Topics: Descriptive Statistics A road map Examining data through frequency distributions Measures of central tendency Measures of variability The normal curve Standard scores and the standard normal distribution
The Role of Description Description as a purpose of research Choosing the right statistical procedures
Raw Data: Overachievement Study
Frequency Distributions A method of summarizing and highlighting aspects of the data in a data matrix, showing the frequency with which each value occurs. Numerical Representations: a tabular arrangement of scores Graphical Representations: a pictorial arrangement of scores
Numerical Frequency Distributions Ungrouped Frequency Distributions Grouped Frequency Distributions Relative Frequency Distributions Cumulative Frequency Distributions
Tabular Frequency Distributions Single-Variable (“Univariate”)
Frequency Distribution: Major MAJOR Valid Cum Value LabelValue FrequencyPercent Percent Percent PHYSICS CHEMISTRY BIOLOGY ENGINEERING ANTHROPOLOGY SOCIOLOGY ENGLISH DESIGN Total Valid cases 40 Missing cases 0
Frequency Distribution: Major Group MAJORGRP Valid Cum Value LabelValueFrequencyPercentPercent SCIENCE & ENGINEERIN SOCIAL SCIENCE HUMANITIES Total
Frequency Distribution: SAT SAT ValidCum ValueFrequencyPercent Percent Total Valid cases 40 Missing cases 0
Grouped Frequency Distribution: SAT
Graphical Frequency Distributions Bar Graphs Histograms Stem and Leaf Frequency Polygons Pie Chart
Graphical Frequency Distributions: Single-Variable (“Univariate”)
Bar Chart: Major
Histogram: SAT (From Grouped Data)
Frequency Polygon Overlay: SAT (From Grouped Data)
Frequency Polygon: SAT (From Grouped Data)
Frequency Polygon: SAT Scores (From Ungrouped Data)
Cumulative Frequency Polygon: SAT Scores
Stem and Leaf: SAT
SAT Stem-and-Leaf Plot Frequency Stem & Leaf Stem width: Each leaf: 1 case(s)
Graphical Frequency Distributions Two-Variable (“Joint” or “Bivariate”)
Relative Frequency Polygon: GPA Comparison of Majors
Relative Frequency Polygon: GPA Comparison of Gender
What Can Be Seen in Frequency Distributions Shape Central Tendency Variability
Shapes of Frequency Polygons
Shapes of Distributions
Descriptive Statistics Central Tendency –Mode –Median –Mean Variability –Range –Standard Deviation –Variance
Definitions: Measures of Central Tendency Mean: –“Arithmetic mean” –“Center of gravity” such that the “weight” of the scores above the mean exactly balances the “weight” of the scores below the mean Median: –The number that lies at the midpoint of the distribution of scores; divides the distribution into two equal halves Mode: –Most frequently occurring score
Mean, Median, Mode: SAT Scores by Gender
Mean, Median, Mode: SAT Scores by Area
Relative Position of Mode, Median, and Mean
Definitions: Measures of Variability Range: –Difference between highest and lowest score Inter-quartile Range: –The spread of the middle 50% of the scores –The difference between the top 25% (Upper Quartile-Q3) and the lower 25% (Lower Quartile-Q1) Standard Deviation: –The average dispersion or deviation of scores around the mean (measured in original score units) Variance: –The average variability of scores (measured in squared units of the original scores (square of the standard deviation)
Range, Interquartile Range, and Standard Deviation: SAT Scores by Area
Range, Interquartile Range, and Standard Deviation: SAT Scores by Gender
Properties of Normal Distribution Bell-shaped (unimodal) Symmetric about the mean Mode, median, and mean are equal (though rarely occurs) Asymptotic (curve never touches the abscissa)
Normal Curve Areas Under the Curve X -1s-2s+1s+2s-3s+3s % 95% 99%
Definitions: Standard Scores Standard Scores: scores expressed as SD away from the mean (z-scores) Obtained by finding how far a score is above or below the mean and dividing that difference by the SD Changes mean to 0 and SD to 1, but does not change the shape (called Standard Normal Distribution)
Uses of Standard Normal Distribution What proportion of scores falls between the mean and a given raw score What proportion of scores falls above or below a given raw score What proportion of scores falls between two raw scores What raw score fall above (or below) a certain percentage of scores