HISTOGRAMS

Representing Data

Why use a Histogram

When there is a lot of data When data is

Continuous a mass, height, volume, time etc

Presented in a Grouped Frequency Distribution Often in groups or classes that are UNEQUAL

Continuous data

NO GAPS between Bars

Histograms look like this......

Bars may be different in width

Determined by Grouped Frequency Distribution

AREA is proportional to FREQUENCY

NOT height, because of UNEQUAL classes!

So we use FREQUENCY DENSITY = Frequency Class width

Grouped Frequency Distribution

Speed, km/h

0< v ≤40 40< v ≤50 50< v ≤60 60< v ≤90 90< v ≤110

Frequency 80 15 25 90 30

Classes

These classes are well defined there are no gaps !

Drawing

Sensible Scales Bases of rectangles correctly aligned

Plot the Class Boundaries carefully Heights of rectangles needs to be correct

Frequency Density

Speed, kph 0< v ≤40 40< v ≤50 50< v ≤60 60< v ≤90 90< v ≤110

Frequency 80 15 25 90 30

Frequency Density

Class width 40 10 10 30 20

2.0 1.5 2.5 3.0 1.5

Frequency Densities

0 4020 60 80 100 120

3.0

2.0

1.0

Fre

q D

en

s

Speed (km/h)

Frequency = Width x Height

Frequency = 40 x 2.0 = 80

Grouped Frequency Distribution

Time taken (nearest minute)

5-9 10-19 20-29 30-39 40-59

Freq 14 9 18 3 5

Speed, kph 0< v ≤40 40< v ≤50 50< v ≤60 60< v ≤90 90< v ≤110

Frequency 80 15 25 90 30

ClassesNo gaps

GAPS! Need to adjust to Continuous

Ready to graph

Adjusting Classes

Class Widths

Time taken (nearest minute)

5-9 10-19 20-29 30-39 40-59

Freq 14 9 18 3 5

9½4½ 19½ 29½ 39½ 59½

105 10 10 20

Frequency Density

Time taken (nearest minute) 5-9 10-19 20-29 30-39 40-59

Freq 14 9 18 3 5

Class width 5 10 10 10 20

Frequency Density 2.8 0.9 1.8 0.3 0.25

Drawing

Sensible Scales Bases correctly aligned

Plot the Class Boundaries Heights correct

Frequency Density

4.5 19.59.5 29.5 39.5 49.5 59.5

3.0

2.0

1.0

Fre

q D

en

s

Time (Mins)

5 10 15 20 25 30 35 40 45 50 55 60

Estimating a Frequency

Imagine we want to Estimate the number of people with a time between 12 and 25 mins

Because we have rounded to nearest minute with our classes we......... Consider the interval from 11.5 to 25.5

4.5 19.59.5 29.5 39.5 49.5 59.5

3.0

2.0

1.0

Fre

q D

en

s

Time (Mins)

11.5 25.5

Frequency = 0.9 x 8 = 7.2

Frequency = 1.8 x 6 = 10.8

Total Frequency = 18

FD Width

We can estimate the Mode

Time taken (nearest minute)

5-9 10-19 20-29 30-39 40-59

Freq 14 9 18 3 5

CF 14 23 41 44 49

Mode is therefore in this Class

4.5 19.59.5 29.5 39.5 49.5 59.5

3.0

2.0

1.0

Fre

q D

en

s

Time (Mins)

Modal class

…and the other one?

Simpler to plot No adjustments required – class widths friendly No ½ values

Estimation from the EXACT values given No adjustment required Estimate 15 to 56 would use 15 and 56!

Appear LESS OFTEN in the exam

Speed, kph 0< v ≤40 40< v ≤50 50< v ≤60 60< v ≤90 90< v ≤110

Frequency 80 15 25 90 30

Why use frequency density for the vertical axes of a Histogram?

The effect of unequal class sizes on the histogram can lead to misleading ideas about the data distribution

widthclass

class offrequency relativeheight rectangledensity

widthclass

class offrequency heightrectangle densityfrequency

The vertical axis is Frequency Density

Example: Misprediction of Grade Point Average (GPA)The following table displays the differences between predicted GPA and actual GPA. Positive differences result when predicted GPA > actual GPA.

Class Interval Frequency Class width

-2.0 to < -0.4 23 1.6

-0.4 to < -0.2 55 0.2

-0.2 to < -0.1 97 0.1

-0.1 to < 0 210 0.1

0 to < 0.1 189 0.1

0.1 to < 0.2 139 0.1

0.2 to < 0.4 116 0.2

0.4 to < 2.0 171 1.6

The frequency histogram considerably exaggerates the incidence of overpredicted and underpredicted values

The area of the two most extreme rectangles are much too large.!!

X 10-3

1000

2.3% of data

17.1% of data

Example: Density Histogram of Misreporting GPA

Class Interval Frequency Class width FrequencyDensity

-2.0 to < -0.4 23 1.6 14

-0.4 to < -0.2 55 0.2 275

-0.2 to < -0.1 97 0.1 970

-0.1 to < 0 210 0.1 2100

0 to < 0.1 189 0.1 1890

0.1 to < 0.2 139 0.1 1390

0.2 to < 0.4 116 0.2 580

0.4 to < 2.0 171 1.6 107

widthclass

class offrequency heightrectangle densityfrequency

Frequency=( rectangle height )x( class width ) = area of rectangle

To avoid the misleading histogram like the one on last slide,

display the data with frequency density

X 10-3

Frequency density x 10-3

Chap 2-24

Principles of Excellent Graphs The graph should not distort the data. The graph should not contain unnecessary things

(sometimes referred to as chart junk). The scale on the vertical axis should begin at zero. All axes should be properly labelled. The graph should contain a title. The simplest possible graph should be used for a

given set of data.

Chap 2-25

Graphical Errors: Chart Junk

1960: $1.00

1970: $1.60

1980: $3.10

1990: $3.80

Minimum Wage

Bad Presentation

Minimum Wage

0

2

4

1960 1970 1980 1990

$

Good Presentation

Chap 2-26

Graphical Errors: No Relative Basis

A’s received by students.

A’s received by students.

Bad Presentation

0

200

300

FD UG GR SR

Freq.

10%

30%

FD UG GR SR

FD = Foundation, UG = UG Dip, GR = Grad Dip, SR = Senior

100

20%

0%

%

Good Presentation

Chap 2-27

Graphical Errors: Compressing the Vertical Axis

Good Presentation

Quarterly Sales Quarterly Sales

Bad Presentation

0

25

50

Q1 Q2 Q3 Q4

$

0

100

200

Q1 Q2 Q3 Q4

$

Chap 2-28

Graphical Errors: No Zero Point on the Vertical Axis

Monthly Sales

36

39

42

45

J F M A M J

$

Graphing the first six months of sales

Monthly Sales

0

39

42

45

J F M A M J

$

36

Good PresentationsBad Presentation