02-15-2012

Median is resistant to outliers (not greatly influenced by outliers)
Mean is not resistant to outliers, prone to follow the tail (at the mercy of outliers)

In class exercise with mean and median using height of class members.
Mean was 66.9 inches
Median was 66 inches

Measures of Variability (Spread)

Range - "Mostly useless, can be beneficial on occasion" To find the range you subtract the highest value of your data set from the lowest value of your data set. Using our class height as an example the tallest member is 78 inches, while the shortest individual is 60 inches. 78 - 60 = 18 inches.

Variance - Not interested in this measure of variability, skip any homework questions asking for variance.

Standard Deviation - "Think of it as the average difference from the mean or the 'acceptable deviation'."

Standard deviation for a parameter:

• σ = Standard Deviation of parameter from a population
• N = Population size
• xi = Individual observation
• μ = Mean of the parameter from a population

Standard Deviation for a statistic:

• s = Standard Deviation of statistic from a sample
• n = Sample size
• xi = Individual observation 
• x-bar = Mean of the statistic from the sample

As complicated as these expressions look they are basically saying: "Sum the squares of all the individual observations (observed value) -  mean (expected value), divide this number by the sample size less one (or the population size), then take the square root of that number. The resulting number is your Standard Deviation.
 

• You cannot know the standard deviation without knowing the following: mean, sample size, individual values.

Going back to the class height data, the standard deviation in height is 3.821 inches. So if you were to grab someone from our class at random you would not be surprised if their height was between 63.1 inches and 70.7 inches.

Using Quiz 1 test results:

• μ = 67%
• Median = 73%
• σ = 23.87%
• range = 100

Describe this distribution.

The mean  is smaller than the median, therefore skewed left.

However, this data is not reflective of the class. Notice that the range is 100, the data set is accounting for people who have not taken the quiz yet (people with zeroes).

Taking this into account and adjusting the data set accordingly:

• μ = 70.5%
• Median = 76.7%
• σ = 18.58%
• range = 67 (from 100-33)

Notice that the Mean has shifted upward because it is no longer being weighed down by the zeroes.
This change in mean also alters the standard deviation because the mean is used to calculate the standard deviation.
If you are unclear on any of this please watch this video