Distributions, Percentiles, and Central Tendency Monday, October 1 Distributions, Percentiles, and Central Tendency

The ogive gives you a good view of the relationship between scores and percentile rank.

Household Income by Language Group (Source: 1990 Census of Population and Housing)

A percentile rank of a score is a single number that gives the percent of cases in the specific reference group scoring at or below that score.

A percentile rank of a score is a single number that gives the percent of cases in the specific reference group scoring at or below that score.

CENTRAL TENDENCY Mean Median Mode

The mode is the score with the highest frequency of occurrences. It is the easiest score to spot in a distribution. It is the only way to express the central tendency of a nominal level variable.

The median. The median is the middle-ranked score (50th percentile). If there is an even number of scores, it is the arithmetic average of the two middle scores. The median is unchanged by outliers. Even if Bill Gates were deleted from the U.S. economy, the median asset of U.S. citizens would remain (more or less) the same.

 _ Xi X The Mean The mean is the arithmetic average of the scores. _________ i X = N

 _ Xi X The Mean The mean is the arithmetic average of the scores. The mean is the center of gravity of a distribution. Deleting Bill Gates’ assets would change the national mean income. _  Xi _________ i X = N

The mean of a group of scores is that point on the number line such that the sum of the squared distances of all scores to that point is smaller than the sum of the squared distances to any other point.

The Mean The sum of squared deviations from the Mean is at the lowest value. _ ( ) 2  Xi - X is lowest

The Mean The sum of squared deviations from the Mean is at the lowest value. _ ( ) 2  Xi - X is lowest _ X

The Mean The mean is the arithmetic average of the scores. The mean is the center of gravity of a distribution. Deleting Bill Gates’ assets would change the national mean! The sum of squared deviations from the Mean is at the lowest value. The mean is not a good measure of central tendency if there are outliers.