MODULE 09 Frequency Distributions

7/23/2019 MODULE 09 Frequency Distributions

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Module 9: Frequency Distributions

This module includes descriptions of frequencydistributions, frequency tables, histograms and

frequency polygons. Also included are stem and

leaf plots.

Reviewed ! "ay ! #"$%&'( 9


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Frequency Distribution

A frequency distribution is the organi*ation of a data

set into contiguous, mutually e+clusive intervals so

that the number or proportion of observations fallingin each interval is apparent.


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Frequency Distribution Example

An e+ample of a frequency distribution is the age distributionof one of the classes for an in-class version of this course.

The class included n 1)1 students, with the age distribution

as shown on a following slide.

Two things are notable about this distribution. $ne has to do

with the way we measure age, specified as age at last

birthday, not age at nearest birthday.

The other is the ages listed as ! years, with )1 studentshaving this age. This is an interval with indefinite length and

it will require special handling.


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Age 0o.

)1

)) 1

) 11)/ 9

)! 12

) 1

)2

)3 !

)9 /

1 1

) /

/ )

! )1

Total 1)1


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Intervals for a Frequency Distribution

4 &se not less than intervals, generally 3-1!

4 &se equal width intervals if feasible and appropriate

4 Avoid intervals with indefinite length, if possible4 5elect interval width by dividing difference between

smallest and largest observation by 1

4 Ta6e into account the manner measurements were

made when setting up intervals


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7or the age distribution e+ample, the ages range from )1

years to ! years and older8 to construct at least intervals

for this age range, each interval has to be at least ) years

wide. To create the intervals, we need to consider themidpoint of the intervals8 The midpoint for the age interval

for persons )1 years old is )1.! years since they will not

become )) until their ))nd birthday. Thus, the age interval

for persons )1 and )) years old goes from )1 to ), with ))

as the midpoint.

)1 )) )

The intervals will be written as )1 ), ) )!, )! )2

etc. :e will deal with the ! interval in a subsequent

slide.

Midpoint


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Age 0o. ;nterval"idpoint

)1 ))

)) 1

) 11)/

)/ 9

)! 12)

) 1

)2 )3)3 !

)9 /

1 1)

) /

/

/ )

! )1 um =

)1 )) )) 13 13

)) 1

) 11)/ ) 12 !

)/ 9

)! 12) )!

) 1

)2 )3 11 9 9)3 !

)9 / 2 2!

1 1

) ! / 29) /

/ ! / 3

/ )

! )1 / )1 12 1

Total 1)1


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4 &se Braph of a frequency distribution4 7or >ontinuous variables

Rules4 0o space between bars

4 (qual areas must represent equal percentages ornumbers

4 Cercentages often preferred to numbers for verticala+is

istogram


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From a Frequency !able to a istogram

Histocomes from the Bree6 word for cell. ;n this case, the

reference is to a cell li6e those in bee honeycombs, all of whichhave an equal area when viewed from the top. That is, a

histogram is an equal cell or equal area graph, with each

percentage represented by the same area. To prepare a histogram

from a frequency table, we must first decide the shape and areafor the cell that represents a single =. :e can then stac6 the

appropriate number of these cells on top of each other to

represent the percentage of the frequency distribution located

within the interval of interest.

7or this e+ample, we will ma6e each cell be two years wide and

one unit high so that each percentage point for a specific interval

will loo6 something li6e


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The )1 to ) years interval of the age frequency distribution includes 13= of

the observations, with a midpoint of ))8 so for this interval, we need to draw

13 cells, as shown below.

)

))

1)

1/

1

13

3

1

/

)

)/

)

/))3))/))) /3

"e

rcentageofstudents

Interval Midpoint

Each yellow bloc# represents $ cell% thus the

&$ to &' years interval requires $( bloc#s

13


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)

))

1)

1/

1

13

3

1

/

)

)/

)

/))3))/))) /3

"ercentageof)tudents

Interval Midpoint

13

12

)!

9

/ /


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Dealing with our Interval with Indefinite *ength

4 The areashown for an indefinite interval should be on

same basis as it is for other intervals. $ur indefinite

interval contains 12= of the distribution so it needs to

include the area equivalent to that of 12 cells, where each

cell is ) years wide by 1= tall.

4 The widthof the indefinite interval is 1 years so that it is

3 cells wide

4 The height of the bar for this interval thus needs to be12#3 or two and 1#3=.


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) )) )/ ) )3 ) / 3 / /) // / /3 ! !) !/

)

))

1)

1/

1

13

3

1

/

)

)/

)

"ercentageofstudents

+ge Interval Midpoint

Indefinite Interval

13

12

)!

9

/ /

1)3

1)3

1)

3

1)3

1)3

1)3

1)3

1)3


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Source: Am J Public Health, April 2004;94:559


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)ource: Am J Public Health, June 2004;94:559


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Data )ource: Am J Public Health, June 2004;94:559

+ revised version of the histogram on the previous slide%to deal

with the indefinite interval for patients who are D 2/ years old, we

made an arbitrary decision that the oldest patient was 3! years old .

"idpoint of interval 19 30 40 50 60 70 80

Age, years

0

20

40

60

Numb

erofPatients

3!

'argest value


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"idpoint of

interval

19 30 40 50 60 70 87

+ge, years

0

20

40

60

-umberof"atients

'argest value

100

Data )ource: Am J Public Health, June 2004;94:559

A histogram of data fromAJPH,Eune )/8 9/!!9 article with oldest

patient assumed to be 1 years old.


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+ge Distribution of the ./)/ "opulation, by )ex: $901

Millions

15 10 5 0 5 10 15

)ource: .) 2E-).) 3.4E+.

!-9

1-1/

1!-19

)-)/

)!-)9

-/

!-9

/-//

/!-/9

!-!/

!!-!9

-/

!-9

2-2/

2!-29

3-3/

F!

+ge

Interval

FemaleMale3!


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10 5 0 5 1015

+ge Distribution of the ./)/ "opulation, by )ex: &111

Millions

Male Female

15

!-9

1-1/

1!-19)-)/

)!-)9

-/

!-9

/-//

/!-/9

!-!/

!!-!9

-/

!-9

2-2/

2!-29

3-3/

F!

+geI

nterval

)ource: .) 2E-).) 3.4E+.

Year 1950

Year 2000

3!


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15 10 5 0 5 10 15

Male Female

Millions

!-9

1-1/

1!-19

)-)/

)!-)9

-/

!-9

/-//

/!-/9

!-!/

!!-!9

-/

!-9

2-2/

2!-29

3-3/

F!

+geInterval

+ge Distribution of the ./)/ "opulation, by )ex: &101

)ource: .) 2E-).) 3.4E+.

Year 2000

Year 2050

3!


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Frequency "olygon


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A frequency polygon is prepared by connecting themidpoints of the tops of the histogram bars with

straight lines in such a manner that the area covered

by the resulting figure includes 1= of the

histogram area.

Frequency "olygon

F " l f + E l


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Frequency "olygon for +ge Example

)

))

1)

1

13

3

1

/

)

"ercentageo

fstudents

) )) )/ ) )3 ) / 3 / /) // / /3 ! !) !/

1/

)/

)

+ge Interval Midpoint


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)ource:Am J Public Health, Aug 200!;9!:!209


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)ource:Am J Public Health, Sept !9"#;##:$"4


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) d * f "l


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)tem and *eaf "lot

5tem# 'eaf >ount

)3 1) )) 1!

)! 111) /

)/ )! !

) /23

)) 1!!!39 2

)1 !!!!!!!! 3

) 12222 919 1)))//!239 1)

13 1)/!!233999 1/

12 )))//!22333 1!

1 111))))////9 1/

1! 111)//9999 1)

1/ 11)!299 1

1 1//2333 3

1) /229 2

11 1)/ !

1 1)22 /

9 )/

2 1) )

The stem contains the

leading digits of the

observations8 because we

have digit numbers from

11 to )3) for this data set

the stem includes the firsttwo digits of the numbers.

;n general the stem consists

of n-1 digits, where n is the

number of digits in the

number.

7or the last two sets of

numbers the stem

contains only 1 digit

because the observations

are ) digit numbers.

The leaf always contains

the last digit of eachobservation with the

same leading digits8

arranged in increasing

order of magnitude from

9 to form the leaf.

Bives the number

of digits in the

leaf of each stem


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4 A chart that displays a frequency distribution

similar to a histogram.

4 A stem and leaf plot shows

4 The spread of the data 4 The mode

4 :hether the distribution is s6ewed

4 :hether there are gaps in the data4 :hether there are any unusual data points

)tem and *eaf "lot


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)tem and *eaf "lot

A stem and leaf plot typically consists of threecolumns, the first two being separated by a single

blan6 space. The first column is the stem or leading

digits. The second is the leaf which represents the

values in the interval following the stem digits. The

third column indicates the number of data values or

count in the interval.


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)tem and *eaf "lot Example

5uestion %isplay the information in the table below in a 5tem and 'eaf Clot.


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)tem *eaf "lot for 3lood 2holesterol Data

5tem#'eaf >ount

)2 1

) 1 1

)!

)/ 2 1

) ! 1

)) 13 )1 2 1

) 1 !

19 1)))//!239 1)

13 1)/!!23399 1

12 )))//!239 1

1 111))))////!!!2339999 )/

1! 111))/////!!!3339999 )!1/ 111)!299 11

1 1/2333 1

1) /29 !

11 1/ )

1

9 ! 1

)+) 6utput


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Y

30

20

10

0

Std. Dev = 28.66

Mean = 167.6

N = 129.00

)

3

)

2

)

1

)

!

)

/

2

)

,

!

)

)

13

)

1

2

)

1

1

9

1))),//!239

1

3

1)/!!23399

1

2

))),,//!239

1

111))))////!!!2339999

1

!

111)),,/////!!!3339999

1

/

111),!299

1

,

1,/2333

1

)

,/29

1

1

1/

1

9

!

0ote The stem G 'eaf plot is plotted in decreasing order of

magnitude, while the @istogram is plotted in increasing order, from

ll h hi h id i

istogram and )tem 7 *eaf plot for the 2holesterol Data

MODULE 09 Frequency Distributions

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