Measures of shape

Measures of shape
• We are going to consider the following measures
of shape
1) Skewness, and
2) Kurtosis
Measures of shape
• We are going to consider the following measures
of shape
1) Skewness, and
2) Kurtosis
Skewness
• To describe skewness, we need
to first understand the concept
of symmetry.
• In statistics, a symmetric
distribution is a distribution in
which the left and right sides
mirror each other.
• The most well-known symmetric
distribution is the normal
distribution, which has a distinct
bell-shape.
Skewness
• If you were to draw a line down
the center of the distribution,
the left and right sides of the
distribution would perfectly
mirror each other.

• In a symmetrical distribution,
the mean, median, and mode
are all equal.
How to Tell if a Distribution is Left Skewed or Right
Skewed
• Let’s start by contrasting characteristics of
the symmetrical normal distribution with
skewed distributions.

• The normal distribution has a central peak

where most observations occur, and the
probability of events tapers off equally in
both the positive and negative directions on
the X-axis. Both halves contain equal
numbers of observations. Unusual values
are equally likely in both tails.
How to Tell if a Distribution is Left Skewed or Right
Skewed
• However, that’s not the case with asymmetrical distributions
where probabilities decrease more slowly in one direction
relative to the other.
• In other words, extreme values that fall further away from
the peak are more likely to occur in one tail than the other.
• That’s why you’ll hear about left and right skewed
distributions, also known as negatively and positively skewed
distributions.
Right skewed (positively skewed)

• Right skewed distributions occur when

the long tail is on the right side of the
distribution. Analysts also refer to them
as positively skewed.
• This condition occurs because
probabilities taper off more slowly for
higher values.
• Consequently, you’ll find extreme
values far from the peak on the high
end more frequently than on the low.
Left skewed (negatively skewed)

• Left skewed distributions occur when the

long tail is on the left side of the
distribution. Statisticians also refer to them
as negatively skewed.
• This condition occurs because probabilities
taper off more slowly for lower values.
• Therefore, you’ll find extreme values far
from the peak on the low side more
frequently than the high side.
• The crucial point to keep in mind is that the
direction of the long tail defines the skew
because it indicates where you’ll find the
majority of exceptional values.
What Skewed Distributions Look Like in Graphs
This histogram displays a right-skewed This histogram displays a left-skewed
distribution of body fat data. distribution.
What Skewed Distributions Look Like in Graphs
This boxplot displays a right-skewed This boxplot displays a left-skewed
distribution of body fat data. distribution.
Skewed Distributions and the Mean, Median, and Mode

• The mean, median, and mode are all

equal in the normal distribution and
other symmetric distributions.
• However, when you have a asymmetric
distribution, it affects the relationship
between these measures of central
tendency.
• The mean is sensitive to extreme
values. Consequently, the longer tail in
an asymmetrical distribution pulls the
mean away from the most common
values.
Skewed Distributions and the Mean, Median, and Mode

• Right skewed: The mean is

greater than the median. The
mean overestimates the most
common values in a positively
skewed distribution.
Skewed Distributions and the Mean, Median, and Mode

• Left skewed: The mean is less than

the median. The mean
underestimates the most common
values in a negatively skewed
distribution.
• Because the mean over or
underestimates the most
frequently occurring values in
asymmetric distributions, analysts
often use the median in these
cases.
• The median is a more robust
statistic in the presence of extreme
values.
Kurtosis: Definition, Leptokurtic & Platykurtic

• Kurtosis is a statistic that measures the extent to which a

distribution contains outliers. It assesses the propensity of a
distribution to have extreme values within its tails.
• There are three kinds of kurtosis: leptokurtic, platykurtic, and
mesokurtic.
• Statisticians define these types relative to the normal distribution.
• Higher kurtosis values indicate that the distribution has more
outliers falling relatively far from the mean.
• Distributions with smaller values have a lower tendency for
producing extreme values. When you’re assessing a sample,
outliers have the greatest impact on this statistic.
Kurtosis: Definition, Leptokurtic & Platykurtic

• Kurtosis is a unitless measure of a distribution’s shape.

Consequently, analysts use the value for a normal distribution
as the baseline for comparing other distributions.
• For instance, statisticians describe leptokurtic distributions as
having higher kurtosis than the normal distribution.
• These distributions have “heavy tails,” indicating that they
have relatively long tails that contain more outliers.
• Conversely, platykurtic distributions have “light tails” that are
shorter and include fewer extreme values. Below, are graphs
for all three types for comparison.
Excess versus The Unstandardized From
• Excess kurtosis is a form of the statistic that helps you compare
the tails of your distribution to those of the normal distribution.
The excess form simply takes the standard statistic and
normalizes it by subtracting 3.
• For instance, a normal distribution has a kurtosis of 3. The excess
form subtracts three, causing a normal distribution to have an
excess value of 3 – 3 = 0. Consequently, positive excess values
indicate heavy tails, while negative values signify light tails.
• To complicate matters, some computer programs, such as Excel,
label their output as kurtosis but actually display the excess form.
Note that the regular kurtosis cannot have negative values.
Consequently, if you see a negative value, you know it is the
excess version.
Graphing the Three Types of Kurtosis

• There are three types of kurtosis, and they all use the normal
distribution as the basis for comparison.
• In the graphs, the blue distributions are normal, while the red
distribution is the comparison.
• Below, are graphs for mesokurtic, leptokurtic, and platykurtic
distributions for comparison.
Mesokurtic distributions

• Mesokurtic distributions have

kurtosis values close to that of a
normal distribution.
• These distributions have a value of
approximately 3 or an excess value
near zero.
• When sample data have kurtosis
values that are notably different
from the normal distribution, it
indicates that the population might
not follow a normal distribution.
Leptokurtic distributions – High Kurtosis

• Leptokurtic distributions have

more kurtosis than the normal
distribution.
• These distributions have heavy
tails that are longer and contain
more extreme values.
• In short, there is a greater
tendency for outliers.
• They have values of greater than
3 or positive excess values (> 0).
Platykurtic distributions – Low Kurtosis

• Platykurtic distributions have

less kurtosis than the normal
distribution.
• They have lighter tails that are
shorter and contain fewer
outliers.
• These distributions have values
of less than 3 or negative excess
values (< 0)