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 tailedLight tailed Left skewedRight skewed

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

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

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

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 error Lack 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 tables Grouped 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) column X = 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: Cumbersome Difficult 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, 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 categories Frequency 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 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)

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 = Upper limit = Figure 1.7 (p 19)

FDG: Histograms (I/R) Bars should be in contact with each other Extend to real limits Figure 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-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)

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

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 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)

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