Measures of Spread

Reading

Sections 1.6.4

Practice Problems

1.9.6 (page 65)

1.47

Notes

• The goal of measures of spread is to assess the “variation” in the data in some way.
  • Are the values spread out and far from each other, or are they all real close?
  • Are the values of one variable less or more spread out than those of another variable?
• Different measures achieve this in different ways.

• The two main measures of spread are:

  Std. Dev.

  Standard Deviation. Measures distance of data from the mean.

  Often denoted by \(s\).

  Not resistant.

  IQR

  Interquartile range. Measures range occupied by the middle 50% of the data.

  Resistant.

• NOTE: The standard deviation and the IQR are in general not comparable to each other, they measure spread in very different ways.

• Let us review how the standard deviation is computed. Its formula is a bit complicated, and it looks something like this:

  \[s = \sqrt{\frac{1}{n-1} \sum (x-\bar{x})^2}\]

  Basically:
  • Compute the mean of all the values (the \(\bar x\)).
  • Subtract the mean from each value (the result of this step is the “deviations”, \(x-\bar x\)).
  • Square all the deviations.
    • Ensures values are positive before computing average.
  • Average these squared deviations: Add them all up (the big “sigma”), then divide by \(n-1\). The result at this stage is called the variance.
    • Why \(n-1\): Technical reason, and won’t really matter for large \(n\).
    • One way to think about it: The deviations always add up to \(0\), so once you know \(n-1\) of them the last one is determined.
    • In this context, \(n-1\) is called the “degrees of freedom”.
  • Take a square root at the end.
    • Fixes the units of measurement.
• Outliers have a considerable effect on this formula.
  • They have a very large deviation, and because they pull the mean towards themselves they cause larger deviations in the other values as well.
  • Because we square them before adding, those large deviations will dominate the equation even more.

• What to use depends on the distribution:

  Symmetric

  Mean for center, Standard Deviation for spread

  Skewed/Outliers

  Median for center, IQR for spread

• Chebyshev’s Rule:
  • At least 75% of the data is within 2 standard deviations from the mean
  • At least 89% of the data is within 3 standard deviations from the mean
• For Bell Shaped data (Empirical Rule):
  • Approximately 68% of the data is within one standard deviation of the mean.
  • Approximately 95% of the data is within two standard deviations of the mean.
  • More than 99% of the data is within three standard deviations of the mean.