2007© BOLD Educational Software Descriptive Statistics: Frequency Distributions and Graphic...

2007©BOLD Educational Software

Descriptive Statistics:

Frequency Distributions

and Graphic Presentations

Note: for more information, print the NOTES PAGES, or view the notes below the PowerPoint slide


What Is the Question?What Is the Question?What Is the Question?What Is the Question?

• Descriptive statistics answers the question:

• What does this data look like?

• Descriptive statistics answers the question:

• What does this data look like?


Frequency DistributionFrequency Distribution

• Frequency distribution: a list of all the scores for a sample or population indicating the number of times that each score occurs.

• The data are grouped into mutually exclusive categories showing the number of observations in each category.


Practice: Describe the Practice: Describe the DistributionDistribution


• 46 students took an 18-question quiz. Describe the distribution.

• Note: read “zero, up to, but not including 3; 3 up to, but not including 6”, etc.

Students f

0 up to 3 1

3 up to 6 1

6 up to 9 4

9 up to 12 5

12 up to 15 10

15 up to 18 25

Total 46




• Out of 46 students, largest number of students (25) got 15, 16, 17, or 18 answers correct. Only 2 students got less than 6 answers correct.

Students f

0 up to 3 1

3 up to 6 1

6 up to 9 4

9 up to 12 5

12 up to 15 10

15 up to 18 25

Total 46


Creating a Creating a Relative Frequency DistributionRelative Frequency Distribution

Creating a Creating a Relative Frequency DistributionRelative Frequency Distribution

Students f Relative Frequency

0 up to 3 1 2.17% 1/463 up to 6 1 2.17 1/466 up to 9 4 8.70 4/469 up to 12 5 10.87 5/4612 up to 15 10 21.74 10/4615 up to 18 25 54.35 25/46Total 46 100.00


Practice: Where Do the Data Practice: Where Do the Data Tend to Cluster?Tend to Cluster?

Practice: Where Do the Data Practice: Where Do the Data Tend to Cluster?Tend to Cluster?

•Data tend to cluster between 40 and 59

Ages f

10 up to 20 2

20 up to 30 1

30 up to 40 5

40 up to 50 20

50 up to 60 25

60 up to 70 3

70 up to 80 4

Total 60




•The ages range from a low of 10 to a high of 79, with some clustering in the 40 to 59 age brackets.

Ages f

10 up to 19 2

20 up to 29 1

30 up to 39 5

40 up to 49 20

50 up to 59 25

60 up to 69 3

70 up to 79 4

Total 60


Ages f Relative Freq.

10 up to 19 2 2/6020 up to 29 1 1/6030 up to 39 5

40 up to 49 20

50 up to 59 25

60 up to 69 3

70 up to 79 4

Total 60 100.00Practice: Determine the Relative Frequency Distribution


Charts and GraphsCharts and GraphsCharts and GraphsCharts and Graphs


Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution

•The three commonly used graphic forms are:–Histograms,– Bar charts,–Frequency polygons.




• Histogram: A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other.


Histogram for Years of Histogram for Years of Experience: Experience:

Note the Normal Curve Note the Normal Curve SuperimposedSuperimposed

Histogram for Years of Histogram for Years of Experience: Experience:

Note the Normal Curve Note the Normal Curve SuperimposedSuperimposed


SymmetricWhen folded vertically, both sides are (more or less) the same.

Common Shapes of Common Shapes of HistogramsHistograms



Also Symmetric




Uniform (Platykurtic – flat)




Leptokurtic (Leptokurtic (high peakhigh peak))Leptokurtic (Leptokurtic (high peakhigh peak))


Non-Symmetric Histograms

These histograms are skewed..




Skewed Histograms

Skewed left (negative skew)

Skewed right (positive skew)




Skewed Histograms

Skewed left (negative skew)

Skewed right (positive skew)

Notice that the Notice that the SKEWSKEW follows the follows the TAILTAIL

Notice that the Notice that the SKEWSKEW follows the follows the TAILTAIL




Bimodal

ff

The two largest rectangles are separated by at least one class.


Bar GraphsBar GraphsBar GraphsBar Graphs

• A bar graph illustrates nominal data with the scores/categories along the x-axis and the frequencies on the y-axis.

• The scores are not ordered.• The bars do not touch

(unlike a histogram).• The heights correspond to the

number of times the score occurs.


Practice: Describe the Practice: Describe the DataData

Practice: Describe the Practice: Describe the DataData

The majority of students in the sample were White. The next largest group was Hispanic. The smallest representative groups were Black, Asian, and Indian.




•A frequency polygon consists of line segments connecting the points formed by the class midpoint and the class frequency.


Frequency Polygon for Frequency Polygon for

Hours Spent Hours Spent StudyingStudying

0

2

4

6

8

10

12

14

10 15 20 25 30 35

Hours spent studying

Fre

qu

ency


Compare the Frequency Polygon Compare the Frequency Polygon to the Histogramto the Histogram

Compare the Frequency Polygon Compare the Frequency Polygon to the Histogramto the Histogram


Histogram Using SPSSHistogram Using SPSSHistogram Using SPSSHistogram Using SPSS


Frequency Polygon Using Frequency Polygon Using SPSSSPSS

Frequency Polygon Using Frequency Polygon Using SPSSSPSS


Stem and LeafStem and LeafStem and LeafStem and Leaf

• A frequency distribution table that provides a visual picture of the distribution


Stem and LeafStem and LeafStem and LeafStem and Leaf

• Each raw score has two parts: a stem, consisting of all but the last digit, and the leaf, the last digit in the number.


Stem and LeafStem and LeafStem and LeafStem and LeafCurrent Salary Stem-and-Leaf Plot

Frequency Stem & Leaf

2.00 1 . 55 2.00 2 . 47 6.00 3 . 001234 3.00 4 . 016 1.00 5 . 0 2.00 Extremes (>=81250)

Stem width: 10000 Each leaf: 1 case(s)


Current Salary Stem-and-Leaf Plot Frequency Stem & Leaf 2.00 1 . 55 2.00 2 . 47 6.00 3 . 001234 3.00 4 . 016 1.00 5 . 0 2.00 Extremes (>=81250) Stem width: 10000 Each leaf: 1 case(s)

Each stem represents 10 thousand, Each stem represents 10 thousand, so the 1 (stem) = 10,000so the 1 (stem) = 10,000

There are two cases (frequency=2) with 15,000, There are two cases (frequency=2) with 15,000, two cases with 20,000 (actually, 24,000 and two cases with 20,000 (actually, 24,000 and

27,000), 27,000), 6 cases with 30,000 (30, 30, 31, 32, 33, and 34 6 cases with 30,000 (30, 30, 31, 32, 33, and 34

thousand) in this data set.thousand) in this data set.

Each stem represents 10 thousand, Each stem represents 10 thousand, so the 1 (stem) = 10,000so the 1 (stem) = 10,000

There are two cases (frequency=2) with 15,000, There are two cases (frequency=2) with 15,000, two cases with 20,000 (actually, 24,000 and two cases with 20,000 (actually, 24,000 and

27,000), 27,000), 6 cases with 30,000 (30, 30, 31, 32, 33, and 34 6 cases with 30,000 (30, 30, 31, 32, 33, and 34

thousand) in this data set.thousand) in this data set.


Stem and Leaf in Stem and Leaf in SPSSSPSS

Stem and Leaf in Stem and Leaf in SPSSSPSS

• To create a stem and leaf in SPSS, select the following:

• Analyze• Descriptives• Explore• Select stem and leaf in “plots”• Click continue• Click OK


Scatter Scatter plotsplots


• A scatter plot illustrates the relationship between two continuous variables.


• A scatter plot illustrates the values of Y (vertical axis) versus the corresponding values of X (horizontal axis)




• Scatter plots can provide answers to the following questions:

• Are variables X and Y correlated? (as one variable goes up, the other variable goes up/down)





• Is there a linear relationship between X and Y? (as one variable goes up, the other variable goes up/down)







• Is there a curvilinear relationship between variables X and Y? (As Y goes up X goes up, then at a peak, as X continues to go up, Y goes down

X

1086420

Y

8

7

6

5

4

3

2

1

0





• Are there outliers? (Do one or more points stray from the trend?)

2007© BOLD Educational Software Descriptive Statistics: Frequency Distributions and Graphic...

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