Answer to Question #249251 in Statistics and Probability for KCK

Question

The following table give the grouped data on the weights of all 100 babies born at Matero level 1
clinic last year.
Weight (Units) Number of
babies
3 to less than 5 5
5 to less than 7 30
7 to less than 9 40
9 to less than 11 20
11 to less than 13 5
(a) Calculate the mean
(b) Calculate the variance and standard deviation
(c) Calculate the mode
(d) Calculate the median
(e) Calculate the coefficient of variation

Answer 1

SOLUTION

I drew a frequency table to represent the data and ease the calculation

(a) Calculate the mean

$Mean(\mu)=\frac{\Sigma{f(x)}}{n}=\frac{780}{100}=7.8$

Answer $=7.8$

(b) Calculate the variance and standard deviation

$\sigma^2=\frac{\Sigma{f(x)^2}-(\Sigma{f(x))^2}}{n}$

$=\frac{6440-(780)^2}{100}=3.56\\ \sigma=\sqrt{3.56}=1.8868$

Answer $=1.8868$

(c) Calculate the mode

Maximum frequency is 40. The mode class is 7-9.

$L=$ lower boundary point of mode class $=7$

$f_1=$ frequency of the mode class$=40$

$f_0=$ frequency of the preceding class $=30$

$f_2=$ frequency of the succeeding class $=20$

$c=$ class length of mode class $=2$

$Mode=Z=L+\Big(\frac{f_1-f_0}{2f_1-f_0-f_2}\Big)*c\\=7+\Big(\frac{40-30}{2(40)-30-20}\Big)*2=7.6667$

Answer $=7.6667$

(d) Calculate the median

value of $(n/2)th$  observation $=$ value of $(100/2)th$  observation $=$

value of $(50)th$  observation

From the column of cumulative frequency $cf,$ we find that the $(50)th$  observation lies in the class 7-9.

The median class is 7-9.

$L=$ lower boundary point of median class $=7$

$n=$ Total frequency $=100$

$cf=$ Cumulative frequency of the class preceding the median class $=35$

$f=$ Frequency of the median class $=40$

$c=$ class length of median class $=2$

$median=M=L+(\dfrac{\dfrac{n}{2}-cf}{f})\cdot c$

$$=7+(\dfrac{50-35}{40})\cdot 2=7.75$$

Answer $=7.75$

(e) Calculate the coefficient of variation

coefficient of variation $=\dfrac{\sigma}{\bar{x}}\cdot100\%=\dfrac{1.8868}{7.8}\cdot100\%\approx24.19\%$

Answer $=24.19\%$