1. MATHEMATICS The Measure of Data Location

2. 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).

3. Example 1 Determine Q1, Q2, and Q3 for the following data.
b. 8, 2, 3, 6, 7, 9, 5, 6, 10
a. Size of data, n = 8.
The order data:

4. 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

5. Example 2
The data:
Determine a. D b. D6
Answers:
Data is firstly ordered from the smallest to the biggest:
Size of data, n = 15
a. The 3rd decile located in the value of
so, D3 = x4 = 0.8(x5 – x4) = (6 – 6) = 6.
b. The 6th decile located in the value of
So, D6 = x (x10 – x9) = (8 – 7) = 7.6

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.
Note: = below side of quartile class
n = Size of data
= cumulative frequency prior to quartile class
f = frequency of quartile class
c = length of class
i = 1, 2, 3

7. Mean while, the decile value is determined by using the following formula.
Note : = the decile i-th , n = size of data, = cumulative frequency prior to decide class , f = frequency of decile class , c = length of class, I = 1, 2, 3, 4, …,9

8. Example 3
Determine the lower and upper quartile, as well as the 6th decile from the following data
Score
Frequency
Cumulative frequency
40 – 49
50 – 59
60 – 69
70 – 79
80 – 89
90 – 99 1 4 8 14 10 3 5 13 27 37 40

9. Answers:
Length of class c = 10 and size of data n = 40,
The order (rank) where 6th decile located is
First decide class is interval 60 – 69, 3rd decile class is interval
80 – 89, and 6th decile class is interval 70 – 79.
Thus,

10. 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, ,
Inter-quartile range or spread, H is difference between 1st quartile and 3rd quartile
Semi Inter-quartile Range or quartile deviation, , is half of the spread.

11. Example 4
Given the following data: 3, 4, 4, 5, 7, 8, 9, 9, 10. Determine range, inter-quartile range, and quartile deviation!
Answers:

12. 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
formula:
Note: n = size; = value of the of data; = mean

13. Example 5
Determine the mean deviation from the data: 1, 3, 4, 5, 8, 10, 12,13.
Answers:
N = 8
Thus, the mean diviation of the data is 3.75
= 3.75

14. c. variance and standard deviation
Definition
Variance is the square distance of a data toward its mean
Suppose that have mean of , then the variance of the data is determined by the formula
: Mean while, the standard deviation (S) is determined by
the following formula
Note: n = size of data; = value of the of data;
= mean

15. 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:
(through up to 2 decimals location)

16. 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

17. 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 ,
so can be obtained:
There fore, the data in table presented in Example 1.14 has
inter-quartile range, H = and quartile deviation,
b. Mean deviation, variance, and standard deviation
For grouped data, mean deviation is determined by the following:

18. 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
following table.
Midpoint
Score
Midpoint
Frequency
40 – 49
50 – 59
60 – 69
70 – 79
80 – 89
90 – 99
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 77 74
10.5
107.5
62.25

19. Thus,
For the grouped data, variable value can be determined by using
the following
formula: variance
With: = frequency of the of class; k = number of class;
= midpoint of the of the class; = average
The other formula in obtaining variance from the grouped data is
Variance
The above formula could be changed by using mean deviation or
coding method

20. 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 = and the
table could completed as follow:

22. Thank You