SP 225Lecture 7
Measures of Center
Using the Sigma Notation
Alternative Sigma Notation
xMeans add the X values for all data points
Sigma Example
Data Point X value
1st 5
2nd 3
3rd 4
4th 2
142435 x
Sigma with Complex Expression
xy )( yx ax 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.
2xFirst, raise all x-values to the 2nd power. Second, add all terms together.
Complex Expressions Examples
Data Point
X-value Y-value
1st 5 2
2nd 3 1
3rd 4 2
4th 2 1
2328310
)1*2()2*4()1*3()2*5(
xy
Complex Expressions Examples
Data Point
X-value Y-value
1st 5 2
2nd 3 1
3rd 4 2
4th 2 1
203647
)12()24()13()25(
)(
yx
Complex Expressions Examples
Data Point
X-value
1st 5
2nd 3
3rd 4
4th 2 15114
1)2435(
1,
ax
aIf
Complex Expressions Examples
Data Point
X-value
1st 5
2nd 3
3rd 4
4th 2 54
416925
2435 2222
2
x
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
µ is pronounced ‘mu’ and denotes the mean of all values in a population
x =n x
is pronounced ‘x-bar’ and denotes the mean of a set of sample valuesx
Nµ =
x
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 multi-modal
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
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