1. Averages and Variability
FSE 200

2. Outline Measures of Central Tendency Descriptive Statistics Mean
Median
Mode
Descriptive Statistics
Range
Standard Deviation
Variance

3. Computing and Understanding averages
Salkind, Chapter 2
Computing and Understanding averages

4. Example Data Set
The following are the number of calls ran per year for Anywhere Fire Department.
Year
Number of Calls
2000
1231
2001
1342
2002
1423
2003
986
2004
1354
2005
1266
2006
1521
2007
1453
2008
1312
2009
1389

5. Measures of Central Tendency
The AVERAGE is a single score that represents a set of scores
Averages are also known as “Measures of Central Tendency”
Three different ways to describe the distribution of a set of scores…
Mean – typical average score
Median – middle score
Mode – most common score

6. Formula for computing the mean
“X bar” is the mean value of the group of scores
“” (sigma) tells you to add whatever follows it
X is each individual score in the group
The n is the sample size

7. Computing the Mean Example
5 students scored the following on their quizzes: 79, 83, 65, 98, and 86
The average (X-bar) is the sum of the scores (ΣX) divided by the number of students (n)
The average quiz score for this group of students was 82.2

8. Using the AVERAGE function
Select the cell for the AVERAGE function
Create a formula to average the three values
=(A1+A2+A3)/3
OR type the AVERAGE function
=AVERAGE(A1:A3)

9. More Excel Arithmetic Mean Geometric Mean Moving Mean Weighted Mean
Sum of the deviation is equal to zero
Geometric Mean
GEOMEAN uses multiplication instead of addition
Moving Mean
More accurate…good for unique distributions
Weighted Mean
Accounts for the frequency of a score’s occurrence

10. Weighted Mean Example
Using Excel to Compute a Weighted Mean

11. Weighted Mean Example
The Computation of a Weighted Mean

12. Computing the Median
Median = point/score at which 50% of scores fall above and 50% fall below
No standard formula
Rank order scores from highest to lowest or lowest to highest
Find the “middle” score
BUT…
What if there are two middle scores?
What if the two middle scores are the same?

13. Using the MEDIAN function
Select the cell and type the MEDIAN function
=MEDIAN(A2:A7)

14. Computing the Mode Mode = most frequently occurring score No formula
List all values in the distribution
Tally the number of times each value occurs
The value occurring the most is the mode
Democrats = 90
Republicans = 70
Independents = 140 – the MODE!!
When two values occur the same number of times -- Bimodal distribution

15. Using the MODE function
=MODE(A2:A20)

16. Descriptive Statistics Toolpak
The Descriptive Statistics Dialog Box

17. Descriptive Statistics Toolpak
The New and Improved Descriptive Statistics Output

18. Descriptive Statistics
Salkind, Chapter 3
Descriptive Statistics

19. Why Variability Is Important
Variability is how different the scores are from one particular score
Spread
Dispersion
What is the score of interest here?
The MEAN!!
So…variability is really a measure of how each score in a group of scores differs from the mean of that set of scores.

20. Measures of Variability
Three types of variability examine the amount of spread or dispersion in a group of scores
Range
Standard Deviation
Variance
Typically report the average and the variability together to describe a distribution

21. Computing the Range Range is the most general estimate of variability
Two types:
Exclusive Range
R = h - l
Inclusive Range
R = h – l + 1

22. Computing Standard Deviation
Standard deviation (SD) is the most frequently reported measure of variability
SD = average amount of variability in a set of scores

23. Using Excel’s STDEV Function
Data for the STDEV Function

24. Using Excel’s STDEV Function
Using the STDEV Function

25. Why n – 1?
The standard deviation is intended to be an estimate of the POPULATION standard deviation
We want it to be an unbiased estimate
Subtracting 1 from n artificially inflates the SD, making it larger
In other words, we want to be conservative in our estimate of the population

26. Why n – 1?
Comparing the STDEV and STDEVP Functions

27. Things to Remember…
Standard deviation is computed as the average distance from the mean
The larger the standard deviation, the greater the variability
Like the mean, standard deviation is sensitive to extreme scores
If s = 0, then there is no variability among scores; they must all be the same value

28. Computing Variance Variance = standard deviation squared
So…what do these symbols represent? Does the formula look familiar?

29. Using Excel’s VAR Function
Computing the Variance

30. Standard Deviation or Variance
Although the formulas are quite similar, the two are also quite different
Standard deviation is stated in original units
Variance is stated in units that are squared
Which do you think is easier to interpret???

31. Acknowledgement
The majority of the content of these slides were from the Sage Instructor Resources Website