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Statistics Josée L. Jarry, Ph.D., C.Psych. Introduction to Psychology Department of Psychology University of Toronto June 9, 2003 Statistical Procedures • Mathematical aids for summarizing and interpreting data • Descriptive statistics – Used to summarize data sets • Inferential statistics – Used to determine what conclusion can be drawn from data sets Organizing and Summarizing Scores Frequency distribution Central tendency Variability Frequency Distribution • Rank order – Organize scores from lowest to highest • Frequency distribution – Divide the range of scores into equal intervals – Determine how many scores fall into each interval • Demonstrates the frequency of occurrence of each response or range of responses Shapes of Frequency Distributions Normal Distribution • Maximum frequency lies in the center of the range of scores • Frequency tapers off symmetrically on both sides • Many measures in nature are normally distributed • Found when a measure is determined by several independent factors Bimodal Distributions • Occurs when scores form two separate groupings • Mode – the most frequently occurring score or range of scores in a frequency distribution • Two separate areas of peak frequencies or two modes • The normal curve is unimodal Skewed Distributions • Positively skewed distribution – Spread of scores above the mode is greater than the spread below – The tail extends in the direction of high scores • Negatively skewed distributions – Spread of scores below the mode is greater than the spread above – The tail extends in the direction of low scores Measures of Central Tendency • Consists of summarizing an entire distribution with a single score that represents the centre of the distribution. • Median – Is the middle score of a set of ranked scores. • Mean – Found by adding all the scores and dividing by the total number of scores. • M = sum of score N Mean, Median, & Distributions • In a normal distribution – Mean and median are identical • In a positively skewed distribution – The mean is greater than the median; • In a negatively skewed distribution – The mean is smaller than the median • Mean is preferred in normal distributions • Median is preferred in highly skewed distributions Measures of Variability (1) • Measures of variability tell us how widely the observations are spread around the centre • Two distributions can both be normal and have the same mean, but have very different variability. Measures of Variability (2) • Range – The difference between the highest and the lowest scores in a distribution • Variance – Takes into account the extent to which all of the scores in the distribution differ from each other • Standard deviation – Expresses variability in the same unit of measurement as the original scores – The square root of the variance Converting Scores for Purposes of Comparison • Allows to compare different kinds of scores to each other • To compare different kind of scores with each other, each score must be converted into a form that expresses directly its relationship to the whole distribution of scores from which it came Percentile Rank • The most straightforward way of comparing one person to another on a given measure • Allows comparison between different measures • Determine that person’s percentile rank on the measures of interest • Percentile rank – the percentage of scores that are equal to that score or lower, out of a whole set of scores for that measure Standardized Scores • z-score – Expresses a score in terms of the number of standard deviations that the original score is away from the mean of the original scores • Convert any score into a z-score – calculate its deviation from the mean – divide the deviation by the standard deviation of the distribution –z= score - mean Standard deviation Correlation Coefficient (1) • Is a mathematical means of describing the strength and direction of the relationship between two variables that have been mathematically measured • Varies between -1 and +1 • The sign (+ or -) of the correlation indicates the direction (positive or negative) of the relationship • The absolute value of the coefficient (from 0 to 1) indicates the strength of the relationship Correlation Coefficient (2) • A positive correlation indicates that an increase in one variable corresponds to an increase in the other • A negative correlation indicates that an increase in one variable corresponds to a decrease in the other • Scatter plot – Visual display of a correlation – Each point represents a subject’s pair of scores