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Chapter 12: Descriptive Statistics Objectives Describe the process of tabulating and coding data. Define frequency and central tendency, and differentiate among mean, median, and mode. Define variability, and differentiate among the range, quartile deviation, variance, and standard deviation. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 1 © 2009 Pearson Education, Inc. All rights reserved. Chapter 12: Descriptive Statistics Describe the major characteristics of normal and skewed distributions. Define and differentiate among measures of relative position, including percentile ranks and standard scores. Define and differentiate among two measures of relationship, the Pearson r and the Spearman rho. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 2 © 2009 Pearson Education, Inc. All rights reserved. Descriptive Statistics Statistics is a set of procedures for describing, synthesizing, analyzing, and interpreting quantitative data. The mean is an example of a statistic. One can calculate statistics by hand or can use the assistance of statistical programs. Excel, SPSS, and many other programs exist. Some programs are also available on the Web to analyze datasets. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 3 © 2009 Pearson Education, Inc. All rights reserved. Preparing Data for Analysis After data are collected, the first step toward analysis involves converting behavioral responses into a numerical system or categorical organization. It is critical that all data are scored accurately and consistently. Data scoring should be doublechecked for consistency and accuracy (i.e., at least 25% of all cases should be checked). Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 4 © 2009 Pearson Education, Inc. All rights reserved. Preparing Data for Analysis Open-ended items should be scored by two scorers to check reliability. All data scoring and coding procedures should be documented and reported in the written report. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 5 © 2009 Pearson Education, Inc. All rights reserved. Preparing Data for Analysis After instruments are scored, the resulting data are tabulated and entered into a spreadsheet. Tabulation involves organizing the data systematically (e.g., by participant). ID Cond. Gender Grade Ach Mot 001 1 2 10 78 52 002 2 2 11 82 62 003 1 1 11 86 74 Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 6 © 2009 Pearson Education, Inc. All rights reserved. Preparing Data for Analysis In this potential dataset, ID represents participant number Cond. is the experimental condition (1 or 2) Gender is represented by female=1; male=2 Achievement (Ach) and motivation (Mot) were also variables assessed ID Cond. Gender Grade Ach Mot 001 1 2 10 78 52 002 2 2 11 82 62 003 1 1 11 86 74 Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 7 © 2009 Pearson Education, Inc. All rights reserved. Types of Descriptive Statistics After data are tabulated and entered, the next step is to conduct descriptive statistics to summarize data. In some studies, only descriptive statistics will be conducted. If the indices are calculated for a sample, they are referred to as statistics. If indices are calculated for the entire population, they are referred to as parameters. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 8 © 2009 Pearson Education, Inc. All rights reserved. Types of Descriptive Statistics Frequencies The frequency refers to the number of times something occurs. Frequencies are often used to describe categorical data. We might want to have frequency counts of how many males and females were in a study or how many participants were in each condition. Frequency counts are not as helpful in describing interval and ratio data. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 9 © 2009 Pearson Education, Inc. All rights reserved. Measures of Central Tendency Measures of central tendency are indices that represent a typical score among a group of scores. Measures of central tendency provide a way to describe a dataset with a single number. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 10 © 2009 Pearson Education, Inc. All rights reserved. Measures of Central Tendency The three most common measures of central tendency are the mean, median, and mode. Mean: Appropriate for describing interval or ratio data Median: Appropriate for describing ordinal data Mode: Appropriate for describing nominal data Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 11 © 2009 Pearson Education, Inc. All rights reserved. (XX ) Measures of Central Tendency The mean is the most commonly used measure of central tendency. The formula for the mean is: X= ∑Xi/n To calculate the mean, all the scores are summed and then divided by the number of scores. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 12 © 2009 Pearson Education, Inc. All rights reserved. (XX ) The Mean or Average Value Example What is the mean of 4 3 6 8 4 Mean = (4+3+6+8+4)/5 = 5 Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 13 © 2009 Pearson Education, Inc. All rights reserved. (XX ) The Median Value The median is the midpoint in a distribution: 50% of the scores are above the median and 50% are below the median. To determine the median, all scores are listed in order of value. If the total number of scores is odd, the median is the middle score. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 14 © 2009 Pearson Education, Inc. All rights reserved. (XX ) The Median Value If the total number of scores is even, the median is halfway between the two middle scores. Median values are useful when there is large variance in a distribution. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 15 © 2009 Pearson Education, Inc. All rights reserved. (XX ) The Median Value Example 1: Use the previous data put in order 3 4 4 6 8 The Median is (the middle value) = 4 Example 2: 3 4 4 6 8 9 The Median is (4+6/2) = 5 Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 16 © 2009 Pearson Education, Inc. All rights reserved. (XX ) The Mode The mode is the most frequently occurring score in a distribution. The mode is established by looking at a set of scores or at a graph of scores and determining which score occurs most frequently. The mode is of limited value. Some distributions have more than one mode (e.g., bi-modal, or multi-modal distributions) Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 17 © 2009 Pearson Education, Inc. All rights reserved. (XX ) Measures of Central Tendency Deciding among measures of central tendency Generally the mean is most preferred. The mean takes all scores into account. The mean, however, is greatly influenced by extreme scores- outlying data. When there are extreme scores present in a distribution, the median is a better measure of central tendency. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 18 © 2009 Pearson Education, Inc. All rights reserved. Measures of Variability Measures of variability provide an index of the degree of spread in a distribution of scores. Measures of variability are critical to examine and report because some distributions may be very different but yet still have the same mean or median. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 19 © 2009 Pearson Education, Inc. All rights reserved. Measures of Variability Three common measures of variability are the range, quartile deviation, and standard deviation. Range: The difference between the highest and lowest score. The range is not a stable measure. The range is quickly determined. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 20 © 2009 Pearson Education, Inc. All rights reserved. Measures of Variability Quartile Deviation: One half the difference between the upper quartile and the lower quartile in a distribution. By subtracting the cutoff point for the lower quartile from the cutoff point for the upper quartile and then dividing by two we obtain a measure of variability. A small number indicates little variability and illustrates that the scores are close together. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 21 © 2009 Pearson Education, Inc. All rights reserved. Measures of Variability Variance: The amount of spread among scores. If the variance is small the scores are close together. If the variance is large the scores are spread out. Calculation of the variance shows how far each score is from the mean. The formula for the variance is: ∑(X–X)2/n Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 22 © 2009 Pearson Education, Inc. All rights reserved. Measures of Variability Standard deviation: The square root of the variance. The standard deviation is used with interval and ratio data. The standard deviation is the most commonly used measure of variability. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 23 © 2009 Pearson Education, Inc. All rights reserved. Measures of Variability If the mean and the standard deviation are known, the distribution can be described fairly well. SD represents the standard deviation of a sample and the symbol (i.e., the Greek lower case sigma) represents the standard deviation of the population. Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 24 © 2009 Pearson Education, Inc. All rights reserved. Standard Deviation – 3 cases with the same Mean Educational Research: Competencies for Analysis and Application, 9th edition. Gay, Mills, & Airasian 25 © 2009 Pearson Education, Inc. All rights reserved.