© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Chapter 12 Describing Data

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Doing Exploratory Data Analysis Use EXPLORATORY DATA ANALYSIS (EDA) to search for patterns in your data Before conducting any inferential statistic, use EDA to ensure that your data meet the requirements and assumptions of the test you are planning to use (e.g., normally distributed)

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Steps involved in the EDA : 1.Organize and summarize your data on a data coding sheet 2.If desired, organize data for computer entry 3.Graph data (bar graph, histogram, line graph, or scatterplot) so that you can visually inspect distributions  This will help you choose the appropriate statistics 4.Display frequency distributions on a histogram, and create a STEMPLOT 5.Examine your graphs for normality or skewness in your distributions

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Graphing Your Data Bar Graph Presents data as bars extending from the axis representing the independent variable Length of each bar determined by value of the dependent variable Width of each bar has no meaning Can be used to represent data from single-factor and two-factor designs Best if independent variable is categorical

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Line Graph Data represented by a series of points connected by a line Most appropriate for quantitative independent variables Used to display functional relationships Line graphs can show different shapes Positively accelerated: Curve starts flat and becomes progressively steeper as it moves along x-axis Negatively accelerated: Curve is steep at first and then “levels off” as it moves along x-axis Once the curve levels off it is said to be asymptotic

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. A line graph can vary in complexity A monotonic function represents a uniformly increasing or decreasing function A nonmonotonic function has reversals in direction Scatterplot Used to represent data from two dependent variables The value of one dependent variable is represented on the x-axis and the value of the other on the y-axis Pie Chart Used to represent proportions or percentages

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. The Frequency Distribution Represents a set of mutually exclusive categories into which actual values are classified Can take the form of a table or a graph Graphically, a frequency distribution is shown on a histogram A bar graph on which the bars touch The y-axis represents a frequency count of the number of observations falling into a category Categories represented on the x-axis

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Histogram Showing a Normal Distribution

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Histogram Showing a Positive Skew

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Histogram Showing a Negative Skew

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. A Bimodal Distribution

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Measures of Center: Characteristics and Applications Mode Most frequent score in a distribution Simplest measure of center Scores other than the most frequent not considered Limited application and value Median Central score in an ordered distribution More information taken into account than with the mode Relatively insensitive to outliers Used primarily when the mean cannot be used

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Mean Average of all scores in a distribution Value dependent on each score in a distribution Most widely used and informative measure of center

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Measures of Center: Applications Mode Used if data are measured along a nominal scale Median Used if data are measured along an ordinal or nominal scale Used if interval data do not meet requirements for using the mean

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Mean Used if data are measured along an interval or ratio scale Most sensitive measure of center Used if scores are normally distributed

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Measures of Spread: Characteristics Range Subtract the lowest from the highest score in a distribution of scores Simplest and least informative measure of spread Scores between extremes are not taken into account Very sensitive to extreme scores Semi-Interquartile Range Less sensitive than the range to extreme scores Used when you want a simple, rough estimate of spread

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Variance Average squared deviation of scores from the mean Standard Deviation Square root of the variance Most widely used measure of spread

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Measures of Spread: Applications The range and standard deviation are sensitive to extreme scores In such cases the semi-interquartile range is best When your distribution of scores is skewed, the standard deviation does not provide a good index of spread With a skewed distribution, use the semi- interquartile range

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. The Five Number Summary and Box Plots Five Number Summary Convenient way to represent a distribution with a few numbers Statistics included Minimum score The first quartile The median (second quartile) Third quartile Maximum score

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Example of a Five Number Summary Maximum132 Third Quartile (Q2) 110 Median (Q2)101 First Quartile (Q1) 90 Minimum67

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Boxplot Graphic representation of the five number summary First and third quartile define the ends of the box A line in the box represents the median Vertical “whiskers” extending above and below the box represent the maximum and minimum scores (respectively) Data from multiple treatments are represented by side- by-side boxplots

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Example of a Boxplot

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. The Pearson Product–Moment Correlation (r) Most widely used measure of correlation Value of r can range from +1 through 0 to –1 Magnitude of r tells you the degree of LINEAR relationship between variables Sign of r tells you the direction (positive or negative) of the relationship between variables

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Presence of outliers affects the sign and magnitude of r Variability of scores within a distribution affects the value of r Used when scores are normally distributed

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Measures of Association Pearson Product-Moment Correlation Index of linear relationship between two continuously measured variables Point-Biserial Correlation Index of correlation between two variables, one of which is measured on a nominal scale and the other on at least an interval scale Spearman Rank-Order Correlation Index of correlation between two variables measured along an ordinal scale

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Phi Coefficient Index of correlation between two variables measured along a nominal scale

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Linear Regression and Prediction Used to find the straight line that best fits the data plotted on a scatterplot The best fitting straight line is known as the least squares regression line The regression line is defined mathematically:

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. The regression weight (b) is based on raw scores and is difficult to interpret The standardized regression weight (beta weight) is based on standard scores and is easier to interpret You can predict a value of Y from a value of X once the regression equation has been calculated The difference between predicted and observed values of Y is the standard error of estimate