Graphs of Frequency Distribution

Introduction to StatisticsChapter 2

Jan 21, 2010Class #2

Two-dimensional graphs: Basic Set-Up

Commonly Used Graphs

Histogram Height of bars proportional to

frequency Width proportional to class

boundaries Bar Chart

Height proportional to frequency Width not really significant

Frequency Polygon Plot points then connect with straight

lines

Histograms

H is t o g ra m s

Simple Bar Graph

Grouped Bar Graph

Frequency Polygons

T he S h ape of D is tr ib utio ns

D istributions c an be e ithe r s ym me tric a lor s ke w e d, de pe nding on w he the r the re are mo re fre que nc ie s a t one e nd of the distributio n tha n the o the r.

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Shape of Frequency Distribution

Symmetrical If you can draw a vertical line

through the middle (so that you have a mirror image)

The scores are evenly distributed Positively skewed

Scores piled up on left with tail on right

Negatively skewed Scores piled up on right with tail on

left

Frequency Distribution: Different Distribution shapes

Be careful…

See next slide for tricks researchers might use with graphs…

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Reifman, Larrick, & Fein, 1991

Plotting Data: describing spread of data

– A researcher is investigating short-term memory capacity: how many symbols remembered are recorded for 20 participants:

4, 6, 3, 7, 5, 7, 8, 4, 5,1010, 6, 8, 9, 3, 5, 6, 4, 11, 6

– We can describe our data by using a Frequency Distribution. This can be presented as a table or a graph. Always presents:

– The set of categories that made up the original category– The frequency of each score/category

• Three important characteristics: shape, central tendency, and variability

Frequency Distribution Tables

– Highest Score is placed at top– All observed scores are listed– Gives information about

distribution, variability, and centrality

• X = score value• f = frequency• fx = total value associated with

frequency f = N X =fX

Frequency Table Additions

– Frequency tables can display more detailed information about distribution

• Percentages and proportions• p = fraction of total group

associated with each score (relative frequency)

• p = f/N• As %: p(100) =100(f/N)

– What does this tell about this distribution of scores?

Steps in Constructing a Grouped Frequency Distribution

• 1. Determine the Class Interval Size Ideally, we wish to generate a frequency

distribution with 10 class intervals.

We would like the size (width) of each class interval to be in units of 1, 2, 3, 5, 10, 20, 30, 50, or multiples (factor of 10) of these values.

Steps in Constructing a Grouped Frequency Distribution

• 1. Determine the Class Interval Size (continued)

To Achieve These Goals, We Employ the Following Procedure:

Calculate the Range (R) of the Data Set Divide the Range by 10 Select the Tentative Class Interval Size

from the Previous Slide Closest to Your Answer in Step 2 Above (i.e. R/10). Make Certain That Your Selection Will Result in approx. 10 Class Intervals for Your Frequency Distribution.

Grouped Frequency Distribution Tables

– Sometimes the spread of data is too wide– Grouped tables present scores as class intervals

• About 10 intervals• An interval should be a simple round number

(2, 5, 10, etc), and same width• Bottom score should be a multiple of the width

– Class intervals represent Continuous variable of X:

• E.g. 51 is bounded by real limits of 50.5-51.5• If X is 8 and f is 3, does not mean they all

have the same scores: they all fell somewhere between 7.5 and 8.5

Percentiles and Percentile Ranks

– Percentile rank = the percentage of the sample with scores below or at the particular value

– This can be represented be a cumulative frequency column

– Cumulative percentage obtained by:

c% = cf/N(100)– This gives information about relative

position in the data distribution– X values = raw scores, without

context

• 1. Determine the Class Interval Size (continued)

Example: Given the following data 100 74 84 95 95 110 99 87

100 108 85 103 99 83 91 91

84 110 113 105 100 98 100 108

100 98 100 107 79 86 123 107

87 105 88 85 99 101 93 99

Steps in Constructing a Grouped Frequency Distribution

• 2. Determine the Starting Point (First Class Interval) of the Frequency Distribution

Start the Frequency Distribution with a Class Interval in Which the Following Guidelines Apply:

The First Number of the Class Interval is a Multiple of the Class

Interval Size. The First Interval Includes the Lowest

Number or Value in the Data Set

Steps in Constructing a Grouped Frequency Distribution

Credits

http://www.statcan.ca/english/edu/power/ch8/frequency.htm http://www.le.ac.uk/pc/sk219/

introtostats1.ppt#259,4,Plotting Data: describing spread of data

http://leeds-faculty.colorado.edu/luftig/Past_Course_Websites/APPM_4570_5570/Website_without_Sound/Lecture_Slides/CHAPTER2/Chap_2.ppt