Measures of Dispersion

9/26/2013

Readings

• Chapter 2 Measuring and Describing Variables (Pollock) (pp.37-44)

• Chapter 6. Foundations of Statistical Inference (128-133) (Pollock)

• Chapter 3 Transforming Variables (Pollock Workbook)

OPPORTUNITIES TO DISCUSS COURSE CONTENT

Office Hours For the Week

• When– Friday 10-12– Monday 10-12– And by appointment

Homework

• Chapter 2 – Question 1: A, B, C, D, E– Question 2: B, D, E (this requires a printout)– Question 3: A, B, D– Question 5: A, B, C, D– Question 7: A, B, C, D– Question 8: A, B, C

Course Learning Objectives

1. Students will learn the basics of research design and be able to critically analyze the advantages and disadvantages of different types of design.

2. Students Will be able to interpret and explain empirical data.

MEASURES OF DISPERSION

What are They?

• these measure the uniformity of the data

• they measure how closely or widely cases are separated on a variable.

The Standard Deviation

• A More accurate and precise measure than dispersion and clustering

• Is the average distance of values in a distribution from the mean

What it tells us

• When the value of the standard deviation is small, values are clustered around the mean.

• When the value of the standard deviation is high, values are spread far away from the mean.

About the Standard Deviation

• its based on the mean

• the larger the standard deviation, the more spread out the values are and the more different they are

• if the standard deviation =0 it means there is no variability in the scores. They are all identical.

From 2008

Who was more divisive?

The Standard Deviation

• It is a standardized measure…. So what?

• This means it has ratio ( the actual value)and ordinal properties (the number of standard deviations 0,1,2,3.. From the mean

• This means we can compare different means (e.g. test scores)

The Standard Deviation and Outliers

• Any case that is more than 2 standard deviations away from the mean

• These cases often provide valuable insights about our distribution

2011 Baseball Salaries

How to determine the value of a standard deviation

How to determine the value of a standard deviation

• The value of +/- 1 s.d. = mean + value of s.d – e.g. if the mean is 8 and the s.d is 2, the value of -1

s.d's is 6, and + 1 s.d.'s is 10

• The value of +/- 2 s.d. = mean + (value of s.d. *2) – e.g. if the mean is 8 and the s.d is 2, the value of -2

s.d's is 4, and + 2 s.d.'s is 12

• Any value in the distribution lower than 4 and higher than 12 is an outlier

ECU POL Sci

An Example from 2008

• States Database• What is the Value of +/- 1 S.D?. (mean+ 1.s.d) • What is the Value of +/-2 S.D? (mean +/- 2 s.d)

Unwrapping The Results

• Which are Outliers• How did that shape the 2012 campaign

THE NORMAL CURVE

Different Kinds of Distributions

Rectangular

Camel Humps

Dromedary (one hump) Bactrian (bi-modal)

The Normal/Bell Shaped curve

• Symmetrical around the mean

• It has 1 hump, it is located in the middle, so the mean, median, and mode are all the same!

Why we use the normal curve

• To determine skewness

• The Normal Distribution curve is the basis for hypothesis/significance testing

SKEWNESS

What is skewness?

• an asymmetrical distribution.

• Skewness is also a measure of symmetry,

• Most often, the median is used as a measure of central tendency when data sets are skewed.

How to describe skewness

The Mean or the Median?

• In a normal distribution, the mean is the preferred measure

• In a skewed distribution, you go with the median

Deviate from the norm?

1. Divide the skewness value

2. By the std. error of skewness

A distribution is said to be skewed if the magnitude of (Skewness value/ St. Error of Skew) is greater than 2 (in absolute

value)

If the Value is Two or More

Use theMedian

2 or More

If the Value Is Two or Less

Mean

Less Than 2

Baseball Salaries again

• Divide the Skewness by its standard error– .800/.427= 1.87

• This value is less than 2 so we use the mean (92 million)

• What does the positive skew value mean???

Lets Try another One (Per Capita income in the states)

• Divide the Skewness by its standard error.817/.337= 2.42

• The value is greater than two, and the skewness value is positive

• What is the better measure and what might cause this distribution shape?

CO2 Emissions by State

Percent Hispanic

World Urban Population

STATISTICAL SIGNIFICANCE

Testing

• Causality

• Statistical Significance

• Practical Significance

Statistical Significance

• A result is called statistically significant if it is unlikely to have occurred by chance

• You use these to establish parameters, so that you can state probability that a parameter falls within a specified range called the confidence interval (chance or not).

• Practical significance says if a variable is important or useful for real-world. Practical significance is putting statistics into words that people can use and understand.

Curves & Significance Testing

What this Tells us

• Roughly 68% of the scores in a sample fall within one standard deviation of the mean

• Roughly 95% of the scores fall 2 standard deviations from the mean (the exact # for 95% is 1.96 s.d)

• Roughly 99% of the scores in the sample fall within three standard deviations of the mean

A Practice Example• Assuming a normal curve compute

the age (value) – For someone who is +1 s.d, from the

mean

– what number is -1 s.d. from the mean

• With this is assumption of normality, what % of cases should roughly fall within this range (+/-1 S.D.)

• What about 2 Standard Deviations, what percent should fall in this range?

Life Expectancy in Latin America and Caribbean

• Compute the estimated values for Average Life Expectancy for +/- 2 standard deviations from the mean.

• With this is assumption of normality, what % of cases should fall within this range (+/-2 s.d).

If you find this amusing or annoying, you get the concept

STANDARD DEVIATION AND CHARTS IN SPSS

Standard Deviation

For Ratio Variables

Step 1

Step 2

Step 3

Step 4

Standard Deviation in SPSS

• Open up the States.Sav dataset and use the union07 variable.

• Analyze – Descriptive Statistics

• Descriptives – Select your options

Testing for Skewness

In the Descriptives Command In the Frequencies Command

Click Here

Simple Bar Charts

• In SPSS• OPEN GSS 2008 • Analyze – Descriptive Statistics

• Frequencies