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MATHEMATICS
The Measure of Data Location
MAIN TOPIC
The Measure of Data Location
1. Quartile and Decile for Single Data
a. Quartile (Q)
Definition
Quartile is the value that divided the ordered or
data into four part same size after the data is order is
ordered from the smallest to the biggest.
There are there quartiles, lower quartile or first is
Q1, second quartile or median is Q2, and upper quartile
or third quartile is Q3. The quartile of the data are
obtained by following ways.
1. Order the data from the smallest to the biggest
(ascending).
2. Determine the data median or the second quartile.
3. Determine first quartile Q1 (the median of the data less
than the second quartile Q2) and third quartile Q3 (the
median of the data above the quartile Q3).
Example 1
Determine Q1, Q2, and
Q3 for the following
data.
a. 4, 8, 3, 1, 6, 9, 5, 1
b. 8, 2, 3, 6, 7, 9, 5, 6,
10
a. Size of data, n = 8.
The order data:
b. Decile
Definition
Decile is the value that divides the ordered
data into ten parts with same size after the data is
ordered from the smallest to the biggest.
For a single data, the decile’s location is
determined by using the following formula.
The order (rank) that Di is located in the value
of
i ( n  1)
10
Example 2
The data: 7 5 8 7 9 6 6 6 8 5 9 8 6 7
Determine a. D3
b. D6
Answers:
Data is firstly ordered from the smallest to the biggest:
5 5 6 6 6 6 7 7 7 8 8 8 9 9 9
3(15  1)
Size of data, n = 15
 4.8
a. The 3rd decile located in the value of 10
so, D3 = x4 = 0.8(x5 – x4) = 6 + 0.8(6 – 6) = 6.
b. The 6th decile located in the value of 6(15  1)
10
So, D6 = x9 + 0.6(x10 – x9) = 7 + 0.6(8 – 7) = 7.6
9
 9.6
2.
Quartile and decile for grouped data
a. Quartile(Q)
For data that already arranged in the list of frequency
distribution, quartile is calculated by the following
formula.  i

 n  fk 
.c
Qi  t b   4
f






Note: t b = below side of quartile class
n = Size of data
f k = cumulative frequency prior to quartile class
f = frequency of quartile class
c = length of class
i = 1, 2, 3
Mean while, the decile value is determined by using the
following formula.
 i

n  fk 

c
Di  t b   10
f






Note : D i = the decile i-th , n = size of data, Di = f k
cumulative frequency prior to decide class Di , f =
frequency of decile class
, c = length of class, I = 1,
2, 3, 4, …,9
Example 3
Determine the lower and upper quartile, as well as the
6th decile from the following data
Score
Frequency
Cumulative
frequency
( fk )
40 – 49
50 – 59
60 – 69
70 – 79
80 – 89
90 – 99
1
4
8
14
10
3
1
5
13
27
37
40
 fi 
Answers:
1
3
3
Length of class c = 10 and size of data n = 140,
n  (40)  10, n  (40)  30
4
4
4
4
The order (rank) where 6th decile located is i n  1  6(40  1)  24.6
10
10
First decide class is interval 60 – 69, 3rd decile class is
interval
80 – 89, and 6th decile class is interval 70 – 79.
1

 n  fk 
.c  59.5   10  5 .10  65.75
Q1  t b   4
f


 8 




3

 n  fk 
.c  79.5   30  27 .10  82.5
Q3  t b   4
f


 10 




Thus,
 6


 i

  .40   13 
 n  fk 
.10  77.36
.c  69.5    10 
D6  t b   10


f
1










F.
The Measure of Data Dispersion
1. Measure of Data Dispersion for Single Data
a. Range, Spread, and Quartile Deviation
Definition
Range of data, J is difference between biggest data,
, and smallest, xmin , x max
J  xmax  xmin
Definition
Inter-quartile range or spread, H is difference between
1st quartile and 3rd quartile
H  Q3  Q1
Definition
Semi Inter-quartile Range or quartile deviation, , is half
of the spread.
Qd 
1
Q3  Q1 
2
Example 4
Given the following data: 3, 4, 4, 5, 7, 8, 9, 9, 10.
Determine range, inter-quartile range, and quartile
deviation!
Answers:
b.
Means Deviation
Definition
Mean deviation is average of the distance of a data
toward its mean The value of mean deviation (SR) for
single data can be determined using the
n
formula: SR  1
|x x|
n

Note: n = size;
i 1
xi
i
= value of the i  th of data;
x
= mean
Example 5
Determine the mean deviation from the data: 1, 3, 4, 5, 8,
10, 12,13.
Answers:
N=8
1  3  4  5  8  10  12  13 56
x

7
8
8
1
SR  1  7  3  7  4  7  5  7  8  7  10  7  12  7  13  7 
8
1
6  4  3  2  1  3  5  6
8
= 3.75

Thus, the mean diviation of the data is 3.75
c. variance and standard deviation
Definition
Variance is the square distance of a data toward its
mean
Suppose that x1 , x2 , x3 ,...xn have mean
of , then the
variance of the data is determined by the formula
x


2
1 n
S   xi  x
2 i 1
: Mean while, the standard deviation (S) is
determined by
n
2
1
2
the following formula
2
S s 

x  x

n
i 1
Note: n = size of data;
data;
= mean
i
i the
th
xi = value of
x
of
Example 6
Calculate the variance and standard deviation from the
data: 1, 3, 4,
5, 8, 10, 12, 13
Answers:
Data: 1, 3, 4, 5, 8, 10, 12, 13
n = 8 and mean of , then:
 x  x 
8
2
1
i 1
 1  7   3  7   4  7   5  7   8  7   10  7   12  7   13  7 
2
2
2
2
2
2
2
 36  16  9  4  1  9  25  36  136

1 8
S   xi  x
8 i 1
2

2

1
136  17
8
(through up to 2 decimals location)
S  S 2  17  4.12
2
2.
Measure of Data dispersion for grouped
data
a. Inter-quartile Range and Quartile Deviations
Inter-quartile range is also called spread, and quartile
deviation is also called semi inter-quartile range as for
single data, the inter-quartile range of H and quartile
deviation of , for grouped data, are determined by the
following formula
H  Q3  Q1
Qd 
1
Q3  Q1 
2
Example 7
Determine the inter-quartile range and quartile deviation
from the
data table presented in Example 1.143
Answers:
In example 1.19 have been calculated the value of ,Q  65.75, andQ
so can be obtained: H  Q3  Q1  82.5  65.75  16.75
1
Qd 
1
Q3  Q1   1 16.75  8.375
2
2
There fore, the data in table presented in Example 1.14
has
inter-quartile range, H = 16.75 and quartile deviation,
Qd  8.375
b. Mean deviation, variance, and standard deviation
For grouped data, mean deviation is determined by the
following:
1 k
SR   f i xi  x
n i 1
3
 82.5
Example 8
Determine the mean deviation from the data in Example 1.14
Answers:
Calculation average from the data on the table is To calculate the
value of mean deviation,
then the can be completed into the
Midpoint
following table.
x  73.75
Score
40 – 49
50 – 59
60 – 69
70 – 79
80 – 89
90 – 99
Midpoint
Frequenc
y  fi 
xi  x
44.5
54.5
64.5
74.5
84.5
94.5
1
4
8
14
10
3
29.25
19.25
9.25
0.75
10.75
20.75
 xi 
f
i
 40
f i xi  x
29.25
77
74
10.5
107.5
62.25
f
i
xi  x  360.5
Thus,
1
1
f
x

x

x360.5  9.0125
 i i
n
40
For the grouped data, variable value can be determined by
using
k
2
1
2
the following
S   f i xi  x
formula: variance 2 i 1
With: f i
= frequency of thei  th of class; k = number of
class;
xi = midpoint of the i  th of the class; x = average
The other formula in obtaining variance from the grouped
data is
2
k
k
1
1


2
f i xi2  
f i xi 
Variance S 
SR 

n

i 1
2


i 1

The above formula could be changed by using mean
deviation or
coding method
Example 9
Determine variance from the data table
presented in Example 1.14 by :
a. Definition,
b. not using average (mean),
c. using mean deviation method, and
d. using coding method
Then, determine the standard the deviation
Answers:
As the average (mean) had already calculated = 73.75 and
the
table could completed as follow:
Thank You