Quartiles, deciles and percentile
For Ungrouped data

Quartiles: divide the distribution into four group Q_{1} , Q_{2} , Q_{3}
Smallest data Q_{1 }Q_{2} Q_{3} Largest data
25% 25% 25% 25%
Median
q =quartile

If c is not whole number, round up to the next whole number

If c is a whole number, take average of c^{th} and (c+1)^{th}
Example:

The weights in pounds in the data set. Find Q_{1} , Q_{2} , Q_{3}.
16 18 22 19 3 21 17 20

The test score in the data set. Find Q_{1} , Q_{2} , Q_{3}.
42 35 28 12 47 50 49

Deciles: divide the distribution into 10 groups
Smallest data D_{1 }D_{2} D_{3} D_{4 } D_{5} D_{6} D_{7} D_{8} D_{9 }Largest data
10% 10% 10% 10% 10% 10% 10% 10% 10%
Median

arrange the data in order

Find location of quartiles, _{ }where ; n = total number of values
d =decile

If c is not whole number, round up to the next whole number

If c is a whole number, take average of c^{th} and (c+1)^{th}
Example:

(from previous example) Find D_{5}.
16 18 22 19 3 21 17 20

(from previous example)Find D_{7}.
42 35 28 12 47 50 49
Smallest data P_{1 }P_{2} P_{3} _{ } P_{97} P_{98} P_{99} Largest data
10% 10% 10% 10% 10% 10% 10% 10% 10%
D_{1 , }D_{2}, D_{3}, … , D_{9 }correspond to P_{10 , }P_{20}, P_{30}, … , P_{90 }
Q_{1} , Q_{2} , Q_{3} correspond to P_{25 , }P_{50}, P_{75}
Median = Q_{2} = D_{5} = P_{50}

arrange the data in order

Find location of quartiles, _{ }where ; n = total number of values
p =percentile

If c is not whole number, round up to the next whole number

If c is a whole number, take average of c^{th} and (c+1)^{th}
Example:

(from previous example) Find P_{33}.
16 18 22 19 3 21 17 20

(from previous example)Find P_{60}.
42 35 28 12 47 50 49
Finding percentile corresponding to given value, X
Example of data set : 1 1 3 4 5
Find percentile for 4.
P_{70} = 4
(round off the answer)
Example:

(from previous example)Find the percentile rank for each test score in the data set.
42 35 28 12 47 50 49
(Data value 47 = P_{64} but previously when we want to find P_{60} the data value is 47b too. So actually P_{60} closer to P_{64} which is data value 47)
For Grouped Data
METHOD 1: (USE PERCENTILE GRAPH)
xaxis: class boundaries
yaxis: relative cumulative frequency (percentage)
Cumulative relative frequency (%) =
Graph:

percentile graph
Relative cumulative frequency (%)
100
25 P_{25}

Ogive using relative frequency (iii) Ogive
Relative cumulative frequency Cumulative Frequency
1.0 75
0.25 P_{25} 18.75 P_{25}
25% x 75 =18.75
METHOD 2: (USE FORMULA)
Example:
This distribution represents the data for weights of fifthgrade boys.
Weights (pounds)

frequency

52.5 – 55.5

9

55.5 – 58.5

12

58.5 – 61.5

17

61.5 – 64.5

22

64.5 – 67.5

15


Find the approximate weights corresponding to each percentile given by constructing a percentile graph.
(i) Q_{1} (ii) D_{8} (iii) Median (iv) P_{95}

Find the approximate percentile ranks of the following weights.
(i) 57 pounds (ii) 64 pounds (iii) 62 pounds (iv) 59 pounds

Find P_{63} by using the formula.
EXERCISE CHAPTER 3

What type of sampling is being employed if a country is divided into economic classes and a sample is chosen from each class to be surveyed?

Given a set of data 5,2,8,14,10,5,7,10,m, n where =7 and mode = 5. Find the possible values of m and n. (ans: m=5, n=4 or m =4 , n =5)

Find the value that corresponds to the 30^{th} percentile of the following data set:
78 82 86 88 92 97 (ans: P_{30} =82)

Given the variance of the set of 8 data x_{1} , x_{2}, x_{3}, … , x_{8} is 5.67. If , find the mean of the data. (ans: 11.09)

Find Q_{3} for the given data set : 18,22,50,15,13,6,5,12 (ans: 20)

The number of credits in business courses that eight applicants took is 9, 12, 15, 27, 33, p, 63, 72. Given the value that corresponds to the 75^{th} percentile is 54, find p. (ans: 45)

The mean of 5, 10, 26, 30, 45, 32, x, y is 25 where x and y are constants. If x = 16, find the median. (ans: 28)

Construct a frequency distribution by using 7 classes (use 3 as lower limit of the first class)

Find the mean, mode and standard deviation. (ans: 28.15 , 31.3 , 14.63)

Draw an ogive by using relative frequency and estimate the median from the graph.
EXERCISE

In four successive history tests, a student received grades of 45, 73, 77 and 86. Which of the following conclusions can be obtained from these figures by descriptive method and inferential method? Explain your answer.

Only one of the grades exceeds 85.

The student’s grades increased from each test to the next.

The student must have studied harder for each successive test.

The difference between the highest and the lowest grade is 41.

State whether the following are nominal, ordinal, interval or ratio data.

A statistics test which a student took was easy, difficult or very difficult and these alternatives are coded 1, 2 and 3.

The temperature if different kilns at the factory.

The bottles on a Chemistry laboratory shelf are numbered 1,2,3 and 4 representing sulfuric acid, hydrochloric acid, nitric acid and sodium hydroxide.

The race of the students in university campus.

The normal operating temperature of a car engine.

Classification of students using an academic program.

Speaker of a seminar rated as excellent, good, average or poor.

Number of hour’s parents spends with their children per day.
EXERCISE

In four successive history tests, a student received grades of 45, 73, 77 and 86. Which of the following conclusions can be obtained from these figures by descriptive method and inferential method? Explain your answer.

Only one of the grades exceeds 85.

The student’s grades increased from each test to the next.

The student must have studied harder for each successive test.

The difference between the highest and the lowest grade is 41.

State whether the following are nominal, ordinal, interval or ratio data.

A statistics test which a student took was easy, difficult or very difficult and these alternatives are coded 1, 2 and 3.

The temperature if different kilns at the factory.

The bottles on a Chemistry laboratory shelf are numbered 1,2,3 and 4 representing sulfuric acid, hydrochloric acid, nitric acid and sodium hydroxide.

The race of the students in university campus.

The normal operating temperature of a car engine.

Classification of students using an academic program.

Speaker of a seminar rated as excellent, good, average or poor.

Number of hour’s parents spends with their children per day.
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