frequency distributions
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Frequency Distributions
Quantitative Methods in HPELS
440:210
Agenda
Basic Concepts Frequency Distribution Tables Frequency Distribution Graphs Percentiles and Percentile Ranks
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
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
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
The Normal Distribution Characteristics:
1. Horizontally symmetrical
2. Unified mode, median and mean
Non-Normal Distributions
Heavy tailed Light tailed
Left skewed Right skewed
Normal Distribution
How to determine if distribution is normal: Several methods:
Qualititative assessmentQuantitative assessment:
Kolmogorov-Smirnov Shapiro-Wilk Q-Q plots
Interpretation of the Q-Q Normal Plot
Normal Heavy tailed Light tailed
Left skew Right skew
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
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
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
Agenda
Basic Concepts Frequency Distribution Tables Frequency Distribution Graphs Percentiles and Percentile Ranks
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
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.
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
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)
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
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).
Agenda
Basic Concepts Frequency Distribution Tables Frequency Distribution Graphs Percentiles and Percentile Ranks
Frequency Distribution Graphs
Graphs contain same information from the frequency distribution table Scale of measurement or measurement
categoriesFrequency of each category
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.
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)
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)
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)
FDG: Histograms (I/R) Bars should be in contact with each other
Extend to real limitsFigure 2.2a (p 44)
FDG: Histograms (I/R) Variations:
Histogram from grouped frequency table Figure 2.2b (p 45)
Modified histogram Figure 2.4 (p 45)
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)
FDG: Polygons (I/R)
Variations:Polygon from grouped data Figure 2.6 (p 46)
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)
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)
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
Agenda
Basic Concepts Frequency Distribution Tables Frequency Distribution Graphs Percentiles and Percentile Ranks
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
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)
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)
Textbook Problem Assignment
Problems: 1, 8, 16, 17, 20a, 20c, 24, 25
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