Moments describe aspects of a distribution and help determine its nature. The first moment is the mean, describing the center. The second moment is the variance, describing the spread. Higher moments describe skew and peak. Moments can be calculated about the mean or origin for ungrouped or grouped data using formulas. Moment ratios like variance divided by mean squared indicate skew. A distribution is symmetrical if equidistant values have equal frequency, and skewed if one tail is longer, measured by the moment ratio β1. Positive skew means the right tail is longer, negative means the left tail is longer.

1. MOMENTS, MOMENT RATIOAND
SKEWNESS
PRESENTED BY: AYESHA KABEER
UNIVERSITY OF GUJRAT SIALKOT SUB CAMPUS

2. Moments
In statistics moments are certain constant values in a given distribution which
help us to ascertain the nature and form of distribution.
The first moment is called the mean which describes the center of the
distribution.
The second moment is the variance which describes the spread of the
observations around the center.
Other moments describe other aspects of a distribution such as how the
distribution is skewed from its mean or peaked.
A moment designates the power to which deviations are raised before
averaging them. 2

4. Central (or Mean) Moments
In mean moments, the deviations are taken from the mean.
For Ungrouped Data:
In General,
4
 
r Population Moment about Mean=
r
i
th
r
x
N





 
r Sample Moment about Mean=
r
i
th
r
x x
m
n




5. Central (or Mean) Moments
Formula for Grouped Data:
5
 
 
r Population Moment about Mean=
r Sample Moment about Mean=
r
i
th
r
r
i
th
r
f x
f
f x x
m
f











6. Moments about (arbitrary) Origin
•
6

7. Moments about zero
•
7

8. Moment Ratios
•
2
3 4
1 2
3 2
2 2
,
 
 
 
 
2
3 4
1 2
3 2
2 2
,
m m
b b
m m
 

9. Skewness
A distribution in which the values equidistant from the mean have equal
frequencies and is called Symmetric Distribution.
Any departure from symmetry is called skewness.
In a perfectly symmetric distribution, Mean=Median=Mode and the two
tails of the distribution are equal in length from the mean. These values are
pulled apart when the distribution departs from symmetry and consequently
one tail become longer than the other.
If right tail is longer than the left tail then the distribution is said to have
positive skewness. In this case, Mean>Median>Mode
If left tail is longer than the right tail then the distribution is said to have
negative skewness. In this case, Mean<Median<Mode

10. Skewness
Skewness is usually measured by the moment ratio β1
2
3 4
1 2
3 2
2 2
,
 
 
 
 
Cases:
If β1=0 then distribution is symmetrical
If β1<0 then distribution is negatively skewed
If β1>0 then distribution is positively skewed