THE MEASURE OF DATA
DISPERSION
UKURAN
Measure of data dispersion is
a measure which states how big the different data value or
varied with central measurement value or how big is the mean
deviation with the central value.
1. Range
Range is difference between the biggest and the smallest data.
Range can be determined by the formula:
R = X max – X min
Example :
Determine the range of data : 10,6,8,2,4
Answer :
R = Xmax – Xmin = 10 – 2 = 8
Hal.: 2
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2. Mean Deviation
Mean deviation of a number set is:
Absolute mean deviation-Deviation
a. Single Data
SR =
 xx
n
Example :
The math score of 6 students are :7,5,6,3,8,7.
Find the mean deviation!
Hal.: 3
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Answer:
x
=
7 5 638 7
6
= 6
SR =
Hal.: 4
7 6  56  66  36  86  7 6
6
=
8
6
=
1,33
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b. Weight Data / Grouped data
SR =
f x x
f
x = data the-ith (data berbobot )
= mid-point of the- i th interval class (grouped data)
f = frequency
Hal.: 5
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Example :
Determine the mean deviation of this data:
Hal.: 6
Data
Frequency
x
3–5
2
4
6–8
4
7
9 – 11
8
10
12 - 14
6
13
Total
20
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Answer :
F.x
xx
F xx
4
8
5,7
11,4
4
7
8
10
2,7
0,3
10,8
9 – 11
28
80
12 - 14
6
13
78
3,3
Total
20
Data
Frequency
x
3–5
2
6–8
x
=
194
 f .x
 f
SR =
194 = 9,7
=
20
Hal.: 7
2,4
19,8
44,4
 f xx
f
44,4
=
20
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= 2,22
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3.Standard Deviation
Standard Deviation (S) of a set number is root of number square
deviation of the numbers that is divided by the total number or root
of average of square deviation.
a. Single Data
 x  x 
2
S =
S =
Hal.: 8
i
or
n
x
 x 


n
n


2
2
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Example :
Find the standard deviation of this data :
2,3,5,8,7.
x
Answer :
x=
2358 7
5
=5
S =
2


x

x
 i
n
=
=
Hal.: 9
26
5
x  x  x  x 
2
2
-3
9
3
-2
4
5
0
0
8
3
9
7
2
4
26
5,2
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b. Weight Data / Grouped
S =
 f x  x 
f
S =
 fx
f
2
Hal.: 10
2
or
  f.x 


  f 
2
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Example:
Find the standard deviation of this data
Hal.: 11
Data
Frequency
x
3–5
2
4
6–8
4
7
9 – 11
8
10
12 - 14
6
13
Total
20
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Answer :
Data
Freq
x
x2
f.x
f.x2
3–5
2
4
16
8
32
6–8
4
7
49
28
196
9 – 11
8
10
100
80
800
12 - 14
6
13
169
78
1014
Total
20
194
2042
S =
 fx 2    f.x 
f  f 
=
2042 194 
 
20  20 
Hal.: 12
2
2
=
8,01
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4. Quartile
Quartile is the value that divided the ordered data into four parts which
has same size after the data is ordered.
By the number line we can show quartile place, as follows:
Q1
Q2
Q3
Determining quartile value
a. Single Data
Place of Qi = data the- i ( n  1)
4
with i = 1, 2, 3
Hal.: 13
and n = number of data
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Example :
The result of ages data collection of 12 babies (in a year),
known as follow: 4, 3, 4, 4, 2, 1, 1, 2,1, 3, 3, 4 ,
Find :
a. lower quartile (Q1)
b. Median quartile (Q2)
c. upper quartile (Q3)
Answer :
The ordered data : 1,1,1,2,2,3,3,3,4,4,4,4
a. Place of Q1 = the – data
1(12  1)
4
= the- 3 ¼ data
Hal.: 14
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Value Q1 = the-3rd data + ¼ (the 4th data – the 3rd data)
= 1 + ¼ (2 – 1) = 1¼
b. Place of Q2 = the- data
2(12  1)
4
= the-6½ data
Value Q2 = the-6th data + ½ (the-7th data – the-6th data)
= 3 + ½ (3 – 3) = 3
Hal.: 15
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c. Place of Q3 = the- data
3(12  1)
4
= the-9th ¾
Value Q3 = the- 9th data + ¾ (the-10th data – the-9th data)
= 4 + ¾ (4 – 4) = 4
Hal.: 16
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Range of inter quartile sprout / Quartile deviation (Qd)
Defined as follow:
Qd = ½ (Q3 – Q1)
b. Grouped Data
 i.n


F


Value Qi = b + p  4

f




and i = 1,2,3
b = below side of Qi class
p = length of class
F = cumulative frequency prior to Qi class
f = frequency of Qi class
n = size of data
Hal.: 17
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Example :
Find the quartile deviation of this data :
Score
45-49
50-54
55-59
60-64
65-69
70-74
f
3
6
10
12
5
4
Total
40
Hal.: 18
Answer :
To determine Q1 we need = ¼ x 40 data
or 10 data, so Q1 is in the 3rd interval class.
And b = 54,5 ; p = 5; F = 9; f = 10
Value Q1 = 54,5 + 5
 1.40

 4  9


 10 


= 54,5 + 0,5 = 55
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To determine Q3 we need = ¾ x 40 data or 30 data,
so Q3 is in the 4th interval class,
and b = 59,5; p = 5; F = 19 ; f = 12
Score Q3 = 59,5 +
= 59,5 + 5
 3.40


19


5  4

12




 11 

12 

= 59,5 + 4,58 = 64,08
Then, range of inter quartile sprout or quartile deviation of the data
above is
Qd = ½ (Q3 –Q1)
= ½ (64,08 – 55) = 4,54
Hal.: 19
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5. Percentile
Percentile of a set number is value which divides that set
number for the same size of 100 parts after the number has
ordered from the smallest until the biggest.
a. Single Data / weight
Place of Pi = the- data i (n  1)
100
and i = 1,2,…,99
Example :
Given that the data are : 9,3,8,4,5,6,8,7,5,7
Find P20 and P70
Hal.: 20
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Answer :
Ordered Data : 3 ,4, 5, 5, 6, 7, 7 ,8, 8, 9
1
Place of P20 = the- data 20(10  1) = the 2 data
5
100
Score P20 = the 2nd data + 1 (the 3rd – the 2nd data)
1
= 4 + (5 – 4)
15
5
=4
5
Hal.: 21
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Place of P70 = the- data 70(10  1)
100
= the 7th data
Value P70 = the
7th
7
10
7
data +
(the 8th data – the 7th data)
10
7
=7+
(8–7)
10
7
=7
10
Hal.: 22
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b. Grouped Data
 in

 100  F 
Value Pi = b + p 

f




, and i = 1,2,..,99
Percentile Range = P90 – P10
Hal.: 23
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Example :
Determine the percentile range of this data :
Hal.: 24
Score
F
50 - 59
60 - 69
70 - 79
80 - 89
90 - 99
7
10
15
12
6
Total
50
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Answer :
To determine P10 we need =
10
x 50 data = 5 data,
100
it means P10 is in the first interval class with
b = 49,5 ; p = 10 ; F =0 ; f = 7
 10.50


0

Score of P10 = 49,5 + 10  100


7




= 49,5 + 7,14
= 56,64
Hal.: 25
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90
x 50 data = 45 data,
100
th
It means P90 is in the 5 interval class,
To determine P90 we need =
with b = 89,5; F = 44; f = 6.
Score of P90
 90.50


44


= 89,5 + 10  100

6




= 89,5 + 1,67 = 91,17
Percentile range = P90 – P10
= 91,17 – 56,64
= 34,53
Hal.: 26
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Hal.: 27
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Exercises:
1. Math test score of 5 students are : 7,6,7,8,7 the mean deviation of that
data is…..
Answer :
x
xx
7
6
7
8
7
0
1
0
1
0
Tot
2
Hal.: 28
x
=
SR =
=
76787
5
=7
 xx
2
5
n
= 0,4
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2. Standard deviation of the data 4,6,7,6,3,4 are…
Answer :
x
4 6 7  63 4
=
6
= 5
 ( x  x)
S =
n
=
12
6
=
2
Hal.: 29
2
x
(x -
4
6
7
6
3
4
Tot
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x
-1
1
2
1
-2
-1
)
(x-
x
)2
1
1
4
1
4
1
12
Adaptif
UKURAN PENYEBARAN DATA
3. The result of test of the new employee recruitment in a
company noted as follow :
Score
Frequency
30-39
40-49
50-59
60-69
70-79
80-89
90-99
3
8
10
20
18
14
7
Hal.: 30
If the company will accept 75% of
appliances who join the test, then
what is the score minimum that can
be accepted?
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Jawab :
Q1
75%
Untuk menentukan Q1 diperlukan ¼ x 80 data = 20 data,
artinya Q1 terletak pada kelas interval ke 3,
dengan b = 49,5; p = 10; F = 11; f = 10;
Nilai Q1 = 49,5 + 10
 1.80


11
 4



10




 9

 10 
= 49,5 + 10 
= 58,5
Hal.: 31
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Jawab :
Q1
75%
Untuk menentukan Q1 diperlukan ¼ x 80 data = 20 data,
artinya Q1 terletak pada kelas interval ke 3,
dengan b = 49,5; p = 10; F = 11; f = 10;
Nilai Q1 = 49,5 + 10
 1.80


11
 4



10




 9

 10 
= 49,5 + 10 
= 58,5
Hal.: 32
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4. The test result of 50 students of the third class
Industry technology program in SMK are:
Score
F
50-59
60-69
70-79
80-89
90-99
7
10
15
12
6
Determine value of P40 of that data!
Hal.: 33
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Answer:
To determine P40 we need =
40
x 50 data or 20 data,
100
it means P40 is in the third interval class,
with b = 69,5 ; p = 10 ; F = 17 and f = 15.
Score of
 40.50


17
P40 = 69,5 + 10  100



15




3
= 69,5 + 10  
 15 
= 72,5
Hal.: 34
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5. The Math test result of 15 students are :
30,45,50,55,50,60,60,65,85,70,75,55,60,35,30.
The range of inter quartile sprout (Qd) of the above data are
Answer :
Ordered Data :
30,30,35,45,50,50,55,55,60, 60,60,65,70,75,85.
Place of Q1 = the- 1(15  1) data
= the 4th data
4
Value of Q1 = the-4th data
Place of Q3 = the- data 3(15  1)
= the-12nd data
4
Value of Q3
Hal.: 35
= the-12nd data = 65
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The range of inter quartile sprout (Qd) = ½ ( Q3 – Q1 )
= ½ ( 65 – 45 )
= 10
Hal.: 36
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6. Variety Coefficient
Variety coefficient is a comparison between standard deviation and
the average value in percentage.
Variety coefficient used to see the data dispersion of the mean.
The variety coefficient stated by the formula,
KV =
S x 100%
x
KV = variety coefficient
S = standard deviation
x = mean
Hal.: 37
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Example 1:
Mean score of math of the third class of Machinery 1 is 80 and the standard
deviation is 4,5. If mean score of the third class of machinery 2 is 70 and the
standard deviation is 5,2.
Find each variety coefficient.
Answer :
S
KV III Machinery 1 =
x 100%
x
= 4,5
x 100% = 5,6%
80
KV III Machinery 2 =
Hal.: 38
5,2
70
x 100% = 7,4%
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Example 2 :
Standard deviation of a grouped data is 1,5 and its variety coefficient is 12,5%.
Mean of that grouped data is….
Answer :
KV =
S x 100%
x
12,5% = 1,5 x 100%
x
x
Hal.: 39
=
150%
12,5%
= 12
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7. Standard Number
Standard number applies for knowing the position of an
object which is being investigated compared with mean of
the object.
Standard Number can be calculated by using formula :
xx
Z =
s
x = data value
x = mean
s = standard deviation
Hal.: 40
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Example 1:
A student got 70 in math and the mean is then the standard deviation is 12.
She got 80 in English and the mean is 75 and its standard deviation is 15,
then which is the best position of value?
Answer :
Zm = 70  60 = 0,83
12
Zb =
80  75
= 0,33
15
So the position value of math is better than English.
Hal.: 41
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Example 2 :
Mean and standard deviation salary of each employees in an office is
Rp 65.000,00 and Rp 1.500,00. If Mr. Darmawan is one of the employees
who gets Rp 67.250,00, then the standard value of Mr. Darmawan’s salary is….
Answer :
Z=
Rp 67.250,00  Rp 65.000,00
Rp 1.500,00
= 1,5
Hal.: 42
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Size of kurtosis
Kurtosis is pointed degree of a distribution if it is compared by normal
distribution
To calculate the pointed degree of a curve (kurtosis coefficient) can be
denoted :
KK =
Hal.: 43
Q3  Q1
2( P90  P10 )
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Note :
If value of KK > 3 curve leptokurtic (the top is very pointed)
KK < 3 curve platikurtic (the top is little flat)
KK = 0 curve mesokurtic (the top is not so pointed or normal
distribution)
Example :
From the grouped data that ordered in the frequency distribution table,
given that value of Q1 = 55,24 ; Q3 = 73,64 ; P10 = 44,5 ;P90 = 82,5.
then the kurtosis coefficient of that data curve is….
Hal.: 44
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Answer :
KK =
=
73,64  55,24
2(82,5  44,5)
18,4
2(38)
= 0,242
Because KK < 3 then the distribution curve is called platikurtic.
Hal.: 45
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