Chapter 2 Review Using graphs/tables/diagrams to show variable relationships Understand cumulative frequency, percentile rank, and cross-tabulations Perform rates of change

Cumulative Frequencies / Percentile Rank # of Arrestsf%cfC% What is the percentile rank for those with 5 arrests?

Cross-Tabulations Attitude towards Lowering the Drinking Age to 19 MaleFemaleTotal Favor %23%24.5% Neutral %27%25.5% Oppose % Total %

Rate of Change Rate of Change = (100) * (time 2f – time 1f) (time1f) Allows us to compare the same population at two points in time. Always be aware of the sign. – A negative percent signifies a reduction – A positive percent signifies an increase

Chapter 3 Measures of Central Tendency

Measures of Central Tendency Three main types – Mode – Median – Mean Choice depends upon level of measurement

The Mode The mode is the most frequently occurring value in a distribution. Abbreviated as Mo Sometimes there is more than one mode EX: 96, 91, 96, 90, 93, 90, 96, 90 Bimodal Mode is the only measure of central tendency appropriate for nominal-level variables

Mode - Example What is the mode for the following set of numbers? 20, 21, 30, 20, 22, 20 Explains nothing about – Ordering of variables – Variation within variables Distributions can be bimodal and/or multimodal – Several categories with same frequencies

The Median The median is the middle case of a distribution Abbreviated as Mdn Appropriate for ordinal data because it only shows direction and not distance Used if distribution is skewed How to find the median? If even, there will be two middle cases – interpolate If odd, choose the middle-most case Cases must be ordered Position of the Mdn

Example of median: Years in Prison What is the median? – odd or even? (7+1)/2=4th case Where is the 4th case? Sort distribution from lowest to highest

Example of median: Years in Prison 4th case *

Example of median with even # of cases (8+1)/2=4.5 Half way between the 4th and 5th case (2 + 3) / 2 = 2.5 Median = Position of the Mdn

The Mean Most popular measure of central tendency Assumes equality of intervals Basis of many higher order formulas for statistical procedures Use either μ or X depending on whether population or sample estimate

The Mean The mean is appropriate for interval and ratio level variables X = raw scores in a set of scores N = total number of scores in a set

Example: Prison Sentences What is the mean?

The Mean What does the mean do? – Center of gravity – Deviation = (Raw Score – Mean) X (Raw Score)Deviation Mean = 6 = (∑X / N) = (30 / 5)

The Weighted Mean The “mean of the means” – overall mean for a number of groups Best used for unequal groups Example: 4, 7, 3, 8 2, 4, 9, 1, 6, 8

Obtaining the Mode, Median, and Mean from a Simple Frequency Distribution XfcffX Position of the Mdn Mo Mdn

An Illustration: Measures of Central Tendency in a Skewed Distribution Salary $120,000 $60,000 $40,000 $30,000  Mean = $50,000  Median = $40,000  Mode = $30,000

Comparing the Mode, Median, and Mean Three factors in choosing a measure of central tendency 1.Level of measurement 2.Shape or form of the distribution of data Skewness Kurtosis 3.Research Objective

Level of Measurement

Shape of the Distribution In symmetrical distribution – mode, median, and mean have identical values In skewed data, the measures of central tendency are different – Skewness relevant only at the interval level Mean heavily influenced by extreme outliers – median best measure in this situation

Research Objective Choice of reported central tendency depends on the level of precision required. Most published research requires median and/or mean calculations. In skewed data, median more balanced view For advanced statistical analyses, mean usually preferred In large data sets, mean most stable measure

Summary Three best known measure of central tendency – mode, median, mode Three factors determine appropriateness – Level of measurement – Shape of the distribution – Research objective