Measures of Dispersion 9/24/2013

Readings Chapter 2 Measuring and Describing Variables (Pollock) (pp.37-44) Chapter 6. Foundations of Statistical Inference ( ) (Pollock) Chapter 3 Transforming Variables (Pollock Workbook)

OPPORTUNITIES TO DISCUSS COURSE CONTENT

Office Hours For the Week When – Wednesday – Thursday 8-12 – And by appointment You will get your exams back on Thursday Homework, now due on October 3rd

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.

DESCRIPTIVE STATISTICS

Descriptive Statistics These simply describe the attributes of a single variable. You cannot test here (you need two variables) Why do them?

Categories of Descriptive Statistics Measures of Central Tendency The most common, the middle, the average Mean, Median and Mode Measures of Dispersion How wide is our range of data, how close to the middle are the values distributed Range, Variance, Standard Deviation

FREQUENCY DISTRIBUTIONS

To Run A Frequency Distribution Open GSS2008.sav Analyze (95% of all our statistics will come from this menu) – Descriptive Statistics Frequencies

Step 2 Select Your VariableHere is the Output

Interpreting the Results Percent- relative frequency for all cases Valid Percent- relative frequency for valid cases (This excludes missing values). Cumulative Percent- %of observations less than or equal to the category What is the Mode (#, cat)? What is the median (#, cat?)

MEASURES OF CENTRAL TENDENCY

First Run A Frequency Distribution Natenvir Variable- Government Spending on Improving and Protecting The Environment The Statistics Window Click on Statistics

The Output

For Ratio Variables Step 1 Step 2 Step 3 Step 4

CASE SUMMARIES

How To Do it (using world Dataset) Step 1Step 2 Check off this box

MEASURES OF DISPERISON

What are They? these measure the uniformity of the data they measure how closely or widely cases are separated on a variable.

The Range The Simplest Measure of Dispersion – Max – Min Range= max-min (only fun for ratio variables)

Back To the Island NameIncome Skipper50.00 Gilligan Mary Ann Professor Mrs. Howell Mr. Howell Ginger What is the – Maximum – Minimum – Range

High Vs. Low Dispersion Polarized Clustered

High Dispersion

Clustering

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.

From 2008 Who was more divisive?

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.

Standard Deviation in SPSS Open up the States.Sav dataset and use the union07 variable. Analyze – Descriptive Statistics Descriptives – Select your options

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

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

2011 Baseball Salaries

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 Pirates

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 Curves

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 significance testing

Testing Causality Statistical Significance Practical Significance

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 # 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?

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.

Different Kinds of Distributions

How to describe skewness