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