© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 5. Measuring Dispersion or Spread in a Distribution of Scores

© 2008 McGraw-Hill Higher Education Measuring Dispersion in a Score Distribution Dispersion: How the scores of an interval/ratio variable are spread out from lowest to highest and the shape of the distribution in between The most commonly used dispersion statistics are the range and standard deviation

© 2008 McGraw-Hill Higher Education The Range The range is an expression of how the scores of an interval/ratio variable are distributed from lowest to highest It is the distance between the minimum and maximum scores in a sample

© 2008 McGraw-Hill Higher Education Calculating the Range 1. Rank the scores from lowest to highest 2. Identify the minimum and maximum scores 3. Identify the value of the rounding unit Range = (Maximum score - Minimum score) + the value of the rounding unit

© 2008 McGraw-Hill Higher Education Limitations of the Range The range is greatly affected by outliers The range has a narrow informational scope. It provides the width of a distribution of scores, but tells us nothing about how they are spread between the maximum and minimum scores

© 2008 McGraw-Hill Higher Education The Standard Deviation The standard deviation describes how scores are spread across the distribution in relation to the mean score It provides a standard unit of comparison – a common unit of measure for comparing variables with very different observed units of measure Its computation centers on how far each score is from the mean – how far it “deviates”

© 2008 McGraw-Hill Higher Education Calculating the Standard Deviation Sort givens and calculate the mean and deviation scores Sum the deviation scores and verify a result of zero Square the deviation scores and sum them to obtain the variation or “sum of squares” Divide the variation by n - 1 to get the variance Take the square root of the variance to get the standard deviation

© 2008 McGraw-Hill Higher Education The Elements of the Standard Deviation (See Table 5-1) We square deviation scores to remove negative signs and to obtain a sum other than zero We divide the sum of squares by n - 1 to adjust for sample size and sampling error We take the square root of the variance to obtain directly interpretable units of measure (units instead of squared units)

© 2008 McGraw-Hill Higher Education Limitations of the Standard Deviation The standard deviation is greatly inflated by outliers It can be misleading if the distribution is skewed

© 2008 McGraw-Hill Higher Education Three Ways to Express the Value of a Score, X 1. As a raw score – the observed value of X in its original units of measure such as inches 2. As a deviation score – the difference between a raw score and its mean, also in original units of measure 3. As a standardized score (Z-score) – as a number of standard deviations (SD) from the mean

© 2008 McGraw-Hill Higher Education Standardized Scores or Z-scores Z-scores express a raw score as a number of standard deviations (SD) from the mean score Divide the deviation score by the standard deviation to produce a measure of X in standard deviation units

© 2008 McGraw-Hill Higher Education The Standard Deviation Is a Part of the Normal Curve For any normally distributed variable: 99.7% of cases fall within 3 SD of the mean in both directions About 95%, within 2 SD of the mean in both directions About 68%, within 1 SD of the mean in both directions

© 2008 McGraw-Hill Higher Education Using the Normal Curve to Partition Areas If a variable is distributed normally, we can use sample statistics and what we know about the normal curve to estimate how many scores in a population fall within a certain range

© 2008 McGraw-Hill Higher Education Statistical Follies Comparing the relative sizes of the mean and standard deviation is a good way to detect skews When the calculated standard deviation is larger than the mean for the variable, the distribution is skewed or otherwise oddly shaped