Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 3.6 | Central Tendency and the Shape of the Distribution 93
(a) (b) (c)
No mode
FIGURE 3.11
Measures of central tendency
for three symmetrical
distributions: normal,
bimodal, and rectangular.
Mean
Median
Mode
Mode Mode Mean
Median
Mean
Median
The positions of the
mean, median, and mode
are not as consistently
predictable in distributions
of discrete variables
(see Von Hippel, 2005).
■ Skewed Distributions
In skewed distributions, especially distributions for continuous variables, there is a strong
tendency for the mean, median, and mode to be located in predictably different positions.
Figure 3.12(a), for example, shows a positively skewed distribution with the peak (highest
frequency) on the left-hand side. This is the position of the mode. However, it should be
clear that the vertical line drawn at the mode does not divide the distribution into two equal
parts. To have exactly 50% of the distribution on each side, the median must be located
to the right of the mode. Finally, the mean is typically located to the right of the median
because it is influenced most by the extreme scores in the tail and is displaced farthest to
the right toward the tail of the distribution. Therefore, in a positively skewed distribution,
the most likely order of the three measures of central tendency from smallest to largest (left
to right) is the mode, median, and mean.
Negatively skewed distributions are lopsided in the opposite direction, with the scores
piling up on the right-hand side and the tail tapering off to the left. The grades on an easy
exam, for example, tend to form a negatively skewed distribution (see Figure 3.12(b)).
For a distribution with negative skew, the mode is on the right-hand side (with the peak),
while the mean is displaced toward the left by the extreme scores in the tail. As before,
the median is usually located between the mean and the mode. Therefore, in a negatively
skewed distribution, the most probable order for the three measures of central tendency
from smallest value to largest value (left to right), is the mean, median, and mode.
(a)
(b)
Frequency
Frequency
Mode
Median
Mean
X
Mean
Median
Mode
X
FIGURE 3.12
Measures of central tendency for skewed distributions.