Goes with Chapter 3 of Gravetter & Wallnau textbook "Essentials of Statistics for Behavioral Sciences"

1. Mean, Median, Mode
Chapter 3

2.  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

3.  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

4. Used with interval or ratio data, except when the
distribution is skewed or indeterminate

5.  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 X but we will not use it in
this course.
Population Mean Sample Mean
X X
M
N n

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

7.  Compute Σf = n
 Compute f · X for each value in the data set
 Compute Value (X) f fX
10 1 10
ΣX = Σ(f · X) 9 2 18
 M = ΣX / n 8 4 32
7 0 0
6 1 6
Total n = Σf = 8 ΣfX = 66
M = ΣX / n = 66/8 = 8.25

8. 1. Determine the combined sum of all the scores
ΣX1 + ΣX2 = ΣXcombined
2. Determine the combined number of scores
n1 + n2 = ncombined
3. Compute new mean
Mcombined = ΣXcombined /ncombined
X1 X2
overall mean M
n1 n2

9.  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.

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

11.  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.

12.  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”

13. Used with nominal data
Often reported as percentage, not category

14.  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

15. The extreme values affect the Mean more than
they affect the Median

16.  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.

17.  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

18. Measure of Appropriate to choose when … Should not be used when…
Central Tendency
Mean •No situation precludes it •Extreme scores
•First choice measure of •Skewed distribution
central tendency •Ordinal scale
•Nominal scale
Median •Extreme scores •Nominal scale
•Skewed distribution
•Ordinal scale
Mode •Nominal scales •Interval or ratio data, except
•Discrete variables to accompany mean or median
•Describing shape of
distribution

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