SP 225
Lecture 7
Measures of Center
Using the Sigma Notation
Alternative Sigma Notation
x
Means add the X values for all data points
Sigma Example
Data Point
X value
1st
5
2nd
3
3rd
4
4th
2
x  5  3  4  2  14
Sigma with Complex
Expression
xy ( x  y) x  a
First, multiply
all x-values by
their
corresponding
y-values.
Second, add
all products
together
First, add all
x-values to
their
corresponding
y-values.
Second, add
all sums
together.
First, add all
x-values
together.
Second, add
the value a to
the sum.
x
2
First, raise all
x-values to the
2nd power.
Second, add
all terms
together.
Complex Expressions
Examples
Data
Point
1st
X-value Y-value
5
2
2nd
3
1
3rd
4
2
4th
2
1
xy 
(5 * 2)  (3 *1)  (4 * 2)  (2 *1)
 10  3  8  2  23
Complex Expressions
Examples
Data
Point
1st
X-value Y-value
5
2
2nd
3
1
3rd
4
2
4th
2
1
( x  y ) 
(5  2)  (3  1)  (4  2)  (2  1)
 7  4  6  3  20
Complex Expressions
Examples
Data
Point
1st
X-value
2nd
3
3rd
4
(5  3  4  2)  1
4th
2
 14  1  15
5
If , a  1
x  a 
Complex Expressions
Examples
Data
Point
1st
X-value
x 
2
5
2nd
3
3rd
4
5 3 4 2
 25  9  16  4
4th
2
 54
2
2
2
2
Sigma Worksheet Answers
1) 13
2) 5
3) 15
4) 51
5) 169
6) 153
7) 16
Upcoming Material
Chapter 4
 What is typical or average?
Chapter 5 & 6
 Is something unusual?
Measures of Center
Value at the center of or middle of a data
set
 Mean
 Median
 Mode
Notation

denotes the sum of a set of values.
x
is the variable usually used to represent the
individual data values.
n
represents the number of values in a sample.
N
represents the number of values in a population.
Mean
 Found by adding all values and dividing
by the number of values
Notation
x
is pronounced ‘x-bar’ and denotes the mean of a
set of sample values
x =
x
n
µ is pronounced ‘mu’ and denotes the mean of all
values in a population
µ =
x
N
Median
 The middle value when the original data are






arranged in increasing or decreasing order
5, 9, 1, 13, 4
1, 4, 5, 9, 13
Median is 5
If even number of data points, average the 2
median points
Not sensitive to outliers
Used when discussing national income
~
Often denoted by x (pronounced ‘x-tilde’)
Mode
 The value that occurs most frequently
 When 2 values occur with greatest frequency,
the data is bi-modal
 When more than two values occur with the
same greatest frequency, the data is multimodal
 When no value is repeated, the value has no
mode
 Only measure that can be used with nominal or
qualitative data
Round-off Rule
 Keep one more decimal place than the
original data
Problems for Discussion
 A sociologist wants to find the mean
commuting time for all working US
residents. She does an internet search
and finds the mean commuting time for
each of the 50 states. She adds those
times and divides by 50. Is the result
likely to be a good estimate of the mean
commuting time for all workers?
Calculating a Measure
What is
average or
typical?
Skewed
 A distribution is skewed if it is not
symmetric and extends more to one side
or the other
 Skew can be determined by examining
differences between the mean, median
and mode
Skewed (cont.)
In Class Activity
 Consensus
 Dissensus
 Polarization
Income Inequality
 Answer the Question: What do you think does or does
not influence opinion on income inequality?
 Justify your response using data from the GSS in 2-3
paragraphs.
 Analyze the response to 2-3 survey questions including a
graph and measure of center for each survey question
 Include all SPSS output for each survey question