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Averages and Variability
FSE 200
Outline
• Measures of Central Tendency
– Mean
– Median
– Mode
• Descriptive Statistics
– Range
– Standard Deviation
– Variance
Salkind, Chapter 2
COMPUTING AND
UNDERSTANDING AVERAGES
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
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
Computing the Mean
Formula for computing the mean
X
X 
n
“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
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
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)
More Excel
Arithmetic 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
Weighted Mean Example
Using Excel to Compute a Weighted Mean
Weighted Mean Example
The Computation of a Weighted Mean
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?
Using the MEDIAN function
– Select the cell and type the MEDIAN function
– =MEDIAN(A2:A7)
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
Using the MODE function
=MODE(A2:A20)
Descriptive Statistics Toolpak
The Descriptive Statistics Dialog Box
Descriptive Statistics Toolpak
The New and Improved Descriptive Statistics Output
Salkind, Chapter 3
DESCRIPTIVE STATISTICS
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.
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
Computing the Range
• Range is the most general estimate of
variability
• Two types:
– Exclusive Range
• R=h-l
– Inclusive Range
• R=h–l+1
Computing Standard Deviation
• Standard deviation (SD) is the most
frequently reported measure of variability
• SD = average amount of variability in a set of
scores
Using Excel’s STDEV Function
Data for the STDEV Function
Using Excel’s STDEV Function
Using the STDEV Function
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
Why n – 1?
Comparing the STDEV and STDEVP Functions
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
Computing Variance
• Variance = standard deviation squared
• So…what do these symbols represent?
Does the formula look familiar?
Using Excel’s VAR Function
Computing the Variance
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???
Acknowledgement
The majority of the content of these slides were
from the Sage Instructor Resources Website