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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 tailedLight tailed Left skewedRight skewed

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Normal Distribution How to determine if distribution is normal: Several methods: Qualititative assessment Quantitative assessment: Kolmogorov-Smirnov Shapiro-Wilk Q-Q plots

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Interpretation of the Q-Q Normal Plot NormalHeavy tailedLight tailed Left skew Right skew

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Bottom Line: Parametric or Nonparametric? Is the scale of measurement at least interval? No Nonparametric Yes Answer next question Is the distribution normal? No Nonparametric Yes 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 error Lack 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 tables Grouped 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) column X = 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: Cumbersome Difficult 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, 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 categories Frequency 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 zero The 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 = Upper limit = Figure 1.7 (p 19)

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FDG: Histograms (I/R) Bars should be in contact with each other Extend to real limits Figure 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-axis Place dot above scores corresponding to frequency Connect dots with continuous line Draw 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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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 table Create new column (cf) cumulative frequency Count # 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

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