AE 9
STATISTICAL ANALYSIS
with Software Applications
AE 9
STATISTICAL ANALYSIS
with Software Applications
Objectives:
At the end of the class, the students should be able
to:
- Understand the Moments of Statistical
Distribution
- Determine the Skewness of a Data Set
- Determine the Kurtosis of a Data Set
- Determine the Descriptive Statistics of a Data Set
Moments of
Statistical
Distribution
Moments of Statistical Distribution
Moments are used to describe the shape of a distribution.
Four moments apply for describing the shape of a distribution. The 1st
moment describes the middle, the 2nd describes the spread from the
middle, the 3rd describes symmetry about the middle, and the 4th
describes the shape.
Moments of Statistical Distribution
1st moment describes the middle - Mean
2nd moment describes the spread Variance
from the middle
3rd moment describes symmetry Skewness
about the middle
4th moment describes the shape Kurtosis
Skewness
Skewness
Skewness is the degree of distortion from the
symmetrical bell curve or the normal distribution. It measures
the asymmetry (lack of symmetry) of a data series’ distribution
about its mean. If the curve is leaning to the left or right, it is
said to be skewed.
Zero Skewness
A distribution has zero skewness
if it has a symmetric distribution.
In a symmetrical distribution, the
Mean, Median and Mode are
equal to each other and the
ordinate at mean divides the
distribution into two equal parts.
Undefined Skewness
A distribution has undefined skewness if a data series is uniform,
rectangular, or constant, the variance is zero.
Example Histogram w/ uniform shape
Two Types of
Skewness
Negatively Skewed/Skewed Left
A distribution is negatively
skewed when the tail of the left
side of the distribution is longer
or fatter than the tail on the right
side. The mean and median will be
less than the mode.
Positively Skewed/Skewed Right
A distribution is positively skewed
when the tail on the right side of
the distribution is longer or fatter.
The mean and median will be
greater than the mode.
Karl Pearson’s
Measure of
Skewness
Karl Pearson’s Measure of Skewness
Notice that the mean, median, and
mode are not equal in a skewed
distribution. Karl Pearson's
measure of skewness is based
upon the divergence of mean
from mode in a skewed
distribution.
Karl Pearson’s Measure of Skewness
𝑆𝑘 is strategically dependent upon Hence Karl Pearson's coefficient of skewness
is defined in terms of median as
mode. If mode is not defined for a
distribution we cannot find 𝑆𝑘 .
.But empirical relation between
mean, median and mode states
that, for a moderately symmetrical
distribution, we have
Mean−Mode ≈ 3 (Mean−Median)
Karl Pearson’s Measure of Skewness
In MS Excel Formula:
For Sample:
“=skew()”
For Population:
“=skew.p()”
General Rule for Skewness
𝐻𝑖𝑔ℎ𝑙𝑦 𝑆𝑘𝑒𝑤𝑒𝑑
𝑆𝑘 < -1 Highly Negatively Skewed 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐
𝑆𝑘 < 1 Highly Positively Skewed
−0.5 < 𝑆𝑘 < 0.5
𝑀𝑜𝑑𝑒𝑟𝑎𝑡𝑒𝑙𝑦 𝑆𝑘𝑒𝑤𝑒𝑑
−1 < 𝑆𝑘 < -0.5 Moderately Negatively Skewed
0.5 < 𝑆𝑘 < 1 Moderately Positively Skewed
Kurtosis
Kurtosis
It is the measure of outliers present in the distribution.
The outliers in a sample, therefore, have even more effect on
the kurtosis than they do on the skewness.
Higher kurtosis means more of the variance is the result of
infrequent extreme deviations, as opposed to frequent
modestly sized deviations. In other words, it’s the tails that
mostly account for kurtosis, not the central peak. The kurtosis
decreases as the tails become lighter. It increases as the tails
become heavier.
Types of Kurtosis
Types of Kurtosis
Mesokurtic
Leptokurtic
Platykurtic
Mesokurtic
This distribution has
kurtosis statistic similar to
that of the normal
distribution. It has a
Kurtosis=3.
Leptokurtic
Peak is higher and
sharper than normal
distribution, which means that
data are heavy-tailed or
profusion of outliers. It has a
Kurtosis>3.
Platykurtic
Compared to a normal
distribution, its tails are shorter
and thinner, and often its
central peak is lower and
broader. It has a Kurtosis < 3.
Fun Fact
Lepto means “thin” or “slender” in
Greek. In leptokurtosis, the kurtosis
value is high.
Platy means “broad” or “flat”—as in
duck-billed platypus. In platykurtosis,
the kurtosis value is low.
Meso means “middle” or “between.” The
normal distribution is mesokurtic.
Percentile Coefficient of Kurtosis
A measure of kurtosis based on quartiles and percentiles is
Coefficient of Kurtosis
In MS Excel Formula:
“=kurt()”
K>1 Leptokurtic
K < -1 Platykurtic
K=0 Mesokurtic
Descriptive Statistics
using Data Analysis
