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Descriptive statistics Statistics • Many studies generate large numbers of data points, and to make sense of all that data, researchers use statistics that summarize the data, providing a better understanding of overall tendencies within the distributions of scores. Types of statistics • Types of statistics: 1. descriptive (which summarize some characteristic of a sample) • • • Measures of central tendency Measures of dispersion Measures of skewness 2. inferential (which test for significant differences between groups and/or significant relationships among variables within the sample • t-ratio, chi-square, beta-value Reasons for using statistics • aids in summarizing the results • helps us recognize underlying trends and tendencies in the data • aids in communicating the results to others Descriptive statistics • If we wanted to characterize the students in this class we would find that they are: – – – – • • • • Young From Kentucky Fit Male How young? How Kentuckian is this class? How fit is this class? What is the distribution of males and females? Frequency distribution • The frequency with which observations are assigned to each category or point on a measurement scale. – Most basic form of descriptive statistics – May be expressed as a percentage of the total sample found in each category Source : Reasoning with Statistics, by Frederick Williams & Peter Monge, fifth edition, Harcourt College Publishers. Frequency distribution • The distribution is “read” differently depending upon the measurement level – Nominal scales are read as discrete measurements at each level – Ordinal measures show tendencies, but categories should not be compared – Interval and ratio scales allow for comparison among categories Ancestry of US residents Source: Protecting Children from Harmful Television: TV Ratings and the V-chip Amy I. Nathanson, PhD Lecturer, University of California at Santa Barbara Joanne Cantor, PhD Professor, Communication Arts, University of Wisconsin-Madison Source: UCLA International Institute Source: www.cit.cornell.edu/computer/students/bandwidth/charts.html Source: www.cit.cornell.edu/computer/students/bandwidth/charts.html Source: Verisign Normal distribution • Many characteristics are distributed through the population in a ‘normal’ manner – Normal curves have well-defined statistical properties – Parametric statistics are based on the assumption that the variables are distributed normally • Most commonly used statistics • This is the famous “Bell curve” where many cases fall near the middle of the distribution and few fall very high or very low – I.Q. I.Q. distribution Measures of central tendency • These measures give us an idea what the ‘typical’ case in a distribution is like • Mode (Mo): the most frequent score in a distribution • good for nominal data • Median (Mdn): the midpoint or midscore in a distribution. • (50% cases above/50% cases below) – insensitive to extreme cases --Interval or ratio Source : Reasoning with Statistics, by Frederick Williams & Peter Monge, fifth edition, Harcourt College Publishers. Measures of central tendency • Mean – The ‘average’ score—sum of all individual scores divided by the number of scores – has a number of useful statistical properties • however, can be sensitive to extreme scores (“outliers”) – many statistics are based on the mean Source: www.wilderdom.com/.../L2-1UnderstandingIQ.html Statistics estimating dispersion • Some statistics look at how widely scattered over the scale the individual scores are • Groups with identical means can be more or less widely dispersed • To find out how the group is distributed, we need to know how far from or close to the mean individual scores are • Like the mean, these statistics are only meaningful for interval or ratio-level measures Estimates of dispersion • Range • Distance between the highest and lowest scores in a distribution; • sensitive to extreme scores; • Can compensate by calculating interquartile range (distance between the 25th and 75th percentile points) which represents the range of scores for the middle half of a distribution Usually used in combination with other measures of dispersion. Range Source: www.animatedsoftware.com/ statglos/sgrange.htm Source: http://pse.cs.vt.edu/SoSci/converted/Dispersion_I/box_n_hist.gif Estimates of dispersion Variance (S2) – Average of squared distances of individual points from the mean • sample variance – High variance means that most scores are far away from the mean. Low variance indicates that most scores cluster tightly about the mean. – The amount that one score differs from the mean is called its deviation score (deviate) – The sum of all deviation scores in a sample is called the sum of squares Standard Deviation (SD) A summary statistic of how much scores vary from the mean Square root of the Variance – expressed in the original units of measurement – Represents the average amount of dispersion in a sample – Used in a number of inferential statistics Skewness of distributions • Measures look at how lopsided distributions are—how far from the ideal of the normal curve they are • When the median and the mean are different, the distribution is skewed. The greater the difference, the greater the skew. • Distributions that trail away to the left are negatively skewed and those that trail away to the right are positively skewed • If the skewness is extreme, the researcher should either transform the data to make them better resemble a normal curve or else use a different set of statistics— nonparametric statistics—to carry out the analysis Distribution of posting frequency on Usenet So • Descriptive statistics are used to summarize data from individual respondents, etc. – They help to make sense of large numbers of individual responses, to communicate the essence of those responses to others • They focus on typical or average scores, the dispersion of scores over the available responses, and the shape of the response curve