Measures of Dispersion

Measures of Dispersion How far the data is spread out

Range Difference between the highest and lowest values

Variance Includes all data values Measures average distance of squared values to mean Population Sample

Standard Deviation Important value to study of statistics Measures average absolute distance of values to the mean Population Sample

Quantiles • Dividing data into equal groups • Percentiles • Deciles • Quartiles • First quartile – Q1 = • Second quartile – Q2 = • Third quartile – Q3 = • Interquartile range • Q3 – Q1

Percentiles Commonly used measure of relative position. Remember the median? For any data set, the pth percentile is a value (x) such that p percent of the data is less than x and 1-p percent of the data is greater than x.

Boxplots (Box and whisker plots) • Visual display of data • 5-number summary • Minimum • Q1 • Q2 • Q3 • Maximum • IQR • Outliers

Boxplots

Correlations • http://www.duxbury.com/authors/mcclellandg/tiein/johnson/correlation.htm • http://www.rossmanchance.com/applets/guesscorrelation/GuessCorrelation.html

z-scores The z-score tells us how far a data value is from the mean in terms of the number of standard deviations This is another one of the fundamental values in statistics that we will use again, later.

Empirical Rule ±1σ= 68% ±2σ= 95% ±3σ= 99.7%

Summary • Mean, median, and mode • What information does each convey? • Which is the most resistant to outliers?

Summary • Range • Variance/Standard deviation • Z-scores (more later) • Empirical rule • Quantiles • IQR • Boxplots • Correlations

Which to use? • Data is symmetric and unimodal • Use the mean and s.d. • Data is skewed • Use the median and 5-number summary • Mode?

Measures of Dispersion