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Measures of Variation

Measures of Variation. Sample range Sample variance Sample standard deviation Sample interquartile range. Sample Range. or, equivalently. R = largest obs. - smallest obs. R = x max - x min. Sample Variance. Sample Standard Deviation. What is a standard deviation?.

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Measures of Variation

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  1. Measures of Variation • Sample range • Sample variance • Sample standard deviation • Sample interquartile range

  2. Sample Range or, equivalently R = largest obs. - smallest obs. R = xmax - xmin

  3. Sample Variance

  4. Sample Standard Deviation

  5. What is a standard deviation? • it is the typical (standard) difference (deviation) of an observation from the mean • think of it as the average distance a data point is from the mean, although this is not strictly true

  6. Sample Interquartile Range or, equivalently IQR = third quartile - first quartile IQR = Q3 - Q1

  7. Measures of Variation -Some Comments • Range is the simplest, but is very sensitive to outliers • Variance units are the square of the original units • Interquartile range is mainly used with skewed data (or data with outliers) • We will use the standard deviation as a measure of variation often in this course

  8. Boxplot - a 5 number summary • smallest observation (min) • Q1 • Q2 (median) • Q3 • largest observation (max)

  9. Boxplot Example - Table 4, p. 30 • smallest observation = 3.20 • Q1 = 43.645 • Q2 (median) = 60.345 • Q3 = 84.96 • largest observation = 124.27

  10. Creating a Boxplot • Create a scale covering the smallest to largest values • Mark the location of the five numbers • Draw a rectangle beginning at Q1 and ending at Q3 • Draw a line in the box representing Q2, the median • Draw lines from the ends of the box to the smallest and largest values • Some software packages that create boxplots include an algorithm to detect outliers. They will plot points considered to be outliers individually.

  11. Boxplot Example Min = 3.20 Max = 124.27 Q2 = 60.345 Q1 = 43.645 Q3 = 84.96 . . . . . 0 10 20 30 40 50 60 70 80 90 100 110 120 130

  12. Boxplot Interpretation • The box represents the middle 50% of the data, i.e., IQR = length of box • The difference of the ends of the whiskers is the range (if there are no outliers) • Outliers are marked by an * by most software packages. • Boxplots are useful for comparing two or more samples • Compare center (median line) • Compare variation (length of box or whiskers)

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