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Chapter

Two

McGraw-Hill/Irwin

© 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.

Describing Data:Frequency Distribution & Graphic Presentation

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‘Organizing’ data – Remember COPAID?

A Frequency DistributionFrequency Distribution is a grouping of data into mutually exclusive

categories showing the number of observations in each class.

Each class has an interval of $3K

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Decide on the number of classes

5 Steps to constructing a frequency distribution

Determine the class interval

Set the individual class limits

Tally the number of items in each class

Count the number of items in each class

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Example

RAW DATA - 80 data points

Let us ‘organize’ this raw data into different price ranges.

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Example 1 continued

Step OneStep One:: Decide on the number of classes using the formula

22kk > n > n (2 to the k rule)(2 to the k rule)

where k=number of classes n=number of observations

There are 80 observations, so n=80. 266 = 64 & 277 = 128. 277 > 80 > 80Therefore, we should have at least 7 classes, i.e., k=7.

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where H=highest value, L=lowest value

35925 – 15546 7

= $2911

Step TwoStep Two: Determine the class interval or width using the formula

H – LH – L kk

i > =

Round up for an interval of $3000.Set the lower limit of the first class at $15000.Guideline: Make the lower limit of the first class a multiple of the class interval.

Example 1 continued

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EXAMPLE 1 continued

Step ThreeStep Three: Set the individual class limits

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Steps FourSteps Four: Tally the number of items in each class.

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Step FiveStep Five: (last step!)Count the number of items in each class.

The above table is called a Frequency Distribution(one way of ‘organizing’ initial raw data – again remember COPAID) .

You can also present in a graph – we will see that later.

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A Relative Frequency DistributionRelative Frequency Distribution shows the fraction of observations in each class. (easier to compare classes)

You can also express Relative Frequency in %ages. For the above data, it will be 10%, 28.75%, 21.25%, …

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Try out

problem #6

in page 31

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Visuals used in StatisticsVisuals used in Statistics

•HistogramsHistograms•Frequency PolygonsFrequency Polygons•Line & Bar graphsLine & Bar graphs•Pie chartsPie charts•Scatter diagramsScatter diagrams•Contingency tablesContingency tables•Pareto chartsPareto charts

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HistogramA graph with

X- axis: ClassesY- axis: Frequency

Histogram gives the frequency distribution of data

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Graphic Presentation of a Frequency Distribution

A Frequency PolygonFrequency Polygon

- a line graph connecting the points formed by the class midpoint and the class frequency.

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Frequency polygons allow comparison of 2 more frequency distributions

Use %age frequencies if actual frequencies vary widely

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A Cumulative Frequency DistributionCumulative Frequency Distribution is used to determine how many or what proportion of the data values are below or above a certain value.

Cumulative Frequency DistributionCumulative Frequency Distribution

Find the price below which 25 vehicles were sold.Find the price below which half the cars were sold.Find %age of vehicles sold priced below $28500.

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Practice time!

(Problem #14 Page 41)

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Line graphs are typically used to show the change or trend of a variable over time.

U.S. median age by gender

25

30

35

40

Med

ian

Age

Males

Females

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Example 3 continued

Another example of Line Graph

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A Bar ChartBar Chart - useful to show data of any level of measurement.

Bar Chart

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Pie Chart

A Pie ChartPie Chart is useful for displaying a relative frequency distribution.

- a circle is divided proportionally to the relative frequency

Along with the % of each slice, you can also show the actual values

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Scatter diagram

Example The twelve days of stock prices and the overall market index on each day are given as follows:

Variables must be at least interval scaled.

Relationship can be positive (direct) or negative (inverse).

A graph showing relationship between two variables

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969291888685848382797869

PriceIndex(000s)

8.07.57.57.37.27.27.17.17.06.26.25.1

Relationship between Market Index and Stock Price

50

60

70

80

90

100

5 6 7 8 9 10

Index

Pri

ce

Scatter Diagram

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A contingency table is a cross tabulation of two variables.

Contingency tables are used when one or both variables are nominal or ordinal in scale.

Contingency table

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Male Female

Smokers 100 60

Non-

smokers

850 1075

Example Contingency Table

Good Bad Ugly

Dumbo 1 25 74

As Good as It Gets

63 22 15

Nominal

Ordinal

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•a type of histogram; arrange the bars from tallest to shortest

•used in a process improvement project

•Eg. If you record the reasons for a machine breaking down, you might find some problems to be more common than others.

•If you record the frequency of each of these reasons, you will notice that a small number of reasons will account for most of the breakdowns.

•Pareto Principle, or the 80-20 rule. In general about 80% of the problems will result from about 20% of the causes.

Pareto Chart

2- 28  Number of Defects

paper particle buildup 74

excessive temperature 38

worn roller 5

defective paper 10

guides misaligned 26

Pareto Chart

Problems of a

photocopier machine

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 Number of Defects

Repair Cost ($) Total Cost

paper particle buildup

74 30 2220

excessive temperature

38 75 2850

worn roller 5 190 950

defective paper 10 20 200

guides misaligned 26 30 780

Pareto Chart