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### Measures of Dispersion

9/25/2012

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)

Homework

- Homework Due: Chapter 2 Pollock Workbook (10/2)
- 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

Office Hours For the Week

- When
- Friday and Wednesday 11-1
- Thursday 8-12
- And appointment

Course Learning Objectives

- Students will learn the research methods commonly used in behavioral sciences and will be able to interpret and explain empirical data.
- Students will achieve competency in conducting statistical data analysis using the SPSS software program.

Categories of Descriptive Statistics

Measures of Central Tendency

Measures of Dispersion

How wide is our range of data, how close to the middle are the values distributed

Range, Variance, Standard Deviation

- The most common, the middle, the average
- Mean, Median and Mode

Case Summaries

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
- Maximum
- Minimum

- Range= max-min (only fun for ratio variables)

Back To the Island

- What is the
- Maximum
- Minimum
- Range

High Vs. Low Dispersion

- Polarized
- Clustered

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

- Descriptives

- Descriptive Statistics

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

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

An Example from 2008

- 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 would that shaper the 2012 campaign

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 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?

What is skewness?

- an asymmetrical distribution.
- Skewnessis 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

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