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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
Measures of Variation
  • Sample range
  • Sample variance
  • Sample standard deviation
  • Sample interquartile range
sample range

Sample Range

or, equivalently

R = largest obs. - smallest obs.

R = xmax - xmin

slide5

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
sample interquartile range

Sample Interquartile Range

or, equivalently

IQR = third quartile - first quartile

IQR = Q3 - Q1

measures of variation some comments
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
boxplot a 5 number summary
Boxplot - a 5 number summary
  • smallest observation (min)
  • Q1
  • Q2 (median)
  • Q3
  • largest observation (max)
boxplot example table 4 p 30
Boxplot Example - Table 4, p. 30
  • smallest observation = 3.20
  • Q1 = 43.645
  • Q2 (median) = 60.345
  • Q3 = 84.96
  • largest observation = 124.27
creating a boxplot
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.
slide11

Boxplot Example

Min = 3.20

Max = 124.27

Q2 = 60.345

Q1 = 43.645

Q3 = 84.96

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boxplot interpretation
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)