Section 2.1
Frequency Distributions
and Their Graphs
Larson/Farber 4th ed.
Section 2.1 Objectives
• Construct frequency distributions
• Construct frequency histograms, frequency polygons,
relative frequency histograms, and ogives
Larson/Farber 4th ed.
Frequency Distribution
Frequency Distribution
Class Frequency, f
Class width 6
• A table that shows
1–5
5
–
1
=
5
classes or intervals of
6 – 10
8
data with a count of the
11 – 15
6
number of entries in
16 – 20
8
each class.
21 – 25
5
• The frequency, f, of a
class is the number of
26 – 30
4
data entries in the class. Lower class
Upper class
limits
Larson/Farber 4th ed.
limits
Constructing a Frequency Distribution
1. Decide on the number of classes.
 Usually between 5 and 20; otherwise, it may be
difficult to detect any patterns.
2. Find the class width.
 Determine the range of the data.
 Divide the range by the number of classes.
 Round up to the next convenient number.
Larson/Farber 4th ed.
Constructing a Frequency Distribution
3. Find the class limits.
 You can use the minimum data entry as the lower
limit of the first class.
 Find the remaining lower limits (add the class
width to the lower limit of the preceding class).
 Find the upper limit of the first class. Remember
that classes cannot overlap.
 Find the remaining upper class limits.
Larson/Farber 4th ed.
Constructing a Frequency Distribution
4. Make a tally mark for each data entry in the row of
the appropriate class.
5. Count the tally marks to find the total frequency f
for each class.
Larson/Farber 4th ed.
Example: Constructing a Frequency
Distribution
The following sample data set lists the amount spent on
books for a semester.
Construct a frequency distribution that has seven
classes.
91 472 279 249 530 376 188 341 266
199 142 273 189 130 489 266 248 101 375
486 190 398 188 269 43 30 127 354 84
Larson/Farber 4th ed.
Solution: Constructing a Frequency
Distribution
91 472 279 249 530 376 188 341 266
199 142 273 189 130 489 266 248 101 375
486 190 398 188 269 43 30 127 354 84
1. Number of classes = 7 (given)
2. Find the class width
Round up to 72
Solution: Constructing a Frequency
Distribution
3. Use 30 (minimum value)
Class width
as first lower limit. Add
= 72
the class width of 72 to
get the lower limit of the
next class.
30 + 72 = 102
Find the remaining lower
limits.
Lower
limit
30
102
174
246
318
390
462
Upper
limit
Solution: Constructing a Frequency
Distribution
The upper limit of the
first class is 101 (one less
than the lower limit of
the second class).
Add the class width of 72
to get the upper limit of
the next class.
101 + 72 = 173
Find the remaining upper
limits.
Larson/Farber 4th ed.
Lower
limit
Upper
limit
30
101
102
174
173
246
317
318
389
390
461
462
533
245
Class width
= 72
Solution: Constructing a Frequency
Distribution
4. Make a tally mark for each data entry in the row of
the appropriate class.
5. Count the tally marks to find the total frequency f
for each class.
Class
Larson/Farber 4th ed.
Tally
Frequency, f
30 - 101
lllll
5
102 - 173
lll
3
174 - 245
lllll
5
246 - 317
lllll ll
7
318 - 389
llll
4
390 - 461
l
1
462 - 533
llll
4
Σf = 29
Determining the Midpoint
Midpoint of a class
Class
Midpoint
Frequency, f
30 - 101
65.5
5
102 - 173
137.5
3
174 - 245
209.5
5
246 - 317
281.5
7
318 - 389
353.5
4
390 - 461
425.5
1
462 - 533
497.5
4
Class width = 72
Determining the Relative Frequency
Relative Frequency of a class
• Portion or percentage of the data that falls in a
particular class.
Larson/Farber 4th ed.
Determining the Relative Frequency
continued
Class
Midpoint
Frequency, f
Relative
Frequency
30 - 101
65.5
5
0.172
102 - 173
137.5
3
0.103
174 - 245
209.5
5
0.172
246 - 317
281.5
7
0.241
318 - 389
353.5
4
0.138
390 - 461
425.5
1
0.034
462 - 533
497.5
4
0.138
29
1.000
Σf =
Sum of Rel.
Freq.
Determining the Cumulative Frequency
Cumulative frequency of a class
• The sum of the frequency for that class and all
previous classes.
Class
Midpoint
30 - 101
65.5
5
5
102 - 173
137.5
3
8
174 - 245
209.5
5
13
246 - 317
281.5
7
20
318 - 389
353.5
4
24
390 - 461
425.5
1
25
462 - 533
497.5
4
29
Σf =
Frequency, f Cumulative
Frequency
29
Sum of class
and previous
class.
Expanded Frequency Distribution
Class
Midpoint
Frequency, f
Relative
Frequency
Cumulative
Frequency
30 - 101
65.5
5
0.172
5
102 - 173
137.5
3
0.103
8
174 - 245
209.5
5
0.172
13
246 - 317
281.5
7
0.241
20
318 - 389
353.5
4
0.138
24
390 - 461
425.5
1
0.034
25
462 - 533
497.5
4
0.138
29
29
1
Σ=
Graphs of Frequency Distributions
frequency
Frequency Histogram
• A bar graph that represents the frequency
distribution.
• The horizontal scale is quantitative and measures the
data values.
• The vertical scale measures the frequencies of the
classes.
• Consecutive bars must touch.
data values
Larson/Farber 4th ed.
Class Boundaries
Class boundaries
• The numbers that separate classes without forming
gaps between them.
• The distance from the upper
Class
Class
Boundaries
limit of the first class to the
29.5 – 101.5
30 - 101
lower limit of the second
102 - 173
class is 102 – 101 = 1.
174 - 245
• Half this distance is 0.5.
• First class lower boundary = 30 – 0.5 = 29.5
• First class upper boundary = 101 + 0.5 = 101.5
Larson/Farber 4th ed.
Class Boundaries
Class boundaries
Frequency, f
30 - 101
29.5 - 101.5
5
102 - 173
101.5 - 173.5
3
174 - 245
173.5 - 245.5
5
246 - 317
245.5 - 317.5
7
318 - 389
317.5 - 389.5
4
390 - 461
389.5 - 461.5
1
462 - 533
461.5 - 533.5
4
Class
Larson/Farber 4th ed.
Example: Frequency Histogram
Construct a frequency histogram for the book costs
frequency distribution.
Larson/Farber 4th ed.
Class
Class
boundaries
30 - 101
29.5 - 101.5
65.5
5
102 - 173
101.5 - 173.5
137.5
3
174 - 245
173.5 - 245.5
209.5
5
246 - 317
245.5 - 317.5
281.5
7
318 - 389
317.5 - 389.5
353.5
4
390 - 461
389.5 - 461.5
425.5
1
462 - 533
461.5 - 533.5
497.5
4
Midpoint
Frequency,
f
21
Solution: Frequency Histogram
(using Midpoints)
Graphs of Frequency Distributions
frequency
Frequency Polygon
• A line graph that emphasizes the continuous change
in frequencies.
data values
Larson/Farber 4th ed.
Example: Frequency Polygon
Construct a frequency polygon for the Books costs
frequency distribution.
Frequency, f
Class
Larson/Farber 4th ed.
Midpoint
30 - 101
65.5
5
102 - 173
137.5
3
174 - 245
209.5
5
246 - 317
281.5
7
318 - 389
353.5
4
390 - 461
425.5
1
462 - 533
497.5
4
Solution: Frequency Polygon
The graph should begin
and end on the
horizontal axis, so
extend the left side to
one class width before
the first class midpoint
and extend the right
side to one class width
after the last class
midpoint.
This is bull! In most
cases it is better to just
show the data and not
have false markers!
Graphs of Frequency Distributions
relative
frequency
Relative Frequency Histogram
• Has the same shape and the same horizontal scale as
the corresponding frequency histogram.
• The vertical scale measures the relative frequencies,
not frequencies.
data values
Larson/Farber 4th ed.
Graphs of Frequency Distributions
cumulative
frequency
Cumulative Frequency Graph or Ogive
• A line graph that displays the cumulative frequency
of each class at its upper class boundary.
• The upper boundaries are marked on the horizontal
axis.
• The cumulative frequencies are marked on the
vertical axis.
data values
Larson/Farber 4th ed.
Constructing an Ogive
1. Construct a frequency distribution that includes
cumulative frequencies as one of the columns.
2. Specify the horizontal and vertical scales.
 The horizontal scale consists of the upper class
boundaries or upper limit.
 The vertical scale measures cumulative
frequencies.
3. Plot points that represent the upper class boundaries
and their corresponding cumulative frequencies.
Larson/Farber 4th ed.
Constructing an Ogive
4. Connect the points in order from left to right.
5. The graph should start at the lower boundary of the
first class (cumulative frequency is zero) and should
end at the upper boundary of the last class
(cumulative frequency is equal to the sample size).
Larson/Farber 4th ed.
Example: Ogive
Construct an ogive for the book cost frequency
distribution.
Class
Midpoint
Frequency, f Cumulative
Frequency
30 - 101
65.5
5
5
102 - 173
137.5
3
8
174 - 245
209.5
5
13
246 - 317
281.5
7
20
318 - 389
353.5
4
24
390 - 461
425.5
1
25
462 - 533
497.5
4
29
Solution: Ogive
From the ogive, you can see that about 25 students spent $461 or
less. The greatest increase in in cost occurs between $245 and
$389.
Larson/Farber 4th ed.
Section 2.1 Summary
• Constructed frequency distributions
• Constructed frequency histograms, frequency
polygons, relative frequency histograms and ogives
Larson/Farber 4th ed.