Statistics Set-1

Q1. The mean of five observations is $12$. If one of the observations equal to $10$ is replaced by $22$, the new mean becomes:

Q2. The following grouped distribution has a missing frequency $x$ in the class $20$–$30$.

If the median of the distribution is $22$, the value of $x$ is:

Q3. The mean of $40$ numbers is $18$. Ten more numbers, each equal to $k$, are added and the mean of all $50$ numbers becomes $20$. The value of $k$ is:

Q4. A frequency distribution has unequal class widths as shown:

Using class midpoints to estimate the mean of the distribution gives:

Q5. For the following grouped data with equal class width $10$, the modal class is $20$–$30$.

Using the formula for mode for grouped data, the mode (approximately) is:

Q6. A teacher records the marks (out of 50) obtained by 10 students as a frequency distribution: marks 10, 20, 30, 40 with frequencies 3, 2, 4, 1 respectively. Find the mean mark of the students.

Q7. The following grouped frequency distribution gives the number of packets sold in different price ranges in a week: classes (Rs): $0$–$10$:5, $10$–$20$:8, $20$–$30$:10, $30$–$40$:7. Estimate the median using linear interpolation (use class boundaries). (Use $N/2$ rule for median position.)

Q8. Two sections of a class have 12 and 18 students with mean marks $40$ and $46$ respectively. If the two sections are combined, what is the mean mark of the combined group?

Q9. Group A has $20$ observations with mean $50$ and standard deviation $4$. Group B has $30$ observations with mean $54$ and standard deviation $6$. If both groups are combined, the combined mean is $52.4$. Find the combined standard deviation (to two decimal places), using the relation

Q10. A continuous grouped distribution has class-intervals and frequencies as: $0$–$9$:2, $10$–$19$:8, $20$–$29$:12, $30$–$39$:6, $40$–$49$:2 (total $30$ observations). Using quartile interpolation with $Q_1$ at $N/4$ and $Q_3$ at $3N/4$, determine how many outliers the dataset has according to the $1.5\times\text{IQR}$ rule.

Q11. A frequency distribution is given by observations $x=1,2,3,4,5$ with corresponding frequencies $f=3,5,7,4,1$. Using the formula

find the mean of the distribution.

Q12. For the following grouped data, find the median using the formula

where $l$ is the lower boundary of median class, $N$ total frequency, $c$ cumulative frequency before median class, $f$ frequency of median class and $h$ class width.
Classes: $0$–$10(5),\;10$–$20(9),\;20$–$30(16),\;30$–$40(10)$.

Q13. For the grouped frequency distribution below, calculate the mode using the formula

Classes: $0$–$10(5),\;10$–$20(7),\;20$–$30(12),\;30$–$40(6)$.

Q14. A frequency distribution has unequal class widths: classes and frequencies are $0$–$10:30,\;10$–$20:20,\;20$–$40:50$. If a histogram is drawn (area of a class rectangle proportional to frequency), which class will contain the modal value (i.e., the most frequent value per unit class width)?

Q15. Assertion (A): For a distribution with mean $50$ and median $60$, the distribution is negatively skewed.
Reason (R): In a negatively skewed distribution, the mode is greater than the median.
Choose the correct option.