Summarizing Quantitative Data Frequency Distribution Relative Frequency and Percent Frequency Distributions Histogram Cumulative Distributions Ogive

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)



Formula Worksheet

A B C D

1Parts Cost


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

)



Value Worksheet

A B C D

1Parts Cost


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.


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

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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

Documents

Summarizing Quantitative Data Frequency Distribution Relative Frequency and Percent Frequency Distributions Histogram Cumulative Distributions Ogive