First, determine the number of classes by using the “2 Selling Price Number of
to the k rule”. To estimate the number of classes, ($000) Tallies Homes
select the smallest integer (whole number) such that $65 up to $70 /// 3
2k n where n is the total number of observations. $70 up to $75 //// 5
Our set of data has 30 observations. If we try k = 5, we $75 up to $80 //// /// 8
get 25 32 , which is more than 30. Thus the $80 up to $85 //// //// 9
recommended number of classes is 5. $85 up to $90 //// 5
Next, observe that the home with the lowest selling price was $67 thousand and the highest was
$89 thousand. Use text formula [2-1] to determine the interval.
highest value lowest value H - L 89-67 22
Class Interval(i) or i = = =4.4
number of classes k 5 5
We round 4.4 up to 5, thus we let the class interval be $5 thousand. We also decide to let $65
thousand be the lower limit of the first class. Thus, the first class will be $65 up to $70 thousand
and the second class $70 up to $75 thousand, and so on.
Next, the selling prices are tallied into each of the classes. The first home sold for $76 thousand,
therefore, the price is tallied into the $75 thousand up to $80 thousand class. The procedure is
continued, resulting in the frequency distribution shown at the right.
Observe that the largest concentration of the data is in the $80 up to $85 thousand class. As noted
before, the class frequencies are the number of observations in each class. For the $65 up to $70
thousand class the class frequency is 3, and for the $70 up to $75 thousand class the class
frequency is 5. This indicates that three homes sold in the $65 up to $70 thousand price range and
five in the $70 up to $75 thousand range.
It is also clear that the interval between the lowest and highest selling price in each category is $5
thousand. How would we classify a home selling for $70 thousand? It would fall in the second
class. Homes selling for $65,000 up through $69,999.99 go in the first class, but a home selling
for more than this amount goes in the next class. So the $70,000 selling price puts the home in
the second class.
The class midpoint is determined by going halfway between the lower limit of consecutive of
classes. Halfway between $65 and $70 is $67.5 thousand, the class midpoint.
The class frequencies are scaled on the vertical axis (Y-axis) and the selling price on the
horizontal (X-axis). A vertical line is drawn from the two class limits of a class to a height
corresponding to the number of frequencies. The tops of the lines are then connected.
62.5 67.5 72.5 77.5 82.5 87.5 92.5
Se lling Price ($000)
Solution 3 Frequency Polygon
Class frequencies are scaled on the vertical
axis (Y-axis) and class midpoints along the 9
horizontal axis (X-axis). The first plot is at 8
point 67.5 on the X-axis and 3 on the Y-axis. 7
Next, the midpoints of the class below the 6
first class and above the last class are added. 5
This allows the graph to be anchored to the
X-axis at zero frequencies. 2
57.5 62.5 67.5 72.5 77.5 82.5 87.5 92.5 97.5
Selling Price ($000)
Cumulative Frequency Polygon
Construct a cumulative frequency distribution by 35 120%
using the class limits. The first step is to determine 30 100%
the number of observations “less than” the upper 25
limit of each class. Three homes were sold for less 80%
than $70 and eight were sold for between $65 and 60%
$75 thousand. The eight is found by adding the 15
three that sold for $65 to $70 thousand and the five 10
that sold for between $70 and $75 thousand. The 5 20%
cumulative frequency for the fourth class is 0 0%
obtained by adding the frequencies of the first four 60 65 70 75 80 85 90
classes. The total is 25, found by (3 + 5 +8 + 9). Selling Price ($000)
The less-than-cumulative frequency distribution
would appear as shown.
a. To construct a cumulative frequency polygon the upper limits are scaled on the X-axis
and the cumulative frequencies on the Y-axis. The cumulative percents are placed
along the right-hand scale (vertical). The first plot is X = 70 and Y = 3. The next plot is 75 and
8. As shown, the points are connected with straight lines (see the above chart).
b. To estimate the amount for which less than 75 percent of the homes were sold, a horizontal
line is drawn from the cumulative percent (75) over to the cumulative frequency polygon. At
the intersection, a line is drawn down to the X-axis giving the approximate selling price. It is
about $85 thousand. Thus, about 75 percent of the homes sold for $85,000 or less.
c. To estimate the percent of the homes that sold for less than $74,000, first locate the value of
$74 on the X-axis. Next, draw a vertical line from the X-axis at 74 up to the graph. Draw a
line horizontally to the cumulative percent axis and read the cumulative percent. It is about
18%. Hence, we conclude that about 18 percent of the homes were sold for less than $74,000.
As noted, an observation is broken down into a leading digit and a Stem Leaf
trailing digit. The leading digit is called the stem and the trailing 6 779
digit the leaf. The first home sold for $67,000. The $000 was 7 0122466778889
dropped, so the stem value is 6 and the leaf value is 7. The actual 8 00022223456689
data ranges from $67 up to $89 so the stem values range from 6 to
8 using an increment of 10. The usual practice is to order the leaf observations from smallest to
The display shows that there is a concentration of data in the $70 up to $80 thousand and the $80
up to $90 thousand group. There were 13 homes that sold for $70 thousand or more but less than
$80 thousand. Specifically one sold for $70 thousand, one for $71 thousand, two for $72
thousand, one for $74 thousand, two for $76 thousand, two for $77 thousand, three sold for $78
thousand, and one sold for $79 thousand.
The time is scaled at five-year intervals on the horizontal or X-axis. The percent of disposable
income spent for groceries is scaled on the vertical or Y-axis. Two different versions are shown.
In each version plot the first point by going up from 1975 on the X-axis to 13%. Plot the second
point by going up from 1980 to 12.3%. This process is continued for the remaining periods. The
dots are connected with straight lines.
Note that in Version 2 the vertical axis did not start from zero. Technically this is called a scale
break. That is, we started at 8 and ended at 14. In Version 1 we scaled the vertical axis from 0 to
14. Both versions are correct and indicate the trend for spending disposable income for groceries
however, the visual impact is somewhat different. Notice the change in emphasis. Version 2
“shows” a more dramatic decline than is shown in Version 1. The more dramatic decline is
brought about because of the use of the scale break.
Version 1 Version 2 – Misleading
Pe rce nt of Disposable Income Spe nt on Percent of Disposable Income Spent on
Groce rie s Groceries
0.00% 75 80 85 90 95 00
75 80 85 90 95 00
The usual practice is to scale time along the horizontal axis. The height of the bars corresponds to
percent of disposable income spent for groceries. Two different versions are shown. In each
version, to form the first bar, draw parallel vertical lines from 1975 up to 13.0%. Draw a line
parallel to the X-axis at 13.0% to connect the lines. This process is continued for the other
periods. Note that in Version 2 the vertical axis did not start from zero. Technically this is called a
scale break. That is, we started at 8 and ended at 14. In Version 1 we scaled the vertical axis from
0 to 14. Both versions are correct and indicate the trend for spending disposable income for
groceries. However, the visual impact is somewhat different. Notice the change in emphasis. As
noted on the previous page, Version 2 “shows” a more dramatic decline than is shown in Version
1. The more dramatic decline is brought about because of the use of the scale break.
Version 1 Version 2 – Misleading
Percent of Disposable Income S pent on Percent of Disposable Income Spent on
75 80 85 90 95 100 75 80 85 90 95 100
The first step is to draw a circle.
Next draw a line from 0 to the Loan Purpose
center of the circle and another
from the center of the circle to Home
32%. Adding the 32% for home Improvement
improvements and the 30% for 32%
debt consolidation gives 62%. A
line is drawn from the center to Education
62%. The area between 32% and 10%
62% represents the percent of
equity loans for debt
consolidation. The process is
continued for the remaining Car Purchase
cumulative percents. Note that
more than 60 percent of the
loans are for either debt
consolidation or home Debt