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QUARTILES, DECILES AND PERCENTILES

 

As the median divides an array into two parts, the quartiles divide the array into four parts, the deciles divide it into ten parts and the percentiles divide it into one hundred parts.

The first quartile/the lower quartile denoted by Q 1 is computed as follows.

where,

L = Lower limit of the class containing Q 

f = Frequency of the class containing Q1 

h = Width of the class containing Q1

C = Cumulative frequency of the class preceding the class containing Q

Here the cumulative frequency just greater than N/4 is the class containing Q1

The second quartile is the median.

The third quartile/the upper quartile denoted by Q3 is computed as follows:

 

where,

L = Lower limit of the class containing Q3

F= Frequency of the class containing Q3

h= Width of the class containing Q3

C=Cumulative frequency of the class preceding the class containing Q3.

 

Here the cumulative frequency just greater 3N/4 than  is the class containing Q3

The k-th decile denoted by Dk is computed as follows:

 

C = Cumulative frequency

Where,

L= Lower limit of the class containing D 

f =Frequency of the class containing Dk 

h= Width of the class containing Dk

C=Cumulative frequency of the class preceding the class containing Dk,

Here the cumulative frequency just greater than (k*N)/4 is the class containing Dk

(k = 1, 2, ...... ,99). 

The k-th percentile denoted by P k is computed as follows:

where,

L = Lower limit of the class containing P 

f = Frequency of the class containing P 

h = Width of the class containing Pk

C = Cumulative frequency of the class preceding the class containing Pk .

Here the cumulative frequency just greater than (k*N)/100  is the class containing P k

(k = 1,2, ...... ,99).

Example 6. Determine (a) Q1, (b) Q3 , (c) D5 (d) P80 from the following distribution :

 

Solution.

 

(a) N = 100,N/4 = 25. The cumulative frequency just greater than 25 is 35. So the class 15- 20 contains Q1. L = 15, f= 15,  h = 5,  C = 20. Therefore,

                      

 (b) Here 3N/4 = 75. The cumulative frequency just greater than 75 is 88. So the class 25 – 30 contains Q3.

 

L = 25, f = 22, h = 5, C = 66. Therefore,

(c) Here  5N/10=50. The cumulative frequency just greater than 50 is 66. So the class 20 – 25 contains D5

.

L = 20, f = 31, 1z = 5, C = 35, Therefore,

(d)  Here 80N/100 = 80. The cumulative frequency just greater than 80 is 88. So the class 25 - 30 contains P 80.

 

L = 25, f = 22, h = 5, C = 66. Therefore,

Example 7. A given machine is assumed to depreciate 30% in value in the first yew; 35% in e second year and 80% per annum for the next three years, each percentage being calculated on the diminishing value. Calculate the average annual rate of depreciation.

 

Solution. The proportional rates of depreciation for the 5 years are 0.30, 0.35, 0.80, 0.80 and 80 respectively.

 

Let r be the average proportional rate of depreciation.

Then 1 - r is the G.M. of (I - 0.30), (l - 0.35), (1 - 0.80), (1 - 0.80) and (1 - 0.80) i.e.,70, 0.65, 0.20. 0.20 and 0.20.

 

 

Hence the average annual rate of depreciation is 67.47% .

PROBLEMS 

1-      Suppose a train moves 5 hrs at a speed of 40 km/hr, then 3 hrs at a speed of 45 km/hr and next 5 hrs at a speed of 60 km/hr. Calculate the average speed.

2-      A factory has 4 sections. the no. of workers in the different sections being 50, I 00. 60 and 150. The average wages per worker are Rs. I 00, Rs. 120, Rs. 130 and Rs. 110 respect1vely, calculate the average wages for all the workers together.

3-      Compute the A.M., G.M. and H.M. from the following:

4. The mean of 15 items is 34. It was found out later on, that the two items 48 and 32 were wrongly copied as 84 and 23 respectively. Find out the correct mean.

5. The number of telephone calls received daily in a marketing department of a company for 200 days are given below:

 

Calculate the mean, median and mode of the telephone calls.

6. Consider the following distribution of humidity readings in a certain place for 60 days.

Calculate (i) Q1 (ii) Q3, (iii) Q5, (iv) P90.

7. From the following distribution of marks, calculate

(i) mean, (ii) median, (iii) mode, (iv) Q5 (v) P85 

8. Find the missing frequencies in the following distribution when it is known that A.M. = 59.5

Where total of percentage of wage earners is 100.

9. An analysis of production rejects resulted in the following figures:

Compute mean, median and mode of the production rejects.

10. An aeroplane travels distances of d" d2 and d3 km at speeds VI' V 2 and V 3 km per hour respectively.

Show that the average speed is given by V,

 Where =

ANSWERS

1. 48.85 km/hr

2. Rs. 102.22

3. A.M. = 33.33, GM = 32.22, H.M. = 30.95

4. Corrected mean = 32.2

5. Mean = 27.875, Median = 27.74, Mode = 27.39

6. Q1 = 27.5, Q3 == 37.5, 0 10 == 45.5, P90 = 51.5

7. Mean = 39.27, median = 45, mode = 56.46 (using empirical relation)