Frequency Distributions
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Transcript of Frequency Distributions
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Frequency Distributions
Quantitative Methods in HPELS
440:210
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Agenda
Basic Concepts Frequency Distribution Tables Frequency Distribution Graphs Percentiles and Percentile Ranks
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Basic Concepts Frequency distribution: An organized tabulation
of the number of individuals located in each category on the scale of measurement
Frequency distributions can be in table or graph format
There are two elements in a frequency distribution: The set of categories that make up the scale of
measurement The record of the frequency of individuals in each
category
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Basic Concepts
There are two reasons to construct frequency distributions:Assists with choosing the appropriate test
statistic (parametric vs. nonparametric)Assists with identification of outliers
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Basic Concepts Parametric statistics require a normal
distribution Frequency distributions provide a “picture”
of the data for determination of normality If data is normal use parametric
statistic, assuming INTERVAL or RATIO If data is non-normal use nonparametric
regardless of scale of measurement
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The Normal Distribution Characteristics:
1. Horizontally symmetrical
2. Unified mode, median and mean
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Non-Normal Distributions
Heavy tailed Light tailed
Left skewed Right skewed
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Normal Distribution
How to determine if distribution is normal: Several methods:
Qualititative assessmentQuantitative assessment:
Kolmogorov-Smirnov Shapiro-Wilk Q-Q plots
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Interpretation of the Q-Q Normal Plot
Normal Heavy tailed Light tailed
Left skew Right skew
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Bottom Line: Parametric or Nonparametric?
Is the scale of measurement at least interval?No NonparametricYes Answer next question
Is the distribution normal?No NonparametricYes Parametric
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Basic Concepts
The frequency distribution can assist with the identification of outliers
Outlier: An individual data point that is substantially different from the values obtained from other individuals in the same data set
Outliers can have drastic results on the test statistic
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Basic Concepts Outliers may occur naturally or maybe due
to some form of error:Measurement error throw out Input error correct the errorLack of effort or purposeful deceit on behalf of
subject throw out.Natural occurrence keep the data
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Agenda
Basic Concepts Frequency Distribution Tables Frequency Distribution Graphs Percentiles and Percentile Ranks
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Frequency Distribution Tables
FDT contain the following information:Scale of measurement (measurement
categories)Frequency of each point along the scale of
measurement FDT are in row/column format
Simple frequency distribution tablesGrouped frequency distribution tables
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Simple Frequency Distribution Tables
Process:List all measurement categories from lowest
to highest (unless nominal) in a column (X)List the frequency that each category
occurred in the next column (f) Example 2.1 (p 37).
Note that f = N where:N = total number of individuals.
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Simple Frequency Distribution Tables
Obtaining the X from a FDT Process:Create a third column called (fX)Multiply (f) column by (X) column product in
a new (fX) columnX = fX See Table on page 38
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Simple Frequency Distribution Tables Obtaining Proportions and Percentages: Proportion (p): The fraction of the total
group associated with each score where,(p) = f/N
Percentage (%) = p*100 Example 2.2 (p 37)
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Grouped Frequency Distribution Tables
If the data covers a wide range of values, there are disadvantages to listing each individual score:CumbersomeDifficult to interpret
Grouped FDT creates groups (class intervals) of scores
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Grouped Frequency Distribution Tables There are several rules to help with the construction
of grouped FDT: Rule 1: Use ~ 10 class intervals
Too few: Lost information Too many: Complicated
Rule 2: Width/size of each class interval should be simple Easy to count by 2, 5 or 10.
Rule 3: The bottom score in each class interval should be a multiple of the width/size of the class interval
Example: Width/size = 5 Each interval should start with 5, 10, 15 . . .
Rule 4: Each class interval should be the same width/size. Example 2.3 (p 40) and Table 2.2 (p 41).
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Agenda
Basic Concepts Frequency Distribution Tables Frequency Distribution Graphs Percentiles and Percentile Ranks
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Frequency Distribution Graphs
Graphs contain same information from the frequency distribution table Scale of measurement or measurement
categoriesFrequency of each category
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Frequency Distribution Graphs
Format is different: Scale of measurement is located along the
horizontal x-axis (abscissa)Values should increase from left to right.
Frequency is along the vertical y-axis (ordinate)Values should increase from bottom to top.
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Frequency Distribution Graphs
Generally speaking:The point where the two axes intersect should
have a value of zeroThe height (y-axis) of the graph should be
approximately 2/3 to 3/4 of its length (x-axis)Figure 2.2 (p 44)
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Frequency Distribution Graphs
There are several types of FDG:Histograms (Interval/Ratio)Polygons (Interval/Ratio)Stem and leaf displays (Interval/Ratio)Bar graphs (Nominal/Ordinal)
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FDG: Histograms (I/R) Process:
List the numerical scores along the x-axis Draw a bar above each X value so that:
Height: Corresponds to the frequency Width: Extends to the real limits of the value
Real limits: Upper and lower Separate adjacent scores along a number line Example The real limits of 150
Lower limit = 149.5 Upper limit = 150.5
Figure 1.7 (p 19)
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FDG: Histograms (I/R) Bars should be in contact with each other
Extend to real limitsFigure 2.2a (p 44)
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FDG: Histograms (I/R) Variations:
Histogram from grouped frequency table Figure 2.2b (p 45)
Modified histogram Figure 2.4 (p 45)
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FDG: Polygons (I/R) Process:
List the numerical scores along the x-axisPlace dot above scores corresponding to
frequencyConnect dots with continuous lineDraw two lines from the extreme dots to the x-
axis One category below the lowest score One category above the highest score Figure 2.5 (p 46)
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FDG: Polygons (I/R)
Variations:Polygon from grouped data Figure 2.6 (p 46)
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FDG: Stem and Leaf Displays (I/R)
Introduction:Simple plot designed by J.W. Tukey (1977)Two parts:
Stem: First digit Leaf: Last digit(s)
Table 2.3 (p 59)
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FDG: Stem and Leaf Displays (I/R)
Process:List all stems that occur (no duplicates)List all leaves by its stem (duplicates)
Variation:Double stems for greater detail
First of two stems associated with leaves (0-4) Second stem with leaves (5-9) Table 2.4 (p 60)
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FDG: Bar Graph (N/O)
Process: Same as histogram Spaces between the bars no real limits Figure 2.7 (p 47)
Nominal vs. Ordinal Data: Nominal data: The order of the categories is arbitrary Ordinal data: Logical progression of categories
Example: Dislike, mod. dislike, no opinion, mod. like, like
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Agenda
Basic Concepts Frequency Distribution Tables Frequency Distribution Graphs Percentiles and Percentile Ranks
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Percentiles and Percentile Ranks
Introduction: Useful when comparing scores relative to other
scores Determine the relative position of scores within the
data set Rank or percentile rank: Percentage of scores at or
below the particular value Percentile: When a score is identified by its percentile
rank
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Percentiles and Percentile Ranks Process:
Within simple distribution tableCreate new column (cf) cumulative
frequencyCount # of scores AT or BELOW
the category Interpretation:
Cumulative frequency of 20 = 20 scores fall at or below the category
Example 2.4 (p 52)
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Percentiles and Percentile Ranks Process continued:
Same table: Add new column (c%) cumulative percentage or percentile rank
Divide (cf) value by N Intepretation:
Percentile rank of 95% = 95% of the scores fall at or below the category
Example 2.5 (p 53)
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Textbook Problem Assignment
Problems: 1, 8, 16, 17, 20a, 20c, 24, 25