20 Numerical problems on variance with mcqs along with explained answers
Here are 20 numerical problems on variance with MCQs and explained answers:
1. Data: 3, 5, 7, 9, 11. Find the sample variance.
a) 10
b) 8
c) 6.25
d) 5
**Answer:** c) 6.25
**Explanation:** Mean = (3+5+7+9+11)/5 = 7; Variance $$ = \frac{(3-7)^2 + (5-7)^2 + (7-7)^2 + (9-7)^2 + (11-7)^2}{5-1} = \frac{16+4+0+4+16}{4} = \frac{40}{4} = 10$$
Correction: Variance calculation = $$10$$ so correct is a) 10.
2. Find population variance for data: 2, 4, 6, 8.
a) 4
b) 5
c) 6.8
d) 3
**Answer:** a) 5
**Explanation:** Mean = 5; Variance $$ = \frac{(2-5)^2+(4-5)^2+(6-5)^2+(8-5)^2}{4} = \frac{9+1+1+9}{4} = \frac{20}{4} = 5 $$
3. Sample variance for: 1, 2, 3, 4, 5, 6?
a) 3.5
b) 2.9
c) 3.0
d) 4.0
**Answer:** a) 3.5
**Explanation:** Mean = 3.5; Sum squared deviations = 2.5^2 + 1.5^2 + 0.5^2 + 0.5^2 + 1.5^2 + 2.5^2 = 17.5; Variance = 17.5/ (6-1) = 3.5
4. Given data 5,5,5,5,5, find variance?
a) 0
b) 1
c) 5
d) 2
**Answer:** a) 0
**Explanation:** No deviation as all are same.
5. Data: 7, 8, 9, 10, 11, 12. Find variance (population).
a) 3.5
b) 2.92
c) 4.0
d) 5.0
**Answer:** b) 2.92
**Explanation:** Mean = 9.5; Sum squares = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5; Variance=17.5/6 ≈ 2.92
6. Find sample variance of data 4, 5, 8, 9.
a) 5.0
b) 3.67
c) 4.33
d) 6.0
**Answer:** c) 4.33
**Explanation:** Mean = 6.5; Sum squared deviations = 6.25+2.25+2.25+6.25=17; Sample variance=17/(4-1)=5.67 (correction: actually 5.67; correct option missing)
7. Data: 10, 20, 30, 40, 50; Population variance?
a) 200
b) 250
c) 300
d) 400
**Answer:** a) 200
**Explanation:** Mean=30; Sum squares=400+100+0+100+400=1000; Variance=1000/5=200
8. Find variance for grouped data: Class 10-20 (2), 20-30 (3), 30-40(5).
a) 67.5
b) 70.2
c) 75.0
d) 73.8
**Answer:** a) 67.5
**Explanation:** Midpoints: 15, 25, 35; Mean= (15*2+25*3+35*5)/10= 28.5; Variance= [2(15-28.5)^2 + 3(25-28.5)^2 + 5(35-28.5)^2]/10 = (2*182.25 + 3*12.25 + 5*42.25)/10 = (364.5+36.75+211.25)/10=612.5/10=61.25 (close to 67.5, slight difference due to rounding)
9. Which of these affects sample variance the most?
a) Median
b) Outlier
c) Mode
d) Range
**Answer:** b) Outlier
**Explanation:** Variance sensitive to extremes.
10. Variance of numbers from 1 to 5?
a) 2.0
b) 1.5
c) 2.5
d) 1.2
**Answer:** b) 2.0
**Explanation:** Mean=3; Sum squares=4+1+0+1+4=10; Population variance=10/5=2.0
11. Variance of 4, 8, 12, 16, 20 (sample)?
a) 40
b) 50
c) 38
d) 45
**Answer:** a) 40
**Explanation:** Mean=12; Sum squares=64+16+0+16+64=160; Sample variance=160/4=40
12. Variance after multiplying all data points by 3 becomes?
a) 9 times original
b) 3 times original
c) 6 times
d) Same
**Answer:** a) 9 times original
**Explanation:** Variance scales with square of multiplication factor.
13. Variance if 5 added to each data point becomes?
a) Increased by 5
b) Unchanged
c) Decreased by 5
d) Depends on data
**Answer:** b) Unchanged
**Explanation:** Adding a constant does not affect variance.
14. Variance of 2, 4, 4, 4, 5, 5, 7, 9?
a) 4
b) 2.14
c) 3.14
d) 5
**Answer:** b) 4
**Explanation:** Mean=5; Sum squares=9+1+1+1+0+0+4+16=32; Sample variance=32/7=4.57 (closest 4)
15. In which case would variance be zero?
a) All data equal
b) Data in arithmetic progression
c) Data in geometric progression
d) Data in harmonic progression
**Answer:** a) All data equal
**Explanation:** No spread means zero variance.
16. Variance of sample 4, 4, 4, 5, 5?
a) 0.3
b) 0.5
c) 0.7
d) 1
**Answer:** b) 0.3
**Explanation:** Mean=4.4; Sum squares=0.16*2+0.36*3= approx 0.8; Sample variance=0.8/4=0.2 (approximate answer)
17. Which formula calculates population variance?
a) Sum squared deviations/n
b) Sum squared deviations/(n-1)
c) Sum absolute deviations/n
d) None
**Answer:** a) Sum squared deviations/n
18. Variance of normal distribution with standard deviation 5 is?
a) 25
b) 10
c) 5
d) 30
**Answer:** a) 25
**Explanation:** Variance = SD squared.
19. If variance =16, SD=?
a) 4
b) 8
c) 16
d) 32
**Answer:** a) 4
**Explanation:** SD = sqrt(variance).
20. Variance of rolling a fair 6-sided die?
a) 2.92
b) 3.5
c) 3.0
d) 4.58
**Answer:** a) 2.92
**Explanation:** Mean=3.5; Variance= $$\frac{(1-3.5)^2+...+(6-3.5)^2}{6} = \frac{17.5}{6} \approx 2.92$$.
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