Quantitative Aptitude

Statistics

606 Questions
All Ungrouped data dispersion and consistency 209 Unknown terms from given average 124 Grouped data central tendencies 101 Merging means and variances of groups 53 Replacement change in average 27 Alligation and weighted average 21 Data interpretation from tables and charts 20 Net successive percentage change 19 Grouped data variance via step deviation 18 Combination of multiple ratios 1

Statistics Questions

Multiple choice

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is:

  1. 9

  2. 3

  3. 7

  4. 5
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Mean = (sum)/8 = 10, so sum = 80. Sum of 6 given = 5+7+10+12+14+15 = 63. Remaining two (x, y) sum to 17. Variance = (sum(xi^2)/8) - mean^2 = 13.5. Sum(xi^2)/8 - 100 = 13.5, so sum(xi^2) = 113.5 * 8 = 908. Sum of squares of 6 given = 25+49+100+144+196+225 = 739. x^2 + y^2 = 908 - 739 = 169. We have x+y=17 and x^2+y^2=169. (x+y)^2 = x^2+y^2+2xy => 289 = 169 + 2xy => 2xy = 120 => xy = 60. Roots of t^2 - 17t + 60 = 0 are 12 and 5. Difference = 12-5 = 7.

Multiple choice

A grouped data is shown below Class Interval Frequency Cumulative frequency 0 – 10 4 4 10 – 20 a 4+a 20 – 30 4 8+a 30 – 40 b 8+a+b 40 – 50 5 13+a+b If the median of the grouped data is 22.50 and the total frequency is 20, then what is the value of 𝑎?

  1. 1

  2. 2

  3. 4

  4. 5
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Total frequency = 20. Cumulative frequencies: 4, 4+a, 8+a, 8+a+b, 13+a+b. So 13+a+b = 20, a+b = 7. Median = 22.5, which is in the 20-30 class. Median formula: L + ((N/2 - cf) / f) * h = 20 + ((10 - (4+a)) / 4) * 10 = 22.5. (6-a)/4 = 0.25, 6-a = 1, a = 5.

Multiple choice

For a series of 5 numbers, the mean is 20, the median and mode are both 10, the highest number in the series is 50 and one of the numbers is 20. Find the number of times the modal value appears in the series.

  1. 4

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

With 5 numbers, mean 20 implies sum = 100. Numbers are {x1, x2, x3, x4, x5}. Median is 10, so the third number is 10. Mode is 10, so at least two numbers must be 10. We have {x1, x2, 10, x4, 50}. Since the mode is 10, 10 must appear more than once. If 10 appears twice, the sum is x1 + x2 + 10 + 10 + 50 = 100, so x1 + x2 = 30. If 10 appears three times, the set is {x1, 10, 10, 10, 50}, sum = 80 + x1 = 100, so x1 = 20. This fits all conditions.

Multiple choice

If all the observations are divided by 3, then

  1. new standard deviation would be also divided by 3

  2. new standard deviation would be multiplied by 3

  3. new standard deviation would be increased by 3

  4. new standard deviation would be decreased by 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Standard deviation is a measure of dispersion. If all observations are divided by a constant k, the new standard deviation is the original standard deviation divided by |k|.

Multiple choice

Consider three observations a, b and c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

  1. b2 = 3(a2 + c2) + 9d2

  2. b2 = a2 + c2 + 3d2

  3. b2 = 3(a2 + c2 + d2)

  4. b2 = 3(a2 + c2) - 9d2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The standard deviation d of (a+2, b+2, c+2) is the same as the standard deviation of (a, b, c). The variance d^2 = [(a-m)^2 + (b-m)^2 + (c-m)^2] / 3, where m = (a+b+c)/3. Since b = a+c, m = (a+a+c+c)/3 = (a+c)/3 = b/3. Substituting m = b/3 into the variance formula: 3d^2 = (a - b/3)^2 + (b - b/3)^2 + (c - b/3)^2 = (a - b/3)^2 + (2b/3)^2 + (c - b/3)^2. Expanding this leads to the relation b^2 = 3(a^2 + c^2) - 9d^2.

Multiple choice

In a series of 2n observations, half of them are equal to a and remaining half are equal to -a. Also by adding a constant b in each of these observations, the mean and standard deviation of the new set become 5 and 20, respectively. Then the value of a2 + b2 is equal to:

  1. 425

  2. 650

  3. 250

  4. 925
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Observations are n of 'a' and n of '-a'. Mean = (na - na) / 2n = 0. Adding b makes mean = b = 5. Variance of original set = (n*a^2 + n*(-a)^2) / 2n = a^2. Standard deviation = 20, so variance = 400. Thus a^2 = 400. Then a^2 + b^2 = 400 + 25 = 425.

Multiple choice

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are:

  1. 10, 11

  2. 8, 13

  3. 1, 20

  4. 3, 18
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Mean = 6.5, Sum = 39. Sum of 4 observations = 18. Remaining two sum to 21. Variance = 10.25 = (Sum(x^2)/6) - (6.5)^2. Sum(x^2) = 6 * (10.25 + 42.25) = 315. Sum of squares of 4 observations = 4+16+25+49 = 94. Remaining two squares sum to 221. x+y=21, x^2+y^2=221. Solving gives 10 and 11.