1. Means & Medians
Chapter 4

2. Parameter -
Suppose we want to know the MEAN length of fish in Lake Lewisville . . .
Fixed value about a population
Typical unknown

3. Statistic -
Value calculated from a sample

4. Measures of Central Tendency
Median - the middle of the data; 50th percentile
Observations must be in numerical order
Is the middle single value if n is odd
The average of the middle two values if n is even
NOTE: n denotes the sample size

5. Measures of Central Tendency
parameter
Mean - the arithmetic average
Use m to represent a population mean
Use x to represent a sample mean
statistic
Formula:
S is the capital Greek letter sigma – it means to sum the values that follow

6. Measures of Central Tendency
Mode – the observation that occurs the most often
Can be more than one mode
If all values occur only once – there is no mode
Not used as often as mean & median

7. Suppose we are interested in the number of lollipops that are bought at a certain store. A sample of 5 customers buys the following number of lollipops. Find the median.
The numbers are in order & n is odd – so find the middle observation.
The median is 4 lollipops!

8. Suppose we have sample of 6 customers that buy the following number of lollipops. The median is …
The numbers are in order & n is even – so find the middle two observations.
The median is 5 lollipops!
Now, average these two values. 5

9. Suppose we have sample of 6 customers that buy the following number of lollipops. Find the mean.
To find the mean number of lollipops add the observations and divide by n.

10. Using the calculator . . .

11. 2 3 4 6 8 20 5 7.17 The median is . . . The mean is . . .
What would happen to the median & mean if the 12 lollipops were 20? 5
The median is . . .
7.17
The mean is . . .
What happened?

12. 2 3 4 6 8 50 5 12.17 The median is . . . The mean is . . .
What would happen to the median & mean if the 20 lollipops were 50? 5
The median is . . .
12.17
The mean is . . .
What happened?

13. Resistant - YES NO Statistics that are not affected by outliers
Is the median resistant?
YES
Is the mean resistant? NO

14. YES Look at the following data set. Find the mean.
Now find how each observation deviates from the mean.
What is the sum of the deviations from the mean?
Will this sum always equal zero?
This is the deviation from the mean.
YES

15. 27 27 Look at the following data set. Find the mean & median. Mean =
Create a histogram with the data. (use x-scale of 2) Then find the mean and median.
Look at the placement of the mean and median in this symmetrical distribution.
Use scale of 2 on graph

16. 28.176 25 Look at the following data set. Find the mean & median.
Create a histogram with the data. (use x-scale of 8) Then find the mean and median.
Look at the placement of the mean and median in this right skewed distribution.
Use scale of 2 on graph

17. Create a histogram with the data. Then find the mean and median.
Look at the following data set. Find the mean & median.
Mean =
Median =
54.588 58
Create a histogram with the data. Then find the mean and median.
Look at the placement of the mean and median in this skewed left distribution.
Use scale of 2 on graph

18. Recap: In a symmetrical distribution, the mean and median are equal.
In a skewed distribution, the mean is pulled in the direction of the skewness.
In a symmetrical distribution, you should report the mean!
In a skewed distribution, the median should be reported as the measure of center!

19. Example calculations
During a two week period 10 houses were sold in Fancytown.
The “average” or mean price for this sample of 10 houses in Fancytown is $295,000

20. During a two week period 10 houses were sold in Lowtown.
The “average” or mean price for this sample of 10 houses in Lowtown is $295,000
Outlier

21. Looking at the dotplots of the samples for Fancytown and Lowtown we can see that the mean, $295,000 appears to accurately represent the “center” of the data for Fancytown, but it is not representative of the Lowtown data.
Clearly, the mean can be greatly affected by the presence of even a single outlier.
Outlier

22. In the previous example of the house prices in the sample of 10 houses from Lowtown, the mean was affected very strongly by the one house with the extremely high price.
The other 9 houses had selling prices around $100,000.
This illustrates that the mean can be very sensitive to a few extreme values.
SOOOO……

23. Describing the Center of a Data Set with the median
The sample median is obtained by first ordering the n observations from smallest to largest (with any repeated values included, so that every sample observation appears in the ordered list). Then

24. Example of Median Calculation
Consider the Fancytown data. First, we put the data in numerical increasing order to get
231, , , ,000
297, , , ,000
315, ,000
Since there are 10 (even) data values, the median is the mean of the two values in the middle.

25. Consider the Lowtown data
Consider the Lowtown data. We put the data in numerical increasing order to get
93, , , ,000
100, , , ,000
122, ,000,000
Since there are 10 (even) data values, the median is the mean of the two values in the middle.

26. mean vs median

27. Typically,
when a distribution is skewed positively, the mean is larger than the median,
when a distribution is skewed negatively, the mean is smaller then the median, and
when a distribution is symmetric, the mean and the median are equal.

28. The Trimmed Mean
A trimmed mean is computed by first ordering the data values from smallest to largest, deleting a selected number of values from each end of the ordered list, and finally computing the mean of the remaining values.
The trimming percentage is the percentage of values deleted from each end of the ordered list.

29. Example of Trimmed Mean

30. Example of Trimmed Mean

31. Trimmed mean: Purpose is to remove outliers from a data set
To calculate a trimmed mean:
Multiply the % to trim by n
Truncate that many observations from BOTH ends of the distribution (when listed in order)
Calculate the mean with the shortened data set

32. So remove one observation from each side!
Find a 10% trimmed mean with the following data.
10%(10) = 1
So remove one observation from each side!

33. Example:
The Highway Loss Data Institute publishes data on repair costs resulting from a 5-mph crash test of a car moving forward into a flat barrier. The following table gives data for 10 midsize luxury cars:
Model Repair Cost in dollars
Audi A6 0
BMW 328i 0
Cadillac Catera 900
Jaguar X
Lexus ES
Lexus IS
Mercedes C
Saab
Volvo S
Volvo S
Compute the mean and median values.
Why are these values so different?
Which of the mean and median do you think is more representative of the data set? Why?

34. Test day: The average for Mrs
Test day: The average for Mrs. Goins’ 4th period Algebra test was 88% for those 24 students. The average for 6th period was 85% for those 20 students. What was the average for the combined classes?