SYMMETRIC & ASSYMETRIC
DISTRIBUTION
GE Mathematics Faculty
OBJECTIVES
1. Differentiate symmetric from asymmetric
distribution.
2. Illustrate and explain skewness.
3. Differentiate
positively
to
negatively
skewed.
4. Explain the coefficient of kurtosis.
Skewness
Symmetric Distribution
Property of a distribution that has the mean as the
center, acting as a mirror image of the two sides of the
distribution
Most of the data values are found near the mean,
tapering off on both sides of the mean.
The mean is equal to the median.
Asymmetric Distribution
Lack of symmetry
• Can be right-skewed distribution or left-skewed distribution
•
• Right-skewed
distribution
• Left-skewed distribution
Examples:
Right-skewed distribution
The scores of students who did not study for a major
examination
The household income of Filipinos in a certain area in the
National Capital Region
Left-skewed distribution
The number of people buying Christmas presents during
December.
The hours that children spend playing with their gadgets
 Skewness is a measure or a
criterion on how asymmetric the
distribution of data is from the
mean.
Pearson Coefficient of
skewness
• is a method developed by Karl
Pearson to find skewness in a
sample using descriptive statistics
like the mean and mode.
Pearson Coefficient of
skewness
Where:
is the sample mean
Md – is the median
s – standard deviation
Sk = 0, symmetrical
Sk = positive; positively
skewed
Sk = negative;negatively
skewed
Symmetrical Distribution
• Symmetrical distribution and mode
occur occurs when the values of
variables occur at regular frequencies
and the mean, median at the same
point.
Positively skewed
a positively
skewed (or
right-skewed)
distribution is a type of distribution in which
most values are clustered around the left tail
of the distribution while the right tail of the
distribution is longer.
Negatively skewed
• is a type of distribution in which more
values are concentrated on the right
side
while
(tail)
of
the
the distribution graph
left
tail
the distribution graph is longer.
of
Example 1
• For 108 randomly selected high school
students, the following
distribution
were
IQ frequency
obtained.
Find
the
coefficient of skewness of the distribution.
Class Limits
Frequency
90 – 98
6
99 – 107
22
108 – 116
43
117 – 125
28
126 – 134
9
Example1
Class Limits
f
x
fx
cf
90 – 98
6
94
564
6
99 – 107
22
103
2,266
28
108 – 116
43
112
4,816
71
117 – 125
28
121
3,388
99
126 – 134
9
130
1,170
108
𝟑 ( 𝑿 − 𝑴𝒅 )
𝒔𝒌=
𝒔
Σ 𝑓𝑥
𝑥=
𝑛
12 ,204
𝑥=
108
𝑥=113
(
𝑛
− 𝑐𝑓
2
𝑀𝑑=
𝑓
(
)
𝑤 +𝑙𝑚𝑑
)
54 − 28
𝑀𝑑=
9+ 107.5
43
𝑀𝑑=112.94
Example 1
Class
Limits
90 – 98
f
x
6
94
-19
361
2,166
99 – 107
22
103
-10
100
2,200
108 – 116
43
112
-1
1
43
117 – 125
28
121
8
64
1,792
126 – 134
9
130
17
289
2,601
f
2
𝑠 =
8802
𝑠 =
107
2
2
𝑠 =82.26
∑
𝑓 (𝑥−𝑥)
𝑛 −1
𝑠=
2
√
∑ 𝑓 (𝑥−𝑥)
𝑛 −1
𝑠=√ 82.86
𝑠=9.07
2
3 ( 𝑋 − 𝑀𝑑 )
𝑠𝑘=
𝑠
𝑥=113
𝑀𝑑=112.94
𝑠=9.07
3 ( 113 − 112.94 )
𝑠𝑘=
9.07
𝑠𝑘=0.02
𝑇h𝑒 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛𝑖𝑠𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝑙𝑦 𝑠𝑘𝑒𝑤𝑒𝑑
BREAK-OUT
SESSIONS
Measure of Kurtosis
Measure of Kurtosis
• Kurtosis is a measure of whether the data are
heavy-tailed or light-tailed relative to a normal
distribution.
• That is, data sets with high kurtosis tend to
have heavy tails, or outliers. Data sets with low
kurtosis tend to have light tails, or lack of
outliers.
Measure of Kurtosis
A distribution is said
to be:
• Mesokurtic if k = 3
• Leptokurtic if k > 3
• Platykurtic if k < 3
Leptokurtic
Leptokurtic
excess
indicates
kurtosis.
distribution
shows
a
The
heavy
positive
leptokurtic
tails
on
either side, indicating large outliers.
Leptokurtic
Platykurtic
• A platykurtic distribution shows a negative
excess kurtosis.
• The kurtosis reveals a distribution with flat
tails.
• The flat tails indicate the small outliers in a
distribution.
Measure of Kurtosis
• For ungrouped data
•
• For grouped data
Example 1
• For 108 randomly selected high school
students, the following
distribution
were
IQ frequency
obtained.
Find
the
coefficient of skewness & kurtosis of the
distribution.
Class Limits
Frequency
90 – 98
6
99 – 107
22
108 – 116
43
117 – 125
28
126 – 134
9
Example1
Class Limits
f
x
fx
cf
90 – 98
6
94
564
6
99 – 107
22
103
2,266
28
108 – 116
43
112
4,816
71
117 – 125
28
121
3,388
99
126 – 134
9
130
1,170
108
Solution:
Σ 𝑓𝑥
𝑥=
𝑛
12 ,204
𝑥=
108
𝑥=113
Example 1
Class
Limits
90 – 98
f
x
6
94
-19
361
2,166
99 – 107
22
103
-10
100
2,200
108 – 116
43
112
-1
1
43
117 – 125
28
121
8
64
1,792
126 – 134
9
130
17
289
2,601
f
2
𝑠
∑
=
𝑓 (𝑥−𝑥)
𝑛 −1
8802
𝑠 =
107
2
2
2
𝑠 =82.26
𝑠=
√
∑ 𝑓 (𝑥−𝑥)
𝑛 −1
𝑠=√ 82.86
𝑠=9.07
2
Example 1
Class
Limits
f
x
90 – 98
6
94
-19
130,321
781,926
99 – 107
22
103
-10
10,000
220,000
108 – 116
43
112
-1
1
43
117 – 125
28
121
8
4,096
114,688
126 – 134
9
130
17
83,521
751,689
f
∑
𝐾=
𝑓 (𝑥−𝑥)
𝑛𝑠
4
4
1868346
𝐾=
4
108( 9.07)
𝑲 =𝟐 . 𝟓𝟔
𝑲 <𝟑
The distribution is
platykurtic
Key Differences Between
Skewness and Kurtosis
• This is the fundamental differences between
skewness and kurtosis:
1. The characteristic of a frequency distribution that
ascertains its symmetry about the mean is called
skewness. On the other hand, Kurtosis means the
relative pointedness of the standard bell curve,
defined by the frequency distribution.
• This
is
the
fundamental
differences
between skewness and kurtosis:
2. Skewness is a measure of the degree of
lopsidedness
in
the
frequency
distribution.
Conversely, kurtosis is a measure of degree of
tailedness in the frequency distribution.
• This
is
the
fundamental
differences
between skewness and kurtosis:
3. Skewness is an indicator of lack of symmetry, i.e. both
left and right sides of the curve are unequal, with respect
to the central point. As against this, kurtosis is a measure
of data, that is either peaked or flat, with respect to the
probability distribution.
• This is the fundamental differences
between skewness and kurtosis:
4. Skewness shows how much and in which
direction, the values deviate from the mean? In
contrast, kurtosis explain how tall and sharp the
central peak is.
BREAK-OUT
SESSIONS
Thank
You!