Answer to Question #197156 in Statistics and Probability for matheus

Question

The following table is a frequency table of the scores obtained in a QTM quiz competition. Classes Frequency(f) 10 - 20 5 20 - 30 7 30 - 40 10 40 - 50 16 50 - 60 2 Total 40 Use the information in the table above and answer the following questions: 2.1 Find the mean [2] 2.2 Find the median [3] 2.3 Find modal score. [3] 2.4 Find the range of the scores. [1] 2.5 Find the variance of the scores. [4] 2.6 Find the standard deviation. [2] 2.7 Compute the coefficient of variation. [2] 2.8 Compute the interquartile range and the quartile deviation. 

Answer 1

(1) Mean $\bar{x} =\dfrac{\sum xf}{\sum f}=\dfrac{1430}{40}=35.75$

(2) Median class corresponds to CF $= \dfrac{N}{2}=20$

Median class =(30-40)

$l=30,h=10, f=10,cf=22$

$M=l+(\dfrac{\frac{N}{2}-cf}{f})\times h\\[9pt]M=30+\dfrac{20-22}{10}\times (10)\\[9pt]M=30-2=28$

(3) Modal score-

$M=l+(\dfrac{f_o-f_1}{2f_o-f_1-f_2})h\\[9pt]\Rightarrow M=30+\dfrac{10-7}{2(10)-16-7})\times 10\\[9pt]\Rightarrow M=30-\dfrac{3}{4}(10)\\[9pt]\Rightarrow M=22.5$

(4) Range of score = Maximum value-minimum value$=60-10=50$

(5) Variance of square $\sigma^2=\dfrac{(x-\bar{x})^2}{n}=\dfrac{1002.813}{40}=25.07$

(6) Standard deviation $=\sqrt{\sigma^2}=\sqrt{25.07}=5$

(7) Coefficient of variation $=\dfrac{\sigma}{\bar{x}}=\dfrac{5}{35.75}=0.14$

(8) First quartile -

$Q_1=\text{size of }\dfrac{(40+1)}{5}^{th} \text{ term }= \text{ size of } 8.2^{th} \text{ term }=25$

$Q_3=\text{ size of } \dfrac{3(40+1)}{5}^{th} \text{ term } =\text{size of } 24.6^{th} term = 45$

Inner Quartile Range $= Q_3-Q_1=45-25=20$

Quartile deviation =$\dfrac{Q_3-Q_1}{2}=\dfrac{45-25}{2}=\dfrac{20}{2}=10$