Frequency Distribution I. How Many People Made Each Possible Score? A. This is something I show you for each quiz.

Frequency Distribution

I. How Many People Made Each

Possible Score?

A. This is something I show you

for each quiz.

U.S. Distribution of Income Over Time

Share of Income:

Richest 1% Poorest 20%

1920 14.4% NA

1970 7.8% 4.2%

2007 18.0% 3.4%

Measurement I. Quantifying and Describing Variables -1

Four Levels of Measurement

Measures of Central TendencyModeMedianMean

Measurement I. Quantifying and Describing Variables - 2

DON’T WRITE – JUST READ!

We need to review the levels of measurement before continuing. Since you have definitions for the levels of measurement on pages 10-12 of the 300Reader, there is NO REASON to write the following definitions

Four Levels of Precision For Measuring Variables

Nominal Measure: You can put cases into a category, but cannot specify an order or relationship between the categories.

Example: The variable “religion” can take on values such as Catholic, Protestant, Mormon, Jewish, etc.

Four Levels of Precision For Measuring Variables

Ordinal Measure: You can put cases into different categories, and order the categories.

Example: The variable “strength of religious belief” can take on values such as devoutly religious, fairly religious, slightly religious, not religious.

Four Levels of Precision For Measuring Variables

Interval Measure: Not only can you order the categories of the variable, you can specify the difference between any two categories.

Example. The variable “temperature on the Fahrenheit scale” can take on values such as 32 degrees, 74 degrees, 116 degrees.

Four Levels of Precision For Measuring Variables

Ratio Measure: You can order categories, specify the difference between two categories, and the value of zero on the variable represents the absence of the variable.

Example. The variable “annual income” can take on the values of $0, $98,000, or $694,294,129.

Example of a Ratio Measure: Income Inequality

The next several slides what groups in the United States, Japan and Sweden

think is the actual and fair degree of income inequality between an executive and an auto worker. Since a score of zero equals the absence of income, and the difference between $1 and $2 is the same as between $1,000 and $1,001, we have a ratio level measure.

United States

Perceived Fair

Income Income

Business 15.1/1 15.6/1

Labor 14.8/1 7.2/1

Republicans 13.2/1 11.3/1

Democrats 15.4/1 8.2/1

Youth 13.4/1 6.0/1

Japan

Perceived Fair

Income Income

Business 9.1/1 8.6/1

Labor 10.1/1 4.1/1

Conservative 7.1/1 5.4/1

Party

Left Parties 10.3/1 3.7/1

Sweden

Perceived Fair

Income Income

Business 2.4/1 3.5/1

Labor 3.2/1 1.9/1

Conservative/ 2.2/1 2.1/1

Center Party

Left Party 3.2/1 1.9/1

Measures of Central Tendency -1

Kobe Bryant $24.8 millionPaul Gasol $17.8 millionAndrew Bynum $13.7 millionLamar Odom $8.2 millionRon Artest $6.3 millionLuke Walton $5.2 millionSteve Blake $4.0 million

Derek Fisher $3.7 millionShannon Brown $2.1 millionMatt Barnes $1.7

millionTheo Ratliff $1.3 millionJoe Smith $1.3

millionDevin Ebanks $0.4 millionDerrick Caracter $0.4 million

Measures of Central Tendency - 2

Mode: The most frequently occurring value. $1.3 million and $0.4 million

Median: The midpoint of the distribution of cases. 1. Arrange cases in order 2. If the number of cases is odd, median is the

value taken on by the case in the center of the list. 3. If the number of cases is even, median is the

average of the two center values. $3.85 million (4.0 + 3.7 = 7.7 and 7.7/2 = 3.85)

Measures of Central Tendency - 3

DO NOT WRITE ANY OF THE FORMULAS THAT APPEAR AHEAD!

THEY ARE IN THE 300READER AND ONLY APPEAR AHEAD FOR PURPOSES OF DISCUSSION.

Measures of Central Tendency - 4

Mean is the arithmetic average of the values that all the cases take on. Formula: Add up all the values and divide this

sum by the number of cases, N. In our Laker example, $6.5 million.

N

X

N

XXXXMean

iN

...21

Measures of Central Tendency - 5

Bush Tax Cut Example A. The “Mean” household tax cut under the Bush Tax

Cuts is $1,199 while the “Median” household only receives $217.

1. Quite a Difference!

2. 75% of Households Actually Lose Under

the Bush Tax Cuts

Measures of Central Tendency - 6

Question: Why can’t the mean tell us everything?

Answer: While the mean tells us the average, it does NOT tell us how accurate the mean is when making predictions.

Measures of Dispersion - 1

Thus, we need to know whether the mean occurred because many scores were quite close to the mean or was the mean an average of scores quite different than the mean?

This is what measures of dispersion tell us.

DON’T WRITE THE FORMULAS AHEAD!!

Measures of Dispersion - 2

The variance is a measure of how spread out cases are, calculated by:

Compute the distance from each case to the mean, then square that distance.

Find the sum of these squared distances, then divide it by N-1. Lakers: $54.1million.

1

)( 2

N

XXVariance

i

Measures of Dispersion - 3

The standard deviation is the square root of the variance. For the Laker data, $7.4 million.

1

)(ˆ

2

N

XXs

i

Measures of Dispersion - 4

According to page 25 of the 300 Reader, what conclusion should I draw about the dispersion of the Laker data when the mean is 6.5 and the standard deviation is

7.4?

Normally Distributed Curve

Skewed Distributions

Characteristics of the Normal Distribution -1It is symmetrical -- Half the cases are to one

side of the center; the other half is on the other side.

The distribution is single peaked, not bimodal or multi-modal

Most of the cases will fall in the center portion of the curve and as values of the variable become more extreme they become less frequent, with “outliers” at each of the “tails” of the distribution few in number.

Characteristics of the Normal Distribution -2

It is only one of many frequency distributions but the one we will focus on for most of this course.

The Mean, Median, and Mode are the same.

Percentage of cases in any range of the curve can be calculated.

Family of Normal Curves

Summarizing Distributions

Two key characteristics of a frequency distribution are especially important when summarizing data or when making a prediction from one set of results to another:

Central Tendency What is in the “Middle”? What is most common? What would we use to predict?

Dispersion How Spread out is the distribution? What Shape is it?

Appropriate Measures of Central Tendency

Nominal variables Mode

Ordinal variables Median

Interval level variables Mean

- If the distribution is normal (median is better with skewed distribution)

Logic of Measures of Dispersion 1

Why not think of dispersion as the difference between the highest and lowest scores?

Logic of Measures of Dispersion 2

If we need a measure based upon all the scores, why not just subtract the mean from each score, add up this total and divide by the total number of scores?

Logic of Measures of Dispersion 3

If the sum of all the deviations from the mean is zero, we could take the absolute value of each deviation from the mean and avoid this problem. So, why don’t we?

So, what do we do?

Logic of Measures of Dispersion 4

What is the utility of Tchebysheff’s Theorem?