Question

Fill in the table and answer the following questions

**** (Use D-method)

Class	Frequency
10 – 12	6
13 – 15	4
16 – 18	14
19 – 21	15
22 – 24	8
25 – 27	2
28 – 30	1
	50

Class	Real limits	f	cf	x	d	fd
10 – 12
13 – 15
16 – 18
19 – 21					0
22 – 24
25 – 27
28 – 30

Mean()	b. Median(Md=)
c. Mode	d. Range
e. Variance:	f. Standard deviation (S):
g.  3rdQuartile (	h. 45th Percentile

Answer 1

The real limit of a particular class is basically lower limit minus 0.5 and uppoer limit plus 0.5. The frequency would be same as before. The commulative frequency is the addition of the frequencies one by one. The mid value is the arithmatic mean of the class interval limits. For A being the assumed mean, , where i is the class interval size, which is 3 (the difference between the lower upper limits of real limit column). The table will be as below. We would assume the assumed mean to be 20, as that is where the d is shown to be zero.

Class	Real limits	f	cf	x	d	f*d
10-12	9.5-12.5	6	6			-18
13-15	12.5-15.5	4	6+4=10			-8
16-18	15.5-18.5	14	10+14=24			-14
19-21	18.5-21.5	15	24+15=39			0
22-24	21.5-24.5	8	39+8=47			8
25-27	24.5-27.5	2	47+2=49			4
28-30	27.5-30.5	1	49+1=50			3

(a) The mean would be hence, or or . Hence, the mean is 18.5.

(b) The median will be the (N/2)th term, which in this case will be 25th term. From the first class interval, there are 24 objects up to class 16-18. Hence, in the next class, there would be the 25th term, and the median class is 19-21. Also, supposing that all the 15 objects are uniformly distributed over the class 19-21, the objects would be at a fraction of . That means, the objects would be distributed as . The 25th object would be however, the first object, as the previous class interval had 24 objects. Hence, the median is 19.

(c) Modal class is the class interval with highest frequency. Hence, the mode is 20 (midpoint of 19-21, the modal class).

(d) Range would be .

(e) Variance is . The table would be as below.

Class	f	x	(x-mean)squared
10-12	6		56.25	337.5
13-15	4		20.25	81
16-18	14		2.25	31.5
19-21	15		2.25	33.75
22-24	8		20.25	162
25-27	2		56.25	112.5
28-30	1		110.25	110.25

Hence . The variance is 17.37.

(f) The standard deviation is . We have . Hence, the standard deviation is 4.167.

(g) The 3rd quartile is the (3N/4)th term, ie 37.5th term. Hence, the corresponding class would be 19-21 class. The specific term will be found analogously as the median, but using the formula as . Hence, 3rd quartile is 20.8.

(h) The 45th percentile's corresponding group would be where the frequency 45% of N lies, ie at the 22.5th term. The 45th percentile class will be 16-18. Hence, we have, by the same formula as above, but with some minor alteration, . Hence, the 45th percentile is 17.79.