1. Means & Medians
Chapter 4

2. Parameter
A value describing a population
Typically unknown

3. Statistic
A value calculated from sample data

4. Measures of Central Tendency
Median: middle of the data (50th percentile)
Put data in numerical order
n is odd  median = middle number
n is even  median = average of middle two numbers
(n = sample size)

5. Measures of Central Tendency
parameter
Mean: arithmetic average
m (mu) - population mean
x (x-bar) - sample mean
statistic
S – capital Greek letter sigma – sum up the values that follow
Formula:

6. Measures of Central Tendency
Mode – observation that occurs most
Can have more than one mode
Can have 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 a sample of 6 customers buy the following numbers of lollipops
Suppose a sample of 6 customers buy the following numbers of lollipops. Find the median.
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. Now 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 if the 12 lollipops became 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 if the 12 lollipops became 50? 5
The median is . . .
12.17
The mean is . . .
What happened?

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

14. Sum of Deviations YES Find the mean of the following data.
Will this sum always equal zero?
Find the sum of the deviations of each value from the mean.
Deviation from the mean
YES
Mean = balance point

15. 27 27 Find the mean and median of the data set. Mean = Median =
Make a histogram with an x-scale of 2
What shape is this distribution?
Use scale of 2 on graph

16. 28.176 25 Find the mean and median of the data set. Mean = Median =
Make a histogram with an x-scale of 8
What shape is this distribution?
Use scale of 2 on graph

17. What shape is this distribution?
Find the mean and median of the data set.
Mean =
Median =
54.588 58
Make a histogram
What shape is this distribution?
Use scale of 2 on graph

18. Recap: Skewed: mean is pulled toward the skewness
Symmetrical: mean = median
Report the mean as the center
Skewed: mean is pulled toward the skewness
Report the median as the center

19. Trimmed mean: List data in order Multiply the % to trim by n
Cut off that many observations from BOTH ends of the data
Calculate the new mean

20. 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