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)

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0.1

0.2

0.3

0.4

0.5

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0 20000 40000 60000 80000 100000 120000 140000 160000 180000 200000

Household Income

Per

cent

age

Hmong

Cambodian

Lao

Spanish

Samoan

Armenian

Tongan

Indonesian

Vietnamese

Rumanian

Portuguese

Russian

Arabic

Korean

Persian

Ukrainian

Chinese

Punjabi

Japanese

Pilipino

Ilocano

Hindi

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.

The 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 Gateswere deleted from the U.S. economy, the median asset of U.S. citizens would remain (more or less) the same.

The Mean

The mean is the arithmetic average of the scores.

X_

= iXi

_________

N

The Mean

The mean is the arithmetic average of the scores.

The mean is the center of gravity of a distribution. DeletingBill Gates’ assets would change the national mean income.

X_

= iXi

_________

N

The mean of a group of scores is that point on the number linesuch that the sum of the squared distances of all scores to that pointis 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.

X_

Xi -( )2

is lowest

The Mean

The sum of squared deviations from the Mean is at the lowest value.

X_

Xi -( )2

is lowest

T

1.61.41.21.0.8.6.4

DIS2

610

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590

580

570

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540

X_

The Mean

The mean is the arithmetic average of the scores.

The mean is the center of gravity of a distribution. DeletingBill 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 thereare outliers.