Measures of Central Tendency

Mean, Median, ModeChapter 3

Overview of central tendency Central tendency

• A statistical measure• A single number to define the center of a

set of scores Purpose

• Find the single number that is most typical or best represents the entire group

Estimate a population• The central tendency (average) of a sample is

sometimes used to estimate the entire population

Three kinds of average

Mode – most common value in the data

Median – middle case (data point) in the data

Mean – balance point of all the data

Choose based on scale (level) of measurement and skewness

THE MEANUsed with interval or ratio data, except when the distribution is skewed or indeterminate

The Mean The mean: sum of all the scores divided by the

number of scores in the data Symbols: Greek μ for population, M for sample Some texts and publications use but we will

not use it in this course.

Population Mean Sample Mean

N

Xn

XM

X

The Mean: Three definitions

The balance point for a distribution of scores, equal weight on either side.

Computing the Weighted Mean from a

Frequency Distribution Table Compute Σf = n Compute f · X for each value in the data

set Compute

ΣX = Σ(f · X) M = ΣX / n

Value (X) f fX

10 1 10

9 2 18

8 4 32

7 0 0

6 1 6

Total n = Σf = 8 ΣfX = 66

M = ΣX / n = 66/8 = 8.25

The Combined Mean of Two Sets of Scores

1. Determine the combined sum of all the scoresΣX1 + ΣX2 = ΣXcombined

2. Determine the combined number of scoresn1 + n2 = ncombined

3. Compute new mean Mcombined = ΣXcombined /ncombined

21

21

nn

XXM

mean overall

Characteristics of the Mean

Changing the value of one score always changes the mean.

Introducing a new score or removing a score usually changes the mean (unless the score is exactly equal to the mean).

Adding or subtracting a constant from each score changes the mean by the same constant.

Multiplying or dividing each score by a constant multiplies or divides the mean by that constant.

THE MEDIAN

Used with ordinal data.Used with skewed interval or ratio data

The Median

The median is the midpoint of the scores in a distribution when they are listed in order from smallest to largest.

The median divides the scores into two groups of equal size.

If the data have an even number of scores, median is the midpoint between two scores.

Median, Mean and “Middle”

Mean is the balance point of a distribution• Defined by distance from center

(“weight”)• Not necessarily the midpoint

Median is the midpoint of a distribution• Defined by number of scores• Usually is not the balance point

Both measure central tendency, using two different concepts of “middle”

THE MODE

Used with nominal dataOften reported as percentage, not category

The Mode

The mode is the score or category that has the greatest frequency of any in the frequency distribution.• Can be used with any scale of

measurement• Corresponds to an actual score in the

data It is possible to have more than one

mode

SKEWED DATA DISTRIBUTIONSThe extreme values affect the Mean more than they affect the Median

Skewed Data Distributions

In positively skewed data (a) Mean is larger than Median In negatively skewed data (b) Mean is less than Median When you do not have a graph of a distribution, comparing the

Mean and the Median tells you if there is skew present.

Central Tendency in Skewed Distributions

Mean, influenced by extreme scores, is pulled toward the long tail a lot (positive or negative)

Median, in order to divide scores in half is less affected by the extreme scores

Mode is not affected by extreme scores If Mean – Median > O, the distribution

is positively skewed. If Mean – Median < O, the distribution

is negatively skewed

Selecting a Measure of Central Tendency

Measure of Central Tendency

Appropriate to choose when … Should not be used when…

Mean •No situation precludes it•First choice measure of central tendency

•Extreme scores•Skewed distribution•Ordinal scale•Nominal scale

Median •Extreme scores•Skewed distribution•Ordinal scale

•Nominal scale

Mode •Nominal scales•Discrete variables•Describing shape of distribution

•Interval or ratio data, except to accompany mean or median

Central Tendency in the News

Central Tendency in the News

Central Tendency

Core concept for rest of statistics.The mean will reappear all semester.