Measures OF Position (Quartile, Decile and Percentile for grouped and ungrouped data) PDF

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Summary

Computation for Quartiles, Deciles and Percentiles for the grouped and ungrouped data...

Description

MEASURES OF POSITION Quantiles are natural extension of the median idea in that they are values which divide a set of data into equal parts. Quantiles are classified as: 1. quartile 2. decile 3. percentile QUARTILE Quartile is a quantile that divides the distribution into four equal parts. The values of the quartiles are denoted by Q1 for first quartile; Qa for second quartile, and Q3 for third quartile.  For the Q1 : Twenty-five percent (25%) of the distribution falls below and seventy-five percent (75%) falls above the value on the Q1 .  For the Q2 : Fifty percent (50%) of the distribution falls below and the other fifty percent (50%) falls above the value on the Q2 .  For the Q3 : Seventy-five (75%) of the distribution is less than the value on the Q3 while the remaining twenty-five percent (25%) is greater. DECILE Decile is a quantile that divides the distribution of the distribution into ten equal parts. The data set has nine deciles which are represented by D1, D2, D3, D4, D5, D6, D6, D7, D8, and D9. Where D is the number that divides the bottom 10 of the data from the top 90 and so on.  For the D1 or first decile: Ten percent (10%) is below the value in the position of D1 while ninety percent (90%) is greater than the value in the D1 position.  For the D2 or second decile: Twenty percent (20%) is below the value in the position of D2 while eighty percent (80%) is greater than the value in the D2 position.  For the D3 or third decile: Thirty percent (30%) is below the value in the position of D 3 while seventy percent (70%) is greater than the value in the D3 position. PERCENTILE Percentile is a quantile that divides the distribution into 100 equal parts. A set of data has 99 percentiles which are denoted by P1, P2, P3, P4, … P99 .  For the P1 or first percentile: One percent (1%) is below the value in the P1 position and ninetynine percent (99%) is above the value in this position.  For the P5 or fifth percentile: Five percent (5%) is below the value in the P5 position and ninetynine percent (95%) is above the value in this position.  For the P60 or sixtieth percentile: Sixty percent (60%) is below the value in the P60 position and forty percent (40%) is above the value in this position. Solving Quartiles To solve any quantile follow the steps below: 1. Arrange the scores according to magnitude or size. 2. Compute the position of the given quantile in the distribution using the appropriate formula. 3. Starting from the lowest score, locate the score corresponding to the obtained position in the array of data. 4. Interpolate to get the score of the obtained position in the distribution.

Quantiles for Ungrouped Data th

Quartile position

=

i n+(1- i) 4 4

Where:

= =

1 for Q1, 2 for Q2, and 3 for Q3 number of items

i n Examples:

Find the first quartile.

item

=== Q1 =

1 n+(1- 1) 4 4

th

item

Find the second quartile.=== Q2 = Decile position

Where:

i n

th

2 n+(1- 2) 4 4

th

item

=

i n+(1- i ) 10 10

item

= =

1 for D1, 2 for D2, 3 for D3, 4 for D4, 5 for D5, … and 9 for D9 number of items

Find the first decile.=== D1 = 1 n + ( 1 - 1 ) 10 10

th

item

Find the fifth decile.=== D5 = 5 n + ( 1 - 5 ) 10 10

th

item

Examples:

=

i 100

Where:

= =

1 for P1, 2 for P2, 3 for P3, 4 for P4, 5 for P5, … and 99 for P99 number of items

i n Examples:

n+(1-

th

Percentile position

i ) 100

item

Find the twenty-fifth percentile. P25 =

25 100

n+(1-

25 ) 100

th

item

Find the thirtieth percentile. P30 = 30 n + ( 1 - 30 ) 100 100 Note:   

th

item

First quartile is equal to twenty-fifth percentile. Second decile is equal to twentieth percentile. Fiftieth percentile is equal to fifth decile.

Example: 1. Consider the following scores below: 50

52

48

45

33

65

42

37

69

40

41

38

41

42

45

48

50

52

65

69

Find the following: a. First quartile b. Third quartile c. Second decile d. Fifth decile e. Twentieth percentile f. Fortieth percentile Solution: Array: 33

37

38

a. Q1 = 1 n + ( 1 - 1 ) 4 4

40 th

item = 1 (12) + ( 1 - 1 ) 4 4

3.75 is between the 3rd and the 4th items: 38 and 40.

= 3.75th item

Since 3.75 has decimal .75, interpolate to find the value in this position: Find the difference between 38 and 40 === Multiply the difference by the decimal .75 === Add the product to the smaller value which is 38 ===

40 – 38 2 x .75 38 + 1.5

=2 = 1.5 = 39.5

This means that Q1 which is the 3.75th item is 39.5. This also means that 25% of the distribution is below 39.5 and that 75% is above 39.5. b. Q3 = 3 n + ( 1 - 3 ) 4 4

th

item =

3 (12) + ( 1 - 3 ) 4 4

= 9.25th item

9.25 is between the 9th and the10th items: 50 and 52. Find the value in the 9.25th position by interpolation:

52 – 50 2 x .25 50 + .5

=2 = .5 = 50.5

This means that 75% of the distribution is below 50.5 and that 25% is above 50.5. c. D2 = 2 n + ( 1 - 2 ) 10 10

th

item =

2 (12) + ( 1 - 2 ) 10 10

= 3.20th

3.20 is between the 3rd and the 4th items: 40 and 38 By interpolation, find the value in the 3.20th position:

40 – 38 2 x .2 38 + .4

=2 = .4 = 38.4

This means that 20% of the distribution is below 38.4 and that 80% is above 38.4. d.

D5 = 5 n + ( 1 - 5 ) 10 10

th

item = 5 (12) + ( 1 - 5 ) 10 10

= 6.50th

6.50 is between the 6th and the 7th items: 45 and 42 By interpolation, find the value in the 6.5th position:

45 – 42 3 x .5 42 + 1.5

=3 = 1.5 = 43.5

This means that 50% of the distribution is below 43.5 and that the other 50% is above 43.5 e. P 20 = 20 n + ( 1 - 20 ) 10 0 100

th

item =

20 (12) + ( 1 - 20 ) 100 100

3.20 is between the 3rd and the 4th items: 40 and 38 By interpolation, find the value in the 3.20th position:

= 3.20th

40 – 38 2 x .2 38 + .4

=2 = .4 = 38.4

This means that 20% of the distribution is below 38.4 and that 80% is above 38.4. f.

P 40 = 40 n + ( 1 - 40 ) 10 0 100

th

item =

40 (12) + ( 1 - 40 ) 100 100

5.40 is between the 5th and the 6th items: 41 and 42 By interpolation, find the value in the 5.40th position:

= 5.40th

42 – 41 1 x .2 41 + .4

This means that 40% of the distribution is below 41.4 and that 60% is above 41.4.

=1 = .4 = 41.4...