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Given a data set = { x 1 , x 2 , . . . , x n }, we discussed three common numerical measures of spread for its frequency distribution:. Range = x max - x min : Measures the length of the interval over which the entire set of data values is spread.

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Summary numerical measures of a distribution s spread

Given a data set = {x1, x2, . . . , xn}, we discussed three common numerical measures of spread for its frequency distribution:

Range = xmax-xmin:

Measures the length of the interval over which the entire set of data values is spread

  • Summary: Numerical Measures of a Distribution’s Spread

Interquartile Range (IQR) = Q3 - Q1:

Measures the length of the interval over which the middle 50% of the data values is spread

Q3 denotes third quartile (aka 75th percentile) and Q1 denotes the first quartile (aka 25th percentile)


Summary numerical measures of a distribution s spread

3. Standard deviation

Gives an average of the squared deviations from the mean in the same units of measure as the data (thus the square root).

Properties of S:

  • S measures the spread of the data about the mean—it should

  • only be used when the mean is chosen as a measure of center

  • S > 0 for any data set:

  • S = 0 if and only if ___?______. Otherwise, S > 0.

  • The farther the data is spread from the mean, the greater S gets.

  • S is thus very sensitive to extreme differences from the mean.


Summary numerical measures of a distribution s spread

  • Why?

  • Because in addition to specifying a center, it can be used to also measure the length of intervals within which the four ordered quarters of the data are contained, and thus measures the variability/spread of those portions: Q1-xmin, M-Q1, Q3-M, xmax -Q3

  • The Five-Number Summary provides a fairly complete summary of a distribution’s center and spread

    • xminQ1 M Q3 xmax

The Five-Number summary forms the skeleton for the boxplot