1. Frequency Distributions
Section 2.1
Frequency Distributions
and Their Graphs
Larson/Farber 4th ed.

2. Section 2.1 Objectives Construct frequency distributions
Construct frequency histograms, frequency polygons, relative frequency histograms, and ogives
Larson/Farber 4th ed.

3. Frequency Distribution
A table that shows classes or intervals of data with a count of the number of entries in each class.
The frequency, f, of a class is the number of data entries in the class.
Class
Frequency, f
1 – 5 5
6 – 10 8
11 – 15 6
16 – 20
21 – 25
26 – 30 4
Class width 6 – 1 = 5
Lower class
limits
Upper class
limits
Larson/Farber 4th ed.

4. Constructing a Frequency Distribution
Decide on the number of classes.
Usually between 5 and 20; otherwise, it may be difficult to detect any patterns.
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.

5. Constructing a Frequency Distribution
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.

6. Constructing a Frequency Distribution
Make a tally mark for each data entry in the row of the appropriate class.
Count the tally marks to find the total frequency f for each class.
Larson/Farber 4th ed.

7. 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
248
101
375
486
190
398
269 43 30
127
354 84
Larson/Farber 4th ed.

8. Solution: Constructing a Frequency Distribution91
472
279
249
530
376
188
341
266
199
142
273
189
130
489
248
101
375
486
190
398
269 43 30
127
354 84
Number of classes = 7 (given)
Find the class width
Round up to 72

9. Solution: Constructing a Frequency Distribution
Use 30 (minimum value) as first lower limit. Add the class width of 72 to get the lower limit of the next class.
= 102
Find the remaining lower limits.
Lower limit
Upper limit 30
102
174
246
318
390
462
Class width = 72

10. 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.
= 173
Find the remaining upper limits.
Lower limit
Upper limit 30
101
102
173
174
245
246
317
318
389
390
461
462
533
Class width = 72
Larson/Farber 4th ed.

11. Solution: Constructing a Frequency Distribution
Make a tally mark for each data entry in the row of the appropriate class.
Count the tally marks to find the total frequency f for each class.
Class
Tally
Frequency, f
lllll 5
lll 3
lllll ll 7
llll 4 l 1
Σf = 29
Larson/Farber 4th ed.

12. Determining the Midpoint
Midpoint of a class
Class
Midpoint
Frequency, f
65.5 5
137.5 3
209.5
281.5 7
353.5 4
425.5 1
497.5
Class width = 72

13. 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.

14. Determining the Relative Frequency continued
Class
Midpoint
Frequency, f
Relative
Frequency
65.5 5
0.172
137.5 3
0.103
209.5
281.5 7
0.241
353.5 4
0.138
425.5 1
0.034
497.5
Σf = 29
1.000
Sum of Rel. Freq.

15. Determining the Cumulative Frequency
Cumulative frequency of a class
The sum of the frequency for that class and all previous classes.
Class
Midpoint
Frequency, f
Cumulative
Frequency
65.5 5
137.5 3 8
209.5 13
281.5 7 20
353.5 4 24
425.5 1 25
497.5 29
Σf =
Sum of class and previous class.

16. Expanded Frequency Distribution
Class
Midpoint
Frequency, f
Relative
Frequency
Cumulative
65.5 5
0.172
137.5 3
0.103 8
209.5 13
281.5 7
0.241 20
353.5 4
0.138 24
425.5 1
0.034 25
497.5 29
Σ =

17. Graphs of Frequency Distributions
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
frequency
Larson/Farber 4th ed.

18. Class Boundaries Class boundaries
The numbers that separate classes without forming gaps between them.
The distance from the upper limit of the first class to the lower limit of the second class is 102 – 101 = 1.
Half this distance is 0.5.
Class
Boundaries
29.5 – 101.5
First class lower boundary = 30 – 0.5 = 29.5
First class upper boundary = = 101.5
Larson/Farber 4th ed.

19. Class Boundaries Class Class boundaries Frequency, f 30 - 101 5 3 7 4 1
Larson/Farber 4th ed.

20. Example: Frequency Histogram
Construct a frequency histogram for the book costs frequency distribution.
Class
Class boundaries
Midpoint
Frequency, f
65.5 5
137.5 3
209.5
281.5 7
353.5 4
425.5 1
497.5
Larson/Farber 4th ed. 21

21. Solution: Frequency Histogram (using Midpoints)

22. Graphs of Frequency Distributions
Frequency Polygon
A line graph that emphasizes the continuous change in frequencies.
data values
frequency
Larson/Farber 4th ed.

23. Example: Frequency Polygon
Construct a frequency polygon for the Books costs frequency distribution.
Class
Midpoint
Frequency, f
65.5 5
137.5 3
209.5
281.5 7
353.5 4
425.5 1
497.5
Larson/Farber 4th ed.

24. 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!

25. Graphs of Frequency Distributions
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
relative frequency
Larson/Farber 4th ed.

26. Graphs of Frequency Distributions
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
cumulative frequency
Larson/Farber 4th ed.

27. Constructing an Ogive
Construct a frequency distribution that includes cumulative frequencies as one of the columns.
Specify the horizontal and vertical scales.
The horizontal scale consists of the upper class boundaries or upper limit.
The vertical scale measures cumulative frequencies.
Plot points that represent the upper class boundaries and their corresponding cumulative frequencies.
Larson/Farber 4th ed.

28. Constructing an Ogive Connect the points in order from left to right.
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.

29. Example: Ogive
Construct an ogive for the book cost frequency distribution.
Class
Midpoint
Frequency, f
Cumulative
Frequency
65.5 5
137.5 3 8
209.5 13
281.5 7 20
353.5 4 24
425.5 1 25
497.5 29

30. 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.

31. Section 2.1 Summary Constructed frequency distributions
Constructed frequency histograms, frequency polygons, relative frequency histograms and ogives
Larson/Farber 4th ed.