Chapter 3 Summary Measures_Complete-1.pdf

CHAPTER 3:
SUMMARY
MEASURES
Mean
Median
Mode
Range
Variance
Standard Deviation
Coefficient of Variation
Measure of Skewness
Measure of Kurtosis

CHAPTER 3: SUMMARY MEASURES
INTRODUCTION
Although a frequency distribution table is certainly useful in providing a general idea
about how data are distributed between two extreme values, it is usually desirable to
further summarize the important characteristics of the univariate data sets.
These summary measures falls under four categories: measure of central tendency,
measures of variability, measures of position, measures of skewness and kurtosis.
Summary measures are useful for exploratory analysis of a data set as well as for the
reporting of final results of a study.

I. MEASURES OF CENTRAL TENDENCY
A single value about which the set of
observations tend to cluster. The
measures of position provide precise,
objectively determined value that can
easily be manipulated, interpreted
and compared with one another than
do the general impression conveyed
by tabular and graphical summaries.
Mean
Median
Mode
Mean

MEASURES OF CENTRAL TENDENCY
A. The Arithmetic Mean
The arithmetic mean or simply the mean, denoted by 𝜇, is defined as the sum of all
observations divided by the total number of observations. In symbols, if we let 𝑿𝒊 be
the value of the ith observation and N the number of observations, then the mean is
computed as
𝜇 =
σ𝑖=1
𝑁
𝑋𝑖
𝑁
Where: i = 1,2,…N
𝑋𝑖 = 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 𝑚𝑎𝑑𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑖𝑡ℎ 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙
Ungrouped data

The Arithmetic Mean
Example: Consider the scores of 10 college students in their midterm exam as
ungrouped data: 89, 90, 85, 80, 95, 88, 86, 90, 87, 90, then the mean
(𝜇) is computed as follows,
𝜇 =
σ𝑖=1
𝑁
𝑋𝑖
𝑁
=
89 + 90 + ⋯ + 90
10
= 88
** 88 is the mean score of the 10 college students

The Arithmetic Mean
Computation for Grouped Data
The arithmetic mean of the grouped data (frequency distribution) is
computed using the formula,
𝜇 =
σ𝑖=1
𝑘
𝑓𝑖𝑋𝑖
σ𝑖=1
𝑘
𝑓𝑖
Where: i = 1, 2, …, k
k = number of categories or classes
𝑋𝑖 = class midpoint of the ith category
𝑓𝑖 = frequency of the ith category

The Arithmetic Mean
Example. Consider the data on the midterm scores of 30 students, then the mean is
computed as follows
CI 𝒇𝒊 CM (𝑿𝒊) 𝒇𝒊𝑿𝒊
40-50 4 45 180
51-61 3 56 168
62-72 6 67 402
73-83 8 78 624
84-94 7 89 623
95-105 2 100 200
Total 30 2197
𝜇𝐺 =
4 45 + 3 56 + 6 67 + 8 78 + 7 89 + 2(100)
30
=
2197
30
= 73.23 (average midterm score of 30 students)

Properties of the Mean
1. The sum of the deviations of the observations from the mean is always equal to
zero, i.e.
෍
𝑖=1
𝑁
(𝑋𝑖 − 𝜇) = 0
2. The sum of the squared deviations of the observations from the mean is the
smallest, i.e.
෍
𝑖=1
𝑁
(𝑋𝑖 − 𝜇)2 = 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡
3. The mean reflect the magnitude of every observation, since every observation
contributes to the value of the mean

Properties of the Mean
4. It is easily affected by the presence of extreme values, and hence not a good
measure of central tendency when extreme observations do occur
5. Means of subgroups may be combined when properly weighted. Combined mean is
called the weighted arithmetic mean

The Median
The median, denoted by 𝑀𝑑, is a single value which divides an array of
observations into two equal parts such as that fifty percent of the observation fall
below it and fifty percent fall above it. An array is usually set a set of observations
arranged in an increasing magnitude. T he median is computed using the formula for
both the ungrouped and grouped data:
For ungrouped data
𝑀𝑑 = 𝑋 𝑁
2 +1
𝑖𝑓 𝑁 𝑖𝑠 𝑶𝑫𝑫 𝑀𝑑 =
1
2
(𝑋𝑁
2
+ 𝑋 𝑁
2 +1
𝑤ℎ𝑒𝑛 𝑁 𝑖𝑠 𝑬𝑽𝑬𝑵

The Median
Example: Given the set of observations arranged in an increasing magnitude: 80, 85,
86, 87, 88, 89, 90, 90, 90, 95, then the median is computed as:
𝑀𝑑 =
1
2
𝑋𝑁
2
+ 𝑋 𝑁
2 +1
=
1
2
𝑋10
2
+ 𝑋10
2 +1
=
1
2
(𝑋5 + 𝑋6)
=
1
2
(88 + 89)
= 88.5

The Median
For the grouped data, the mean is computed using the formula:
𝑀𝑑 = 𝐿𝑚𝑑 +
𝑁
2
− 𝐹𝑏)
𝑓𝑚𝑑
𝑐
Where:
Lmd = lower TCB of the median class, where the median class is a class
whose < CF is greater or equal to ½ of N
C = class size
N = total number of observations
Fb = less than cumulative frequency of the class immediately preceding the
median class
fmd = frequency of the median class

The Median
Where:
Lmd = lower TCB of the median class, where the median class
is a class whose < CF is greater or equal to ½ of N
C = class size
N = total number of observations
Fb = less than cumulative frequency of the class immediately
preceding the median class
fmd = frequency of the median class
CI 𝒇𝒊 TCB CM <CF
40-50 4 39.5-50.5 45 4
51-61 3 50.5-61.5 56 7
62-72 6 61.5-72.5 67 13
73-83 8 72.5 -83.5 78 21
84-94 7 83.5-94.5 89 28
95-105 2 94.5-105.5 100 30
Total 30
Median class

The Median
CI 𝒇𝒊 TCB CM <CF
40-50 4 39.5-50.5 45 4
51-61 3 50.5-61.5 56 7
62-72 6 61.5-72.5 67 13
73-83 8 72.5 -83.5 78 21
84-94 7 83.5-94.5 89 28
95-105 2 94.5-105.5 100 30
Total 30
Lmd
N
fmd
Fb
𝑀𝑑 = 𝐿𝑚𝑑 +
𝑁
2
− 𝐹𝑏)
𝑓𝑚𝑑
𝑐 = 72.5 +
30
2
− 13
8
11
= 75.25
Thus, half of the 30 students got scores lower than 75.25
and the other half got scores higher than 75.25 or the
median score of the 30 students is 75.25

Properties of Median
1. It is a propositional value and hence is not affected by the presence of extreme values unlike the
mean
2. The sum of the absolute deviations from a point, say, a, is smallest when a is equal to the median, i.e.
෍
𝑖=1
𝑁
𝑋𝑖 − 𝑀𝑑 𝑖𝑠 𝑚𝑖𝑛𝑖𝑚𝑢𝑚
For grouped data
෍
𝑖=1
𝑘
𝑓𝑖 𝑋𝑖 − 𝑀𝑑𝐺 𝑖𝑠 𝑚𝑖𝑛𝑖𝑚𝑢𝑚
3. The median is not suitable to further computations and hence medians of subgroups cannot be
combined in the same manner as the mean
4. The median of grouped data can be calculated even with open-ended intervals provided the median
class is not open-ended

The Mode
The mode, denoted by Mo, is the value which occurs the most frequently in the given
data set .
Example 1: Given the set of observations arranged in an increasing magnitude: 80,
85, 86, 87, 88, 89, 90, 90, 90, 95, then the median is computed as:
𝑀𝑜 = 90
Example 2: The values 50, 55, 70, 76, 79 has no mode because there is no value that
occurred more than once
𝑀𝑜 = 𝑑𝑜 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡

The Mode
Computation for grouped data
The mode of the grouped data is
computed as:
𝑀𝑜 = 𝐿𝑚𝑜 + 𝑐 (
𝑓𝑚𝑜 − 𝑓𝑏
2𝑓𝑚𝑜 − 𝑓𝑏 − 𝑓𝑎
)
Where:
𝐿𝑚𝑜 = lower TCB of the modal class, where the modal
class is a class with the highest frequency
C = class size
𝑓𝑚𝑜 = frequency of the modal class
𝑓𝑏 = frequency of the class immediately preceding the
modal class
𝑓𝑎 = frequency of the class immediately following the
modal class

The Mode
Where:
𝐿𝑚𝑜 = lower TCB of the modal class, where the modal
class is a class with the highest frequency
C = class size
𝑓𝑚𝑜 = frequency of the modal class
𝑓𝑏 = frequency of the class immediately preceding the
modal class
𝑓𝑎 = frequency of the class immediately following the
modal class
CI 𝒇𝒊 TCB
40-50 4 39.5-50.5
51-61 3 50.5-61.5
62-72 6 61.5-72.5
73-83 8 72.5 -83.5
84-94 7 83.5-94.5
95-105 2 94.5-105.5
Total 30
Modal class

The Mode
CI 𝒇𝒊 TCB
40-50 4 39.5-50.5
51-61 3 50.5-61.5
62-72 6 61.5-72.5
73-83 8 72.5 -83.5
84-94 7 83.5-94.5
95-105 2 94.5-105.5
Total 30
𝐿𝑚𝑜
𝑓𝑏
𝑓𝑎
𝐿𝑚𝑜
𝑀𝑜 = 𝐿𝑚𝑜 + 𝑐
𝑓𝑚𝑜 − 𝑓𝑏
2𝑓𝑚𝑜 − 𝑓𝑏 − 𝑓𝑎
= 72.5 + 11
8 − 6
2 8 − 6 − 7
= 79.83
Thus, most of the 30 students got modal score close to 79.83

Properties of the Mode
1. The mode is determined by the frequency and not by the values of the observations
2. It cannot be manipulated algebraically and hence modes of subgroups cannot be
combined
3. The mode can be defined with qualitative or quantitative variables
4. The mode is very much affected by the method of grouping data
5. It can be computed with open-ended intervals provided the modal class is not open-
ended
Properties of the Mode

II. Measures of Dispersion or Variability
The measures of dispersion or
variability simply describe the
spread or variability of the
observations in the data set.
Standard Deviation
Coefficient of variation
Variance
Range

Measures of Dispersion or variability
Range
The range is the difference
between the highest value and
the lowest value. In the grouped
data set, the range is the
difference between the upper
limit of the highest class interval
and the lower limit of the
lowest class interval
In symbol:
For ungrouped data
𝑅 = 𝐻𝑉 − 𝐿𝑉
Example:
Consider the weights of 5 female students:
49, 55, 57, 50, 60. Find the range.
𝑅 = 60 − 49 = 11

Range
In symbol:
For grouped data
𝑅𝐺 = 𝑈𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝐶𝐼 − 𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑜𝑤𝑒𝑠𝑡 𝐶𝐼
Example: CI 𝒇𝒊
40-50 4
51-61 3
62-72 6
73-83 8
84-94 7
95-105 2
Total 30
𝑅𝐺 = 105 − 40 = 65

Properties of Range
1.It is quick but rough measure of dispersion
2.The larger the value of the range, the more dispersed are the
observations
3.It considers only the lowest and the highest values in the
population

The Variance
The variance, denoted by 𝜎2
, is the mean of the squared deviation of the observations
from their arithmetic mean. In symbol, the variance is given by:
𝜎2 =
σ𝑖=1
𝑁
(𝑋𝑖 − 𝜇)2
𝑁
(by definition)
𝜎2 =
σ𝑖=1
𝑁
𝑋𝑖
2
−
(σ𝑖=1
𝑁 𝑋𝑖)
2
𝑁
𝑁
(machine formula)

The Variance
Example: Consider the weights of 5 female students: 49, 55, 57, 50, 60. Find the variance.
𝜇 =
49 + 55 + 57 + 50 + 60
5
= 54.2
𝜎2 =
σ𝑖=1
𝑁
(𝑋𝑖 − 𝜇)2
𝑁
=
(49−54.2)2+(55−54.2)2+ ⋯+(60−54.2)2
5
=
86.8
5
= 17.36

The Variance
Computation for grouped data
𝜎2 =
σ𝑖=1
𝑁
𝑓𝑖(𝑋𝑖 − 𝜇)2
σ𝑖=1
𝑘
𝑓𝑖

The Variance
Example: Find the variance of the grouped midterm scores of the 30 students
CI 𝒇𝒊 CM (Xj) (𝑿𝒋 − 𝝁𝒈) (𝑿𝒋 − 𝝁𝒈)𝟐
𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟐
40-50 4 45 -28.23 796.9329 3187.7316
51-61 3 56 -17.23 296.8729 890.6187
62-72 6 67 -6.23 38.8129 232.8774
73-83 8 78 4.77 22.7529 182.0232
84-94 7 89 15.77 248.6929 1740.8503
95-105 2 100 26.77 716.6329 1433.2658
Total 30 7667.3670
𝜎2 =
σ𝑖=1
𝑁
σ𝑖=1
𝑘
𝑓𝑖
=
7667.3670
30
= 255.58

Properties of the Variance
1. The variance is always non-negative
2. The variance is easy to manipulate for further mathematical treatment
3. The variance makes use of all observations
4. The unit of measurement for the variance is the square of the unit of measure of
the given set of values. Thus, for example, if a data set has an inch as the unit of
measure, the unit of its variance will be squared inches.

Standard Deviation
The standard deviation, denoted by 𝜎, is the positive square root of the variance. In
symbol,
𝜎 = 𝜎2 =
σ𝑖=1
𝑁
(𝑋𝑖 − 𝜇)2
𝑁
for ungrouped data
Example: Consider the weights of 5 female students: 49, 55, 57, 50, 60. Find the
standard deviation
𝜎 = 𝜎2 = 17.36 = 4.17

Standard Deviation
Example: Find the standard deviation of the grouped midterm scores of the 30 students
40-50 4 45 -28.23 796.9329 3187.7316
51-61 3 56 -17.23 296.8729 890.6187
62-72 6 67 -6.23 38.8129 232.8774
73-83 8 78 4.77 22.7529 182.0232
84-94 7 89 15.77 248.6929 1740.8503
95-105 2 100 26.77 716.6329 1433.2658
Total 30 7667.3670
𝜎 =
σ𝑖=1
𝑁
σ𝑖=1
𝑘
𝑓𝑖
= 255.58 = 15.99

Standard Deviation
The coefficient of variation, denoted by CV, is the ratio of the standard deviation and the
mean, which is expressed in percent. In symbol,
𝐶𝑉 =
𝜎
𝜇
𝑋100%
Example: Consider the weights of 5 female students: 49, 55, 57, 50, 60. Find the standard
deviation
𝐶𝑉 =
𝜎
𝜇
𝑋100% =
4.17
52.4
100% = 7.96%

Properties of Standard Deviation
1. It is a quantity without units
2. It can be used to compare the dispersion of two or more sets of data
measured in the or different units

Measure of Skewness and Kurtosis
Measure of Skewness
Describes the extent to which the items are symmetrically distributed.
Skewness is the degree of its departure from symmetry which depends on the
comparative positions of the mean, median and mode with differing degrees
and directions of asymmetry.

Measure of Skewness
Symmetric Distribution
A symmetric distribution is a distribution with one peak or unimodal
with mean, median and mode coincide.
Fig. 1. Symmetric Distribution
𝜇 = 𝑀𝑑 = 𝑀𝑜

Measure of Skewness
Asymmetrical Distribution
Positively Skewed Distribution
When the distribution is skewed to the right or it has a long tail
extending off to the right (indicating the presence of small proportion of
relatively large extreme values) but short tail extending to the left, then the
distribution is known to be positively skewed distribution.

Measure of Skewness
𝑀𝑜 < 𝑀𝑑 < 𝜇
Fig. 2. Positively Skewed Distribution

Measure of Skewness
Negatively Skewed Distribution
When the distribution is skewed to the left or it has a long tail extending
off to the left but short tail extending to the right, then the distribution is
known to be negatively skewed distribution.

Measure of Skewness
𝜇 < 𝑀𝑑 < 𝑀𝑜
Fig. 3. Negatively Skewed Distribution

Measure of Skewness
Example: Recall the computed mean, median, mode of the FDT as shown
below from the previous discussion
𝜇= 73.23
𝑀𝑑 = 75.25
𝑀𝑜 = 79.83
CI 𝒇𝒊
40-50 4
51-61 3
62-72 6
73-83 8
84-94 7
95-105 2
Total 30
Thus, the distribution of the
midterm scores of the 30
students is considered as
negatively squared

Measure of Skewness
Coefficient of Skewness
The coefficient of skewness, denoted by 𝑆𝑘 , for a data set is defined,
𝑆𝑘 =
𝑚3
𝑚2 𝑚2
Where:
For ungrouped data
𝑚3 =
σ(𝑋𝑖−𝜇)3
𝑁
, the third moment about the mean of a set values, X1, X2,….Xn

Measure of Skewness
𝑆𝑘 =
𝑚3
𝑚2 𝑚2
Where:
For grouped data
𝑚3 =
σ 𝑓𝑖(𝑋𝑖−𝜇)3
𝑁
, the third moment about the mean of a set values, X1, X2,….Xn
fi = frequency of the ith category
Xi = class mark of the ith category

Measure of Skewness
𝑆𝑘 =
𝑚3
𝑚2 𝑚2
Where:
Ungrouped data
𝑚2 =
𝑁
, the second moment about the mean of a set values, X1, X2,….Xn
Grouped data
𝑚2 =
𝑁

Measure of Skewness
A rough estimate of the coefficient of skewness is computed as:
𝑆𝑘 =
3 (𝑚𝑒𝑎𝑛 − 𝑚𝑒𝑑𝑖𝑎𝑛)
𝑚2

Measure of Skewness
Example: Let us consider the FDT of the midterm scores of the 30 students
CI 𝒇𝒊 CM (Xj) (𝑿𝒋 − 𝝁𝒈) (𝑿𝒋 − 𝝁𝒈)𝟐 𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟐 𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟑
40-50 4 45 -28.23 796.9329 3187.7316 -89989.7
51-61 3 56 -17.23 296.8729 890.6187 -15345.4
62-72 6 67 -6.23 38.8129 232.8774 -1450.83
73-83 8 78 4.77 22.7529 182.0232 868.2507
84-94 7 89 15.77 248.6929 1740.8503 27453.21
95-105 2 100 26.77 716.6329 1433.2658 38368.53
Total 30 7667.3670 -40095.9

𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟑
40-50 4 45 -28.23 796.9329 3187.7316 -89989.7
51-61 3 56 -17.23 296.8729 890.6187 -15345.4
62-72 6 67 -6.23 38.8129 232.8774 -1450.83
73-83 8 78 4.77 22.7529 182.0232 868.2507
84-94 7 89 15.77 248.6929 1740.8503 27453.21
95-105 2 100 26.77 716.6329 1433.2658 38368.53
Total 30 7667.3670 -40095.9
𝑚3 =
σ 𝑓𝑖(𝑋𝑖 − 𝜇)3
𝑁
=
−40095.9
30
= −1336.53
𝑚2 =
𝑁
=
7667.367
30
= 255.5789
𝑆𝑘 =
𝑚3
𝑚2 𝑚2
=
−1336.53
255.5789 255.5789
= −0.3277 , 𝑏𝑦 𝑑𝑒𝑓𝑖𝑛𝑡𝑖𝑜𝑛

40-50 4 45 -28.23 796.9329 3187.7316 -89989.7
51-61 3 56 -17.23 296.8729 890.6187 -15345.4
62-72 6 67 -6.23 38.8129 232.8774 -1450.83
73-83 8 78 4.77 22.7529 182.0232 868.2507
84-94 7 89 15.77 248.6929 1740.8503 27453.21
95-105 2 100 26.77 716.6329 1433.2658 38368.53
Total 30 7667.3670 -40095.9
𝑆𝑘 =
𝑚3
𝑚2 𝑚2
=
−1336.53
255.5789 255.5789
𝑆𝑘 =
𝑚2
=
3(73.23 − 75.25)
255.5789
= −0.37906
𝜇= 73.23
𝑀𝑑 = 75.25
𝑀𝑜 = 79.83

40-50 4 45 -28.23 796.9329 3187.7316 -89989.7
51-61 3 56 -17.23 296.8729 890.6187 -15345.4
62-72 6 67 -6.23 38.8129 232.8774 -1450.83
73-83 8 78 4.77 22.7529 182.0232 868.2507
84-94 7 89 15.77 248.6929 1740.8503 27453.21
95-105 2 100 26.77 716.6329 1433.2658 38368.53
Total 30 7667.3670 -40095.9
𝑆𝑘 =
𝑚3
𝑚2 𝑚2
=
−1336.53
255.5789 255.5789
𝑆𝑘 =
𝑚2
=
3(73.23 − 75.25)
255.5789
= −0.37906
Since the computed coefficient is
negative, hence, negative skewness in
confirmed

Measure of Kurtosis
Flatness or peakedness of any distribution in relation to another distribution.
Three types of kurtosis may be observed:
Leptokurtic : means that the data are very similar (homogenous)
Platykurtic: means that data are very different (heterogenous)
If one distribution is more peaked than the other, it is leptokurtic (K>3). If less
peaked, it is said to be platykurtic (K<3)
Mesokurtic: falls between leptokurtic and platykurtic

Measure of Kurtosis

Measure of Skewness
Coefficient of Kurtosis
𝑆𝑘 =
𝑚4
(𝑚2)(𝑚2)
Where:
For ungrouped data
𝑚4 =
𝑁
, the fourth moment about the mean of a set values, X1, X2,….Xn

Measure of Skewness
𝑆𝑘 =
𝑚4
(𝑚2)(𝑚2)
Where:
For grouped data
𝑚4 =
𝑁
, the fourth moment about the mean of a set values, X1, X2,….Xn
fi = frequency of the ith category
Xi = class mark of the ith category

Measure of Skewness
𝑆𝑘 =
𝑚4
(𝑚2)(𝑚2)
Where:
𝑚2 =
𝑁

Measure of Skewness
𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟑 𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟒
40-50 4 45 -28.23 796.9329 3187.7316 -89989.7 2540408.188
51-61 3 56 -17.23 296.8729 890.6187 -15345.4 264400.5563
62-72 6 67 -6.23 38.8129 232.8774 -1450.83 9038.6472
73-83 8 78 4.77 22.7529 182.0232 868.2507 4141.5557
84-94 7 89 15.77 248.6929 1740.8503 27453.21 432937.1096
95-105 2 100 26.77 716.6329 1433.2658 38368.53 1027125.4270
Total 30 7667.3670 -40095.9 4278051.4840

CI 𝒇𝒊 CM (Xj) (𝑿𝒋 − 𝝁𝒈) (𝑿𝒋 − 𝝁𝒈)𝟐 𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟐 𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟑 𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟒
40-50 4 45 -28.23 796.9329 3187.7316 -89989.7 2540408.188
51-61 3 56 -17.23 296.8729 890.6187 -15345.4 264400.5563
62-72 6 67 -6.23 38.8129 232.8774 -1450.83 9038.6472
73-83 8 78 4.77 22.7529 182.0232 868.2507 4141.5557
84-94 7 89 15.77 248.6929 1740.8503 27453.21 432937.1096
95-105 2 100 26.77 716.6329 1433.2658 38368.53 1027125.4270
Total 30 7667.3670 -40095.9 4278051.4840
𝑚4 =
𝑁
=
42478051.4840
30
= 142601.7161
𝑚2 =
𝑁
=
7667.367
30
= 255.5789
𝑆𝑘 =
𝑚4
(𝑚2)(𝑚2)
=
142601.7161
(255.5789) (255.5789)

CI 𝒇𝒊 CM (Xj) (𝑿𝒋 − 𝝁𝒈) (𝑿𝒋 − 𝝁𝒈)𝟐 𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟐 𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟑 𝒇𝒊(𝑿𝒋 − 𝝁𝒈)𝟒
40-50 4 45 -28.23 796.9329 3187.7316 -89989.7 2540408.188
51-61 3 56 -17.23 296.8729 890.6187 -15345.4 264400.5563
62-72 6 67 -6.23 38.8129 232.8774 -1450.83 9038.6472
73-83 8 78 4.77 22.7529 182.0232 868.2507 4141.5557
84-94 7 89 15.77 248.6929 1740.8503 27453.21 432937.1096
95-105 2 100 26.77 716.6329 1433.2658 38368.53 1027125.4270
Total 30 7667.3670 -40095.9 4278051.4840
𝑆𝑘 =
𝑚4
(𝑚2)(𝑚2)
=
142601.7161
(255.5789) (255.5789)
= 2.18
Since, the computed value is less than 3, then it is a platykurtic

Chapter 3 Summary Measures_Complete-1.pdf

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Chapter 3 Summary Measures_Complete-1.pdf