What values are most common or most likely?

Three ways to measure:

• mode
• mean
• median

Mode: the most commonly occurring value.

• Used when working with categorical variables.

Mean: this is what we usually think of as the “average”.

• Denoted \(\bar{x}\).
• Add up all of the values and divide by the number of observations (\(n\)): \[ \bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n} = \sum_{i=1}^n \frac{x_i}{n} \]
• \(x_i\) is the \(i\)th observation a
• \(\sum_{i=1}^n\) is the sum of all observations from 1 through \(n\).
  • This is called summation notation.

Median: the middle number when the data are ordered from smallest to largest.

• If there are an odd number of observations, this will be the number in the middle: {1, 3, 7, 9, 9} has median 7
• If there are an even number of observations, there will be two numbers in the middle. The median will be their average. {1, 2, 4, 7, 9, 9} has median \(\frac{4+7}{2}=5.5\)

The mean is sensitive to extreme values and skew. The median is not!

Changing that 9 out for a 45 changes the mean a lot! But the median is 7 for both \(x\) and \(y\).

Because the median is not affected by extreme observations or skew, we say it is a resistant measure or that it is robust.

• Mean: symmetric, numeric data
• Median: skewed, numeric data
• Mode: categorical data

Note: If the mean and median are roughly equal, it is reasonable to assume the distribution is roughly symmetric.