PsychStats- Chapter 3: Central Tendency

1. Overview 2. The Mean 3. The Median 4. The Mode 5. Selecting a Measure of Central Tendency 6. Central Tendency & the Shape of the Distribution

The Mean

Also known as the arithmetic average, for a population is represented by (mu); for samples it is denoted by M orX (x-bar); can be computed by taking the sum of all the scores, X divided by the total number of scores, N.

Formula for the "population mean"

Population mean= sum of all the items/ number of items

Formula for the "sample mean"

M=Ex/n : Sample mean= sum of all the items in sample/ (number of items in sample- 1)

Characteristics of a Mean

a.) Changing any score in the distribution will change the mean b.) Introducing or removing a score will change the mean (Exception: the score introduced or removed is equal to the mean) c.) If a constant is added/subtracted to/from each score in a distribution, the same constant will be added/subtracted to/from the mean. d.) If every score is multiplied/divided by a constant, the mean will change in the same way

The Median

The _________ is the midpoint of a distribution, it is important to arrange the scores in ascending or descending order first, and the score which is at the middle of the ordered scores is the median

The Mode

the most frequently occurring score(s) in a distribution

Extreme Scores or Skewed Distributions

-When a distribution has a few extreme scores, scores that are very different in value from most of the others, then the mean may not be a good representative of the majority of the distribution.

Undetermined values

This occurs usually occurs when you are measuring the number of error (or amount of time) required for an individual to complete a task. When the individual can not (or refuses) to complete the task. - Because it is relatively unaffected by extreme scores, the median commonly is used when reporting the average value for a skewed distribution.

Ordinal Scale.

When scores are measured in an __________, the median is always appropriate and is usually the preferred measure of central tendency.

Nominal Scale

Recall that the mode is not a numerical value, but a category name or score. For nominal scales, where categories are made up of names, such as classifying people by occupation or college major, and do not measure the distance or direction, it is difficult to compute the mean or median. Therefore the only option to represent the central tendency of the data is the Mode

Describing Shape

The mode gives an indication of the shape of the distribution, as well as a measure of a central tendency.

The Symmetrical Distribution.

If a distribution is exactly symmetrical, the median is exactly at the center because exactly half of the area of the graph will be on either side of the center.

Types of Symmetrical Distribution

a.) Perfectly Symmetrical b.) Distribution with one mode c.)Roughly Symmetrical Distribution d.) Binomial Distribution that is Symmetrical e.) Rectangular Distribution

Perfectly Symmetrical

Mean, media, and mode have exactly the same location / value.

Roughly symmentrical distribution

Mean, median, and mode clustered together in the center of the distribution

Binomial Distribution that is Symmetrical

Mean and median together in the center of the distribution, with modes on each side.

Rectangular Distribution

Has no mode, because there are equal frequencies for each x-value. Mean and median still together in the center of the distribution.

positively skewed distribution

the order of the central tendency as they appear from left to right on the distribution is: mode, median, and mean

negatively skewed distribution

the order of the central tendency as they appear from left to right on the distribution is: mean, median, and mode.