Skewness & Kurtosis
Lecture 15
In the modern world, data is becoming more and more
significant. The study, comprehension, manipulation, and processing of data are
the focus of whole professions. Therefore, it is crucial to understand the
various kinds of data and the characteristics that go along with them.
The normal distribution is the most common type of data and probability
distribution. It is defined by a symmetrical bell-shaped curve. However, the
normal distribution may also become skewed when important reasons are present.
Kurtosis and skewness can be used to compute this distortion. This lesson,
"The Simplified and Complete Guide to Skewness and Kurtosis," will
walk you through some of the various kinds of distortion that can happen to a
normal curve.
It is necessary to understand moments, particularly mean moments, before learning skewness and kurtosis. The skewness and kurtosis measurements are based on the moments about the mean.
Moments are measurements used in statistics to
describe the variability and form of a data set. They serve to explain the
data's location and distribution. Moments of various kinds can be computed, and
each one offers unique insights into the data set. Let's examine a few of these
instances, together with their definitions, uses in statistical analysis, and
formulas.
The arithmetic means of the specified power of
deviation from the mean, provisional mean, or zero are called moments.
There are three types of moments.
1. Mean Moments
These moments are denoted by
When frequencies are given, then moments
can be computed as:
Example 4.14: Compute the first four
mean moments of the following data.
5, 6, 7, 8, 10, 12
Solution:
Example 4.15: compute the first four moments of the following frequency
distribution.
|
Class |
0
- 10 |
10
– 20 |
20- 30 |
30
– 40 |
40
– 50 |
|
f |
2 |
10 |
6 |
4 |
2 |
Solution:
2. Moments about Provisional Mean
The moments about the provisional mean (let
a) are denoted by
When frequencies are given, then raw
moments can be calculated as:
These moments (raw moments) can be
transformed into mean moments by using the following relations:
Example 4.16: Compute
the first four raw moments and then convert into mean moments.
5, 6, 7, 8, 9, 11
Solution: Let a = 8
Example 4.17: Compute the first four raw
moments and then convert into mean moments.
|
X |
1 |
3 |
5 |
7 |
8 |
|
|
8 |
5 |
4 |
3 |
1 |
Solution: Let a = 5
3. Moments about Origin
The moments about the provisional mean (let
a) are denoted by
The first moment about origin is equal to the arithmetic mean.
When frequencies are given, then raw moments can be calculated as:
- Read: Measures of Skewness
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