Summarizing Quantitative Data
• Frequency Distribution• Relative Frequency and Percent Frequency
Distributions• Histogram• Cumulative Distributions• Ogive
Constructing a Frequency Distribution for Quantitative Data
3 initial steps
1. Determine the number of nonoverlapping classes.
2. Determine the width of each class.
3. Determine the class limits
Example: Hudson Auto RepairThe manager of Hudson AutoThe manager of Hudson Auto
would like to have a betterwould like to have a better
understanding of the costunderstanding of the cost
of parts used in the engineof parts used in the engine
tune-ups performed in thetune-ups performed in the
shop. She examines 50shop. She examines 50
customer invoices for tune-ups. The costs of customer invoices for tune-ups. The costs of parts,parts,
rounded to the nearest dollar, are listed on the rounded to the nearest dollar, are listed on the nextnext
slide.slide.
Example: Hudson Auto RepairExample: Hudson Auto Repair
Sample of Parts Cost for 50 Tune-upsSample of Parts Cost for 50 Tune-ups
91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73
91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73
Frequency Distribution• Guidelines for Selecting Number of
Classes
• Use between 5 and 20 classes.Use between 5 and 20 classes.
• Data sets with a larger number of elementsData sets with a larger number of elements usually require a larger number of classes.usually require a larger number of classes.
• Smaller data sets usually require fewer classesSmaller data sets usually require fewer classes
Frequency Distribution• Guidelines for Selecting Width of Classes
Largest Data Value Smallest Data ValueNumber of Classes
Largest Data Value Smallest Data ValueNumber of Classes
•Use classes of equal width.Use classes of equal width.
•Approximate Class Width =Approximate Class Width =
Frequency Distribution
•We decide that SIX (6) classes are appropriate for our purposes
•Thus the approximate class width is given by:
Approximate Class Width = (109 - 52)/6 = 9.5 Approximate Class Width = (109 - 52)/6 = 9.5 1010
Thus we have:
Parts Cost($) Frequency
50-59 2
60-69 13
70-79 16
80-89 7
90-99 7
100-109 5
Total 50
Using Excel’s Frequency Function to Construct a Frequency Distribution
A B C D
1Parts Cost
Parts Cost Frequency
2 91 50-593 71 60-694 104 70-795 85 80-896 62 90-997 78 100-1098 69
Note: Rows 9-51 are not shown.Note: Rows 9-51 are not shown.
Formula Worksheet (showing data entered)
Using Excel’s Frequency Function to Construct a Frequency Distribution
Note: Rows 9-51 are not shown.Note: Rows 9-51 are not shown.
Formula Worksheet
A B C D
1Parts Cost
Parts Cost Frequency
2 91 50-59 =FREQUENCY(A2:A51,{59,69,79,89,99,109})3 71 60-69 =FREQUENCY(A2:A51,{59,69,79,89,99,109})4 104 70-79 =FREQUENCY(A2:A51,{59,69,79,89,99,109})5 85 80-89 =FREQUENCY(A2:A51,{59,69,79,89,99,109})6 62 90-99 =FREQUENCY(A2:A51,{59,69,79,89,99,109})7 78 100-109 =FREQUENCY(A2:A51,{59,69,79,89,99,109})8 69
Relative Frequency and Percent Frequency
Parts Cost
($)
Relative Frequency
Percent Frequency
50-59 .04 4
60-69 .26 26
70-79 .32 32
80-89 .14 14
90-99 .14 14
100-109 .10 10
Total 1.00 100
2/50.04(100
)
Using Excel’s Frequency Function to Construct a Frequency Distribution
Note: Rows 9-51 are not shown.Note: Rows 9-51 are not shown.
Value Worksheet
A B C D
1Parts Cost
Parts Cost Frequency
2 91 50-59 23 71 60-69 134 104 70-79 165 85 80-89 76 62 90-99 77 78 100-109 58 69
•5 classes should be sufficient, given the size of our data set.
•Approximate class width is given by:
2.45
12-33 WidthClass eApproximat
Audit example
We round up to the next highest integer, so our class interval is 5.
Using Excel’s Frequency Function to Construct a Frequency Distribution
1. Step 1: Select cells D2:D6, the cells in which we want the frequencies to appear
2. Step 2: Type the following formula in the formula bar:
=FREQUENCY(A2:A21,{14,19,24,29,34})
3. Step 3: Press Ctrl-Shift-Enter
The Audit Example
Audit Time Frequency Relative Frequency Percent Frequency
10-14 4 0.2 20
15-19 8 0.4 40
20-24 5 0.25 25
25-29 2 0.1 10
30-34 1 0.05 5
Total 20 1 100
The HistogramThis is a bar graph of
a frequency distribution for
quantitative data.
•The variable of interest is measured on the horizontal axis.
•Frequency, relative frequency, or percent frequency for each class is measured on the vertical axis.
•There are no spaces between the bars on a histogram—the rectangles are adjacent.
Histogram
PartsCost ($)PartsCost ($)
22
44
66
88
1010
1212
1414
1616
1818
Fre
qu
en
cy
Fre
qu
en
cy
50 60 70 80 90 100 110 12050 60 70 80 90 100 110 120
Tune-up Parts CostTune-up Parts Cost
Using Excel’s Chart Wizard to Construct a Histogram: Audit Example
1. Step 1: Select cells C1:D6.
2. Click the Chart Wizard button.
3. When Chart Type dialog box appears, select Clustered Column from the Chart sub-type display. Click Next>.
4. At Step 2 —Chart Data Source —Click Next>.
5. At Step 3 – Chart Options – select Titles Tab and then type Histogram for Audit Time Data in the Chart title box. Type Audit Time in Days in the Category (X) axis box and type Frequency in the Value (Y) axis box. Select the Legend tab and then remove the check in the Show Legend box. Click Next>.
6. Step 6: Click Finish
Removing the Gaps
Between Rectangles (bars)
• Step 1: Click right on any rectangle (bar) in the chart to produce a list of options.
• Step 2: Select the Format Data Series option• Step 3: Select the Options tab and then enter 0 in the
Gap width box.Click OK
Histogram for Audit Time Data
0
2
4
6
8
10
10-14 15-19 20-24 25-29 30-34
Audit Time in Days
Fre
qu
ency
It looks like this:
Cumulative Distributions
These show the number of data items with values less than or equal to the class limit of each class.
•Cumulative relative frequency distributions show the proportion of data items with values less than or equal to the class limit of each class.
•Cumulative frequency distributions show the proportion of data items with values less than or equal to the class limit of each class.
Cumulative Distributions• Hudson Auto Repair
<< 59 59
<< 69 69
<< 79 79
<< 89 89
<< 99 99
<< 109 109
Cost ($)Cost ($)
CumulativeCumulativeFrequencyFrequency
CumulativeCumulativeRelativeRelative
FrequencyFrequency
CumulativeCumulativePercentPercent
FrequencyFrequency
22
1515
3131
3838
4545
50 50
.04.04
.30.30
.62.62
.76.76
.90.90
1.00 1.00
44
3030
6262
7676
9090
100 100
2 + 2 + 1313
15/5015/50 .30(10.30(100)0)
PartsPartsCost Cost ($)($) PartsPartsCost Cost ($)($)
2020
4040
6060
8080
100100
Cu
mu
lati
ve P
erc
en
t Fr
eq
uen
cyC
um
ula
tive P
erc
en
t Fr
eq
uen
cyC
um
ula
tive P
erc
en
t Fr
eq
uen
cyC
um
ula
tive P
erc
en
t Fr
eq
uen
cy
50 60 70 80 90 100 11050 60 70 80 90 100 11050 60 70 80 90 100 11050 60 70 80 90 100 110
(89.5, (89.5, 76)76)
Ogive withOgive with
Cumulative Percent Frequencies Cumulative Percent Frequencies
Tune-up Parts CostTune-up Parts CostTune-up Parts CostTune-up Parts Cost
Ogive for Audit Time Data
0
5
10
15
20
25
1-14 15-19 20-24 25-29 30-34
Audit Time in Days
Fre
qu
ency
Ogive for Audit Example
Cumulative Percent Frequency for the Audit Example
0
20
40
60
80
100
120
1-14 15-19 20-24 25-29 30-34
Audit Time in Days
Cu
mu
lati
ve P
erce
nt
Fre
qu
ency