Mathematics : Statistics And Probability : Measures of Dispersion : Exercise Questions with Answers
Largest value:
$$ L = 125 $$
Smallest value:
$$ S = 63 $$
Range:
$$ = L-S $$
$$ =125-63 $$
$$ =62 $$
Coefficient of range:
$$ =\frac{L-S}{L+S} $$
$$ =\frac{125-63}{125+63} $$
$$ =\frac{62}{188} $$
$$ \approx0.33 $$
Answer
Range:
$$ 62 $$
Coefficient of range:
$$ 0.33 $$
Largest value:
$$ 61.4 $$
Smallest value:
$$ 13.6 $$
Range:
$$ 61.4-13.6 $$
$$ =47.8 $$
Coefficient of range:
$$ =\frac{47.8}{61.4+13.6} $$
$$ =\frac{47.8}{75} $$
$$ \approx0.64 $$
Answer
Range:
$$ 47.8 $$
Coefficient of range:
$$ 0.64 $$
Range:
$$ =L-S $$
Given:
$$ 36.8=L-13.4 $$
$$ L=36.8+13.4 $$
$$ L=50.2 $$
Answer
$$ 50.2 $$
Range:
$$ =L-S $$
After identifying the largest and smallest observations from the table,
$$ \text{Range}=250 $$
Answer
$$ 250 $$
Pages yet to complete:
$$ 28,25,23,30,27,24,25,23 $$
Mean:
$$ \bar{x}=\frac{28+25+23+30+27+24+25+23}{8} $$
$$ =\frac{205}{8} $$
$$ =25.625 $$
Using
$$ \sigma=\sqrt{\frac{\sum(x-\bar{x})^2}{n}} $$
After calculation,
$$ \sigma \approx 2.34 $$
Answer
$$ 2.34 $$
Mean:
$$ \bar{x}=300 $$
Using variance formula,
$$ \sigma^2=\frac{\sum(x-\bar{x})^2}{n} $$
$$ =222.22 $$
Standard deviation:
$$ \sigma=\sqrt{222.22} $$
$$ \approx14.91 $$
Answer
Variance:
$$ 222.22 $$
Standard deviation:
$$ 14.91 $$
Total strikes in 12 hours:
$$ 1+2+3+\cdots+12 $$
$$ =\frac{12(13)}{2} $$
$$ =78 $$
In one day:
$$ 2\times78=156 $$
Answer
Total strikes in one day:
$$ 156 $$
Natural numbers:
$$ 1,2,3,\ldots,21 $$
Mean:
$$ \bar{x}=\frac{21+1}{2}=11 $$
Using standard deviation formula,
$$ \sigma=\sqrt{\frac{\sum(x-\bar{x})^2}{n}} $$
$$ \sigma\approx6.05 $$
Answer
$$ 6.05 $$
Subtracting a constant from every observation does not change standard deviation.
Answer
$$ 4.5 $$
New standard deviation:
$$ =\frac{3.6}{3} $$
$$ =1.2 $$
New variance:
$$ =(1.2)^2 $$
$$ =1.44 $$
Answer
Variance:
$$ 1.44 $$
Standard deviation:
$$ 1.2 $$
$$ 7.76 $$
$$ 14.6 $$
$$ 6 $$
$$ 1.24 $$
Corrected mean:
$$ 60.5 $$
Corrected standard deviation:
$$ 14.61 $$
Answer
Mean:
$$ 60.5 $$
Standard deviation:
$$ 14.61 $$
Let remaining observations be $x$ and $y$.
Mean:
$$ \frac{2+4+10+12+14+x+y}{7}=8 $$
$$ 42+x+y=56 $$
$$ x+y=14 $$
Using variance formula:
$$ \sigma^2=\frac{\sum x^2}{n}-\bar{x}^2 $$
$$ 16=\frac{2^2+4^2+10^2+12^2+14^2+x^2+y^2}{7}-64 $$
$$ x^2+y^2=100 $$
Using:
$$ (x+y)^2=x^2+y^2+2xy $$
$$ 196=100+2xy $$
$$ xy=48 $$
Thus,
$$ t^2-14t+48=0 $$
$$ (t-6)(t-8)=0 $$
Hence,
$$ t=6,8 $$
Answer
$$ 6 \text{ and } 8 $$
Answer Key
| Q.No | Answer | |---|---| | 1(i) | 62 ; 0.33 | | 1(ii) | 47.8 ; 0.64 | | 2 | 50.2 | | 3 | 250 | | 4 | 2.34 | | 5 | 222.22 ; 14.91 | | 6 | 6.9 | | 7 | 6.05 | | 8 | 4.5 | | 9 | 1.44 ; 1.2 | | 10 | 7.76 | | 11 | 14.6 | | 12 | 6 | | 13 | 1.24 | | 14 | 60.5 ; 14.61 | | 15 | 6 and 8 |
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