Stats Chapter 3: Central Tendency

Definition of a symmetrical distribution

A distribution where the left-hand side is a mirror image of the right-hand side.

Characteristics of a rectangular distribution

No mode, mean and median the same and in the center

The APA style uses the letter ____ as the symbol for the sample mean.

M

The formula for the SAMPLE mean is:

M equals sigma times x over n (note little n)

Abbreviation for median

Mdn

Characteristics of a symmetrical distribution

Mean, median, and mode will be the same and in the center.

Definition of mode

Most common observation among a group of scores. Score or category with the greatest frequency.

For a distribution with negative skew, the mode is on the ___-hand side with the peak, whereas the mean is displaced toward the ______ by the extreme scores in the tail.

right hand side, left.The median is located between the mean and the mode.

If a mean is identified with the symbol M, you should realize that we are dealing with a ________

sample

Definition of a weighted mean

the average of two means, calculated so that each mean is weighted by the number of scores it represents.

In order from smallest value to largest value, or left to right, the three measures of central tendency for a negatively skewed distribution are:

the mean, the median, and the mode.

In a positively skewed distribution, the order of the three measures of central tendency from smallest to largest, or left to right, is:

the mode, the median, and the mean. Think of the mode being at the peak, and the mean being closest to the tail.

The precise median and the 50th percentile are both defined as:

the point that separates the top 50% of a distribution from the bottom 50%.

An alternate definition for the mean: the amount each individual gets when...

the total is distributed eually

The goal of central tendency

to findthe single score that is most typical or most representative of the entire group.

The goal of the median:

to locate the midpoint of the distribution.

3 situations in which the mode is commonly used as an alternative to the mean or in conjunction with the mean

1. Nominal scales (do not measure quantity) 2. Discrete variables (whole, indivisible categories) 3. Describing shape (location of peaks)

4 situations in which the median serves as a valuable alternative to the mean

1. extreme scores or skewed distributions 2. undetermined values (did not finish) 3. open-ended distributions (no upper or lower limits for one of the categories) 4. ordinal scale (greater than, less than)

When is it appropriate to calculate a precise median? When is it inappropriate?

Appropriate with continuous variables; inappropriate for discrete variables (think seconds versus family members).

How is the mean computed?

By adding all the scores in the distribution and diving by the number of scores.

Central tendency definition

Central tendency is a statistical measure to determine a single score that defines the center of a distribution.

True or false. There is an equation for computing the median.

False

True of false: A distribution with a mean of 70 and a median of 75 is probably positively skewed.

False.

True or false: In a perfectly symmetrical distribution, the mean, the median, and the mode will all have the same value.

False. If the distribution is bimodal, the modes will be different than the median and mode.

True or false: adding a new score to a distribution always changes the mean.

False. If the score is equal to the mean, it does not change the mean.

Where do independent variables go on a graph?

Horizontal axis (treatment conditions)

Discuss adding or subtracting a constant from each score and its effect on the mean

If you add or subtract the same value from each score, you can also just add or subtract that same value to the mean. So if you add two points to each value, you can also add 2 points to the mean and it will be correct.

Discuss multiplying or subtracting a constant from each score and its effect on the mean

If you multiply or divide each value, you also do the same to the mean.

In general, the mode is the only measure of central tendency that can be used with data from a _____ ______ of measurement.

Nominal scale

Why is it usually considered inappropriate to compute a mean for scores measured on an ordinal scale?

The definition of the mean is based on distances (the mean balances the distances) and ordinal scales do not measure distance.

Adding a new score to a distribution, or removing an existing score, usually changes the mean. What is the exception?

The exception is when the new score, or the removed score, is exactly equal to the mean.

Definition of median

The median is the midpoint in a list when the scores in a distribution are listed in order from smallest to largest. 50% of scores are below the median and 50% are above.

True or false: Changing any score in the distribution causes the mean to be changed.

True

True or false: changing the value of a score in a distribution always changes the mean.

True

Where do dependent variables go on a graph?

Vertical axis (scores)

Describe the expression "number crunching"

When you take a distribution consisting of many scores and "crunch" them down to a single value that describes them all.

The mean is also know as the _________ _______

arithmetic average

A second alternate definition for the mean: a ______ point for the distribution

balance

A distribution with two modes is said to be _______

bimodal

The mode is the only measure of central tendency that:

corresponds to an actual score in the data

In general, greek letters are used for _________ and letters of the alphabet are used for _______

greek letters for a population letters of alphabet for a sample value (statistic)

You can say there are two modes even if they don't have the same frequency. When two modes have unequal frequencies, researches occasionally differentiate the two values by calling the taller peak the _____ mode and the shorter one the ______ mode.

major; minor

In most situations with numerical scores from an interval or a ratio scale, the ____ is the preferred measure of central tendency.

mean

The preferred measure of central tendency

mean

Which measure of central tendency is most affected if one extremely large score is added to a distribution?

mean

Characteristics of a bimodal, symmetrical distribution

mean and median are the same in the center, two modes on the ends

Provide the symbols for the following: mean for a population mean for a sample

mean for a population: greek letter mu mean for a sample: M or x-bar

Statisticians have developed 3 different methods for measuring central tendency:

mean, median, and mode

The formula for the POPULATION mean is:

mu equals sigma times x over N (note big N)

A distribution with more than two modes is called _______

multimodal