 1
 Works well with a small data set

10. Random Values:
3,7,4,6,0,7,2,0,7,5,8,2,4,0,3,1,9,9,3,4,2,3,7,9,2,4,6,7,
7,1,3,9,2,9,4,1,6,7,9,9,2,5,4,9,2,3,5,6,8,7

11. What is the shape, center, and spread of the dot plot?
Check For Understanding - 1

12. Stemplot
Things to remember:
 Separate each piece of data into a stem (all but the
rightmost digit) and a leaf (the final digit) Write the
stems vertically in increasing order from top to
bottom.
 Be very neat and make sure you leave the same
amount of space in between leaves.
 Title your graph
 Include a key identifying what the stem and leaves
represent.
 Works well with a small data set

13. Random Values:
35,40,43,49,51,54,57,58,58,64,68,68,75,77
Stemplot of Random Values
stem leaves
3 5
4 0 3 9
5 1 4 7 8 8
6 4 8 8
7 5 7
Key: 3 5 represent 35│

14.  Splitting the stem makes it easier to see the
distribution.
Stemplot of random numbers
2 12
2 577
3 44
3
4 01
4 6788
5 233344
5 55567788899 key:2 1= 21

15. The data below give the amount of caffeine content
(in milligrams) for an 8-ounce serving of popular
soft drinks.
20 15 23 29 23 15 23 31 28 35 37 27 24 26 47 28 24 28 28
16 38 36 35 37 27 33 37 25 47 27 29 26 43 43 28 35 31 25
(a) Construct a stemplot.
(b) Construct a split stemplot.
Check For Understanding - 2

16. Histogram
Things to remember:
 It is the most common graph of a quantitative
variable.
 The x-axis is continuous, so there should be
no gaps between the bars (unless a class has
zero observations).
 Title your graph.

17. More about histograms
 Be sure to choose classes that are all the
same width.
 Use your judgement in choosing classes
to display the shape.

18. How to make a histogram:
STEP 1: Divide the data into "classes“ of equal
size.
STEP 2: Find the frequencies. Count the number
of individuals in each class.
STEP 3: Draw your histogram. Classes go on the
horizontal and frequencies go on the vertical.

19. Histogram of Random Values
Frequency Table
Class Frequency
1 - 5 4
6 - 10 30
11 - 15 26
16 - 20 50
21 - 25 12

20. Interpreting a Histogram
Describe the Histogram:
• Shape: roughly symmetric and unimodal
• Outlier: no outliers
• Center: midpoint around 110
• Spread: about 80 to 150

21. Check For Understanding - 3
The following data represents scores of 50
students on a calculus test.
72 72 93 70 59 78 74 65 73 80
57 67 72 57 83 76 74 56 68 67
74 76 79 72 61 72 73 76 67 49
71 53 67 65 99 83 69 61 72 68
65 51 75 68 75 66 77 61 64 74
(a) Construct a frequency histogram for this
data set.
(b) Describe the shape, center, and spread of
the distribution of test scores.

22. Cumulative Frequency Plot (Ogive)
The Ogive is plotted by
graphing a dot at each
class end point for the
cumulative frequency
value and connecting
the dots.

23. Check For Understanding - 4
The following data represents the percentage of people
aged 65 or older in each state.
(a) Construct a relative cumulative frequency graph (ogive) for these data.
(b) Use your ogive to answer the following questions:
• In what percentage of states was the percentage of “65 or older” less than 15%?
• What is the 40th percentile of this distribution, and what does it tell us?
• What percentile is associated with Missouri?

24. Time Plots
 Plots each observation against the time at which
it was measured. Always mark the time scale on
the horizontal axis and the variable being
measured on the y axis.
 Trend: A common overall pattern.
 Seasonal Variations: A pattern that repeats itself
at regular time intervals

25. Identify any trends and describe the time plot.
Check For Understanding - 5

26. Comparing Two Data Sets
 If you want to compare two data sets, then make sure that
the two graphic displays are as alike as possible—the scale of
the axes, the groups for the histogram, etc. If you are using
stemplots, use the same set of stems for both sets (thus
creating a back-to-back stemplot).
 When describing a single distribution, you want to comment
on its center, spread and shape. When comparing two
distributions, you will want to compare center, spread and
shape! Don't forget to make the comparisons in context.

27. Back-to-Back Stemplot of Random Values
• A back-to-back
stemplot is used to
compare two data
sets.
Key: 2 0 = 20

28. Check For Understanding - 6
During the early part of the 1994 baseball season, many
sports fans and baseball players noticed that the
number of home runs being hit seemed to be unusually
large. Here are the data on the number of home runs hit
by American and National League teams:
American League
35, 40, 43, 49, 51, 54, 57, 58, 58, 64, 68, 68, 75, 77
National League
29, 31, 42, 46, 47, 48, 48, 53, 55, 55, 55, 63, 63, 67
(a) Construct a back-to-back stemplot to compare the
number of home runs hit in the two leagues.
(b) Write a few sentences comparing the distributions of
home runs in the two leagues.