Statistics : Exercise - 14.3 (Mathematics NCERT Class 10th)


Like the video?  Subscribe Now  and get such videos daily!

 

Q.1      The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

29

Sol.      Calculation of median
             First, we prepare the following table to compute the median :

30
              We have n = 68  \Rightarrow {n \over 2} = {{68} \over 2} = 34
               The cumulative frequency just greater than {n \over 2}, is 42 and the corresponding class is 125 – 145.
               Thus, 125 – 145
is the median class such that {n \over 2} = 34, \ell = 125, c.f. = 22, f  = 20, h = 20 .
               Substituting these values in the formula
               Median = \ell + \left( {{{{n \over 2} - cf} \over f}} \right) \times h
               Median = 125 + \left( {{{34 - 22} \over {20}}} \right) \times 20 = 125 – 12 = 137 units
               Calculation of mean
               Let the assumed mean, A = 135,class interval, h = 20
               so {u_i} = {{{x_i} - A} \over h} = {{{x_i} - 135} \over {20}}

208

              \bar x = A + h \times {{\Sigma {f_i}{u_i}} \over {\Sigma {f_i}}}
              = 135 + 20  \times {7 \over {68}}
             135 + 2.05 = 137. 05
             Therefore, Mean = 137.05 units
             Calculation of mode
             The class 125 – 145 has the maximum frequency,therefore, this is the modal class.
              Here \ell =125, h =20, {{f_1}} =20, {{f_0}} = 13 and {{f_2}} =14
              Now, let us substitute these values in the formula
              Mode = \ell + \left( {{{{f_1} - {f_0}} \over {2{f_1} - {f_0} - {f_2}}}} \right) \times h
                        = 125 + {{20 - 13} \over {40 - 13 - 14}} \times 20
                        = 125 + {7 \over {13}} \times 20
                       = 125 + 10.76 = 135.76
              Therefore, mode = 135.76 units
              Clearly, the three measures are approximately the same in this case.


Q.2       If the median of the distribution given below is 28.5, find the values of x and y.

106
Sol.       Here it is given that median is 28.5 and n = {\Sigma {f_i}} = 60
             We now prepare the following cumulative frequency table :

107
             Here n = 60           \Rightarrow {n \over 2} = 30
             Since the median is given to be 28.5, thus the median class is 20 – 30.
            Therefore, \ell = 20, h = 10, f = 20 and cf = 5 + x
            Therefore, median = \ell + \left( {{{{n \over 2} - cf} \over f}} \right) \times h
             \Rightarrow 28.5 = 20 + {{30 - \left( {5 + x} \right)} \over {20}} \times 10
             \Rightarrow 28.5 = 20 + {{25 - x} \over 2}
             \Rightarrow 57 = 40 + 25 – x
             \Rightarrow x = 65 – 57 = 8
             \Rightarrow Also, 45 + x + y = 60
             \Rightarrow 45 + 8 + y = 60                   [As x = 8]
             \Rightarrow y = 60 – 53 = 7
            Hence, x = 8, y = 7


Q.3     A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 year.
34
Sol.          We are given the cumulative frequency distribution. So, we first construct a frequency table from the given cumulative frequency distribution and then we will make necessary computations to compute median .

35
               Here n = \Sigma {f_i} = 100 \Rightarrow {n \over 2} = 50
               We see that the cumulative frequency just greater than {n \over 2}, i.e., 50 is 78 and the corresponding class is 35 – 40. So, 35 – 40 is the median class.
               Therefore, {n \over 2} = 50,\ell = 35, c.f = 45 , f = 33, h = 5
               Now, let us substitute these values in the formula
               Median = \ell + \left( {{{{n \over 2} - cf} \over f}} \right) \times h
                           = 35 + {{50 - 45} \over {33}} \times 5  = 35 + {5 \over {33}} \times 5
                           = 35 + 0.76 = 35.76
              Hence, the median age = 35.76 years .


Q.4      The lengths of 40 leaves of a plant are measured correct to the nearest millimeter, and the data obtained is represented in the following table:

36
              Find the median length of the leaves.

Sol.           Here the frequency table is given in the inclusive form. So, we first convert it into exclusive form by subtracting and adding {h \over 2} to the lower and upper limits respectively of each class, where h denotes the difference of lower limit of a class and the upper limit of the previous class. Converting the given table into exclusive form and preparing the cumulative frequency table, we get

37
               We have n = 40  \Rightarrow {n \over 2} = 20
               The cumulative frequency just greater than {n \over 2} is 29 and the corresponding class is 144.5 – 153.5.
               So, 144.5 – 153.5 is the median class.
               Here {n \over 2} = 20,\ell = 144.5, h = 9, f = 12, cf =17
               Substituting these values in the formula
               Median = \ell + \left( {{{{n \over 2} - cf} \over f}} \right) \times h, we have
               Median = 144.5 + \left( {{{20 - 17} \over {12}}} \right) \times 9
                           = 144.5 +{3 \over {12}} \times 9
                           = 144.5 + 2.25
                           = 146.75 mm


Q.5     The following table gives the distribution of the life time of 400 neon lamps:

221
          Find the median life time of a lamp.

Sol.       First, we prepare the following table to compute the median:

121121209121             
               We have n = 400  \Rightarrow {n \over 2} = 200

              The cumulative frequency just greater than {n \over 2} is 216 and the corresponding class is 3000 – 3500.
              Thus, 3000 – 3500 is the median class such that {n \over 2} = 200,\ell = 3000, cf = 130, f = 86 and h = 500.
              Substituting these values in the formula
              Median = \ell + \left( {{{{n \over 2} - cf} \over f}} \right) \times h, we have
              Median = 3000 + \left( {{{200 - 130} \over {86}}} \right) \times 500
                         = 3000 + {{70} \over {86}} \times 500
                         = 3000 + 406.98 = 3406.98
              Hence, median life = 3406.98 hours


Q.6     100 surnames were randomly picked up from a local telephone directory and the frequency
distribution of the number of letters in English alphabets in the surnames was obtained as follows:

43
          Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? 
Also, find the modal size of the surnames.
Sol.      Calculation of median
             First, we prepare the following table to compute the median

210             
              We have n = 100  \Rightarrow {n \over 2} = 50

              The cumulative frequency just greater than {n \over 2} is 76 and the corresponding class is 7 – 10 . Thus 7 – 10 is the median class such that
              {n \over 2} = 50,\ell = 7, cf = 36, f = 40 and h = 3
              Substitute these values in the formula
              Median = \ell + \left( {{{{n \over 2} - cf} \over f}} \right) \times h, we have
              Median =  7 + \left( {{{50 - 36} \over {40}}} \right) \times 3
                           = 7 + {{14} \over {40}} \times 3
                          = 7 + 1.05 = 8.05
             Calculation of mean

45
             Therefore, Mean, \bar x = {{\Sigma {f_i}{x_i}} \over {\Sigma {f_i}}} = {{832} \over {100}} = 8.32
             Calculation of mode
             The class 7 – 10 has the maximum frequency therefore, this is the modal class.
             Here \ell = 7 , h = 3, {{f_1}} = 40, {{f_0}} = 30 and {{f_2}} = 16
             Now, let us substitute these values in the formula
             Mode = \ell + \left( {{{{f_1} - {f_0}} \over {2{f_1} - {f_0} - {f_2}}}} \right) \times h
                       = 7 + {{40 - 30} \over {80 - 30 - 16}} \times 3
                       = 7 + {{10} \over {34}} \times 3 = 7 + 0.88 = 7.88
             Hence, median = 8.05, mean = 8.32 and mode = 7.88


Q.7     The distribution below gives the weights of 30 students of a class. Find the median weight of the students.
120

Sol.      Let us prepare the following table to compute the median :

119
              We have n = 30  \Rightarrow {n \over 2} = 15

              The cumulative frequency just greater than {n \over 2} is 19 and the corresponding class is 55 – 60.
              Thus, 55 – 60 is the median class such that {n \over 2} = 15,\ell = 55, f = 6, cf = 13 and h = 5
              Substituting these values in the formula
              Median = \ell + \left( {{{{n \over 2} - cf} \over f}} \right) \times h
                          = 55 + \left( {{{15 - 13} \over 6}} \right) \times 5
                          = 55 + {2 \over 6} \times 5 = 55 +1.67 = 56.67
             Hence the median weight = 56.67 kg .