Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 1 Statistical Distributions Describe the Normal Curve and its characteristics Describe the characteristics of skewed distributions. Define and differentiate among measures of relative position, including percentile ranks and standard scores.

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 2 The Normal Curve

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 3 The Normal Curve If a variable is normally distributed then several things are true about the distribution of the variable. Fifty percent of the scores are above the mean (µ) and 50% are below the mean. The mean, median, and mode have the same value. Most scores are near the mean.

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 4 The Normal Curve 68.3% of all the scores fall within ± one standard deviation of the mean. More than 99% of the scores fall within three standard deviations above and below the mean.

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 5 The Normal Curve Skewed distributions When a distribution is not normally distributed, it is said to be skewed. A skewed distribution is not symmetrical. The mean, median, and mode are not the same value. The farther apart the mean and the median, the more skewed the distribution.

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 6 The Normal Curve Skewed distributions

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 7 The Normal Curve Skewed distributions A negatively skewed distribution has extreme scores at the lower end of the distribution. Mean<Median<Mode A positively skewed distribution has extreme scores at the higher end of the distribution. Mean>Median>Mode

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 8 Measures of Relative Position Measures of relative position indicate where a score falls in the distribution relative to all the other scores. Measures of relative position indicate how well an individual has scored in comparison to others in the distribution. Measures of relative position express different scores on a common scale.

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 9 Measures of Relative Position The two most frequently used measures of relative position are percentile ranks and standard scores. Percentile ranks indicate the percentage of scores that fall at or below a given score. Percentile ranks are appropriate for ordinal data and are also used for interval data. A percentile rank of 50 indicates the median score.

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 10 Measures of Relative Position A standard score uses standard deviation units to express how far an individual student’s test score is from the mean. i.e., a standard score reports how many standard deviations a given score is from the mean of a distribution. Standard scores allow scores from different tests to be compared on a common scale and therefore allow for calculations on those data.

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 11 Measures of Relative Position A z-score is the most basic and most often used standard score. A z-score is directly tied to the standard deviation. A score that represents the mean has a z-score of 0. A score at 1 standard deviation above the mean has a z-score of 1.

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 12 Z Score

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 13 Measures of Relative Position To convert a raw score to a z-score we use the following formula where X is the raw score. Z =X-X SD The characteristics of the normal distribution can be used to approximate where a score falls based upon a standard score.

Educational Research: Competencies for Analysis and Application, 9 th edition. Gay, Mills, & Airasian © 2009 Pearson Education, Inc. All rights reserved. 14 Measures of Relative Position A T-score is a standard score sometimes used instead of a z-score. A T-score is calculated by multiplying a z-score by 10 and adding 50. T-scores transform the scores such that there are no negative values.