Internal Assessment / Mathematics Test
Statistics & Probability
Time Allowed: 30 Minutes
Maximum Marks: 05
Q.
If the median of the distribution given below is 525 and the total frequency is 100, find the values of the missing frequencies x and y.
[5 Marks]
| Class Interval | Frequency |
|---|---|
| 0 - 100 | 2 |
| 100 - 200 | 5 |
| 200 - 300 | x |
| 300 - 400 | 12 |
| 400 - 500 | 17 |
| 500 - 600 | 20 |
| 600 - 700 | y |
| 700 - 800 | 9 |
| 800 - 900 | 7 |
| 900 - 1000 | 4 |
| Total |
SECTION F — Assertion and Reasoning Type Questions [1 Marks Each]
Directions: Each of the following questions consists of two statements, namely, Assertion (A) and Reason (R). Select the correct option from the choices given below:
- (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c) Assertion (A) is true, but Reason (R) is false.
- (d) Assertion (A) is false, but Reason (R) is true.
-
Q1. Assertion (A): If the median of a data is $15$ and the mean is $12$, then its mode is $21$.
Reason (R): The empirical relationship between mean, median, and mode for any distribution is given by $\text{Mode} = 3(\text{Median}) - 2(\text{Mean})$. -
Q2. Assertion (A): The sum of deviations of all observations of a data from their mean is always zero.
Reason (R): Algebraic sum of deviations from the arithmetic mean ($\sum (x_i - \bar{x})$) is zero by definition. -
Q3. Assertion (A): An ogive is a graphical representation of a cumulative frequency distribution.
Reason (R): Both "less than type" and "more than type" ogives can be used to determine the median graphically. -
Q4. Assertion (A): If each observation of a raw dataset is multiplied by $3$, then the mean of the new dataset also gets multiplied by $3$.
Reason (R): Arithmetic mean is affected by a change of scale (multiplication/division), whereas it is unaffected by a change of origin (addition/subtraction) -
Q5. Assertion (A): The mode of the data $2, 3, 4, 5, 2, 3, 2, 6, 7$ is $2$.
Reason (R): Mode is the value of the observation that has the maximum frequency -
Q6. Assertion (A): If the class intervals of a distribution are $1-10, 11-20, 21-30$, they must be converted into continuous exclusive classes before calculating the median or mode
Reason (R): Formulas for median and mode of grouped data require continuous class boundaries to avoid gaps between intervals -
Q7. Assertion (A): The empirical formula does not always give the exact value of the mode for a multi-modal distribution
Reason (R): The formula $\text{Mode} = 3(\text{Median}) - 2(\text{Mean})$ is strictly designed for moderately skewed distributions. -
Q8. Assertion (A): If the total frequency ($N$) of a grouped data is $50$, then the median position is given by the $25^{\text{th}}$ item.
Reason (R): The median position for any grouped frequency distribution is always located at $N/2$. -
Q9. Assertion (A): The arithmetic mean of first $n$ natural numbers is $\frac{n+1}{2}$.
Reason (R): The sum of the first $n$ natural numbers is given by $\frac{n(n+1)}{2}$ -
Q10. Assertion (A): The "less than" ogive curve always slopes upwards from left to right.
Reason (R): Cumulative frequencies in a "less than" table keep increasing or remain constant as the upper class limits increase. -
Q11. Assertion (A): If the mean of $5$ observations $x, x+2, x+4, x+6, x+8$ is $11$, then the value of $x$ is $7$.
Reason (R): The mean of consecutive terms can be found directly by taking the middle term if the number of observations is odd -
Q12. Assertion (A): The class mark of a class interval is calculated as $\frac{\text{Lower Limit} + \text{Upper Limit}}{2}$.
Reason (R): Class mark represents the exact mid-point of a class interval and is used as a representative value ($x_i$) for calculating the mean. -
Q13. Assertion (A): A "more than" ogive curve slopes downwards from left to right
Reason (R): Cumulative frequencies in a "more than" table decrease as we move from lower class limits to higher class limits -
Q14. Assertion (A): The median of a set of data is always one of the numbers in the dataset.
Reason (R): If the number of observations $n$ is even, the median is the mean of the two middle terms, which may or may not be an element of the original data. -
Q15. Assertion (A): In the formula for the mode of grouped data, $l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$, $f_1$ stands for the frequency of the preceding class.
Reason (R): $f_1$ represents the frequency of the modal class itself, while $f_0$ represents the frequency of the preceding class. -
Q16. Assertion (A): Changing raw data into an assumed mean ($A$) or step-deviation ($u_i$) method alters the final calculated value of the arithmetic mean
Reason (R): Shortcut methods use mathematical transformations that yield the exact same mean as the direct method while reducing computation size. -
Q17. Assertion (A): If the median of a distribution is $30$ and $N = 60$, the cumulative frequency just greater than $N/2$ determines the median class.
Reason (R): The median class is the class whose cumulative frequency is first greater than or equal to $N/2$. -
Q18. Assertion (A): The value of the mode can be determined graphically by drawing a histogram.
Reason (R): Mode is found by drawing lines from the top corners of the modal rectangle to the adjacent rectangles' corners, intersecting at a point whose $x$-coordinate gives the mode. -
Q19. Assertion (A): If each observation in a dataset is increased by $5$, the median of the dataset also increases by $5$.
Reason (R): Measures of central tendency like median, mean, and mode are all affected uniformly by a change of origin -
Q20. Assertion (A): The sum of the frequencies ($\sum f_i$) in a grouped frequency distribution is always equal to the total number of observations ($N$).
Reason (R): Frequency denotes the number of times a particular observation or class interval occurs in the dataset.
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