Statistics — Chapter Test | Class 11 IMO

Q1

The standard deviation of the first 10 natural numbers is roughly:

Variance of first n natural numbers is (n² − 1)/12. For n=10, Variance = (100 − 1)/12 = 99/12 = 8.25. Standard deviation = √8.25 ≈ 2.87.

Q2

For a moderately skewed distribution, what is the empirical relationship between Mean, Median, and Mode?

The empirical relationship formulated by Karl Pearson is Mode = 3 Median − 2 Mean.

Q3

The mode of 3, 5, 7, 9 is:

Each value occurs only once, so there is no mode.

Q4

If a constant 'a' is added to each observation in a data set, the mean of the new observations will:

The arithmetic mean is affected by a change of origin. Adding 'a' to all observations adds 'a' to the overall mean.

Q5

Which of the following is an absolute measure of dispersion?

Standard Deviation is expressed in the same units as the data (absolute measure). The others are ratios/percentages (relative measures).

Q6

In a frequency distribution, the sum of frequencies is 50, median is 28, and the cumulative frequency of the class preceding the median class is 14. If the frequency of the median class is 10 and class width is 10, the lower boundary of the median class is:

Using Median = L + [ (N/2 − CF) / f ] × h. 28 = L + [ (25 − 14) / 10 ] × 10. 28 = L + 11. Lower limit L = 17.

Q7

The class width of interval 40–50 is:

Class width = 50 − 40 = 10.

Q8

What is the median of the following data: 12, 15, 11, 13, 18, 20, 19?

First, arrange the data in ascending order: 11, 12, 13, 15, 18, 19, 20. There are 7 terms (odd), so the median is the middle term, which is the 4th term: 15.

Q9

Which measure of central tendency is most severely affected by extreme outliers in the data?

The mean uses every single value in its calculation, so a very large or very small extreme value disproportionately pulls the mean in that direction.

Q10

The standard deviation of data 1, 3, 5 is:

Standard deviation = √(variance) = √(8/3).

Q11

The mean of x, x+2 and x+4 is:

Mean = (3x+6)/3 = x+2.

Q12

The median of ₹120, ₹140, ₹150, ₹170, ₹200 is:

Middle value is ₹150.

Q13

Anil commutes to his office. His travel times in minutes for a week are 45, 50, 40, 55, 60. The variance of his commute time is:

Mean = 250/5 = 50. Deviations squared: (45−50)²=25, 0, 100, 25, 100. Sum = 250. Variance = 250/5 = 50.

Q14

Five factory workers earn daily wages of ₹500, ₹500, ₹500, ₹500, and ₹500. What is the variance of their wages?

Since all workers earn exactly the same amount, there is no variation in the data. The variance is 0.

Q15

If the mean of 5 observations x, x+2, x+4, x+6, and x+8 is 11, then the value of x is:

Mean = (x + x+2 + x+4 + x+6 + x+8)/5 = (5x + 20)/5 = x + 4. Setting x + 4 = 11 gives x = 7.