Central Tendency

Variables have distributions

A variable is something that changes or has different values (e.g., anger).

A distribution is a collection of measures, usually across people.

Distributions of numbers can be summarized with numbers (called statistics or parameters).

Central Tendency refers to the Middle of the Distribution

Middle of the Distribution

Mode Most common score

Median Top from bottom 50 percent

Mean Arithmetic mean or average

(Common Statistics)

Mode

The most frequently occurring score. Can have bimodal and multimodal distributions. Modal psychology student is female. Modal number of pubs from grad school is zero.

Median

Score that separates top 50% from bottom 50%

Even number of scores, median is half way between two middle scores. 1 2 3 4 | 5 6 7 8 – Median is 4.5

Odd number of scores, median is the middle number 1 2 3 4 5 6 7 – Median is 4

Mean

Sum of scores divided by the number of people. Population mean is (mu) and sample mean is (X-bar).

We calculate the sample mean by:

We calculate the population mean by:

X

N

XX

N

X

Raw score is X. N is number of people. Sigma (Greek symbol like big E) is summation sign. Add up scores and divide by the number of people.

Computation of Mean

X (scores) Sum = 2+4+6 = 12

2 Mean = 12 / 3 = 4

4

6

12642X

43

12

N

X

Deviations from the Mean

Deviation defined. Deviations sum to zero.

Raw scores:

Deviation scores:

XXx

0x

9

8 9 10

7 8 9 10 11

0

-1 0 1

-2 -1 0 1 2

Comparison of stats (1)

Mode Good for nominal variables Good if you need to know most frequent

observation Quick and easy

Comparison of stats (2)

Median Good for “bad” (skewed) distributions Good for distributions with arbitrary ceiling or floor Often used with distributions of money

Comparison of stats (3)

Mean Used for inference as well as description; best

estimator of the parameter Based on all data in the distribution Generally preferred except for “bad” distribution. Most commonly used statistic for central

tendency.

Effects of Distribution Shape

Review

What is central tendency? Mode Median Mean

Computation

Consider the following scores: 1, 2, 2, 3, 3, 3, 4, 5

For the above set of scores, what is N? Cannot be determined 2 3 8

Computation

Consider the following scores: 1, 2, 2, 3, 3, 3, 4, 5

For the above set of scores, what is the percentage (relative frequency) of 2s? 2 10 20 25

Computation

Consider the following scores: 1, 2, 2, 3, 3, 3, 4, 5

For the above set of scores, what is the mode? 2 3 4 5

Discussion Questions

Name a variable where it would be better to find the median than the mean.

Why is it misleading to say that the average person has 1.2 brothers? Why might it be useful or helpful to say it anyway?