CBSE Class 9 Mathematics Measures of Central Tendency Assignment Set 02

IX - Mathematics Assignment No - 04 - Statistics

Measure of Central Tendency

M. C. Q. Type

Choose the Correct Ans. from 4 choices.

Question 1. The mean of 1st ten natural numbers is
(i) 5.5
(ii) 4.5
(iii) 6.5
(iv) 5
Answer: (i) 5.5
The first 10 natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Sum = \(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55\)
Number of observations = 10
Mean = \(\frac{55}{10} = 5.5\)
In simple words: Add the first ten numbers and divide the sum by 10 to find the mean.

Exam Tip: Use the formula for the sum of the first \(n\) natural numbers: \(\frac{n(n+1)}{2}\).

Question 2. Find mean of prime numbers from 1 to 10
(i) 3.25
(ii) 4.25
(iii) 5.25
(iv) 5
Answer: (ii) 4.25
Prime numbers from 1 to 10 are 2, 3, 5, and 7.
Sum = \(2 + 3 + 5 + 7 = 17\)
Number of prime numbers = 4
Mean = \(\frac{17}{4} = 4.25\)
In simple words: Identify the prime numbers up to 10, add them up, and divide by 4.

Exam Tip: Remember that 1 is neither prime nor composite, so prime numbers from 1 to 10 start with 2.

Question 3. If mean of 3, 5, 7, 10, 12, \(x\) is 10. The value of \(x\) is
(i) 25
(ii) 24
(iii) 23
(iv) 22
Answer: (iii) 23
Number of observations = 6
Mean = 10
Sum of observations = \(10 \times 6 = 60\)
Sum = \(3 + 5 + 7 + 10 + 12 + x = 37 + x\)
\(37 + x = 60\)
\(x = 60 - 37 = 23\)
In simple words: Multiply the mean by the total count to get the total sum, then subtract the known numbers to find \(x\).

Exam Tip: Always check the total count of numbers carefully to find the correct total sum.

Question 4. Median of 3, 5, 2, 10, 15, 17, 19 is
(i) 15
(ii) 17
(iii) 5
(iv) 10
Answer: (iv) 10
Arrange the given data in ascending order: 2, 3, 5, 10, 15, 17, 19
Number of terms \(n = 7\) (which is odd)
Median = Value of the \(\left(\frac{n+1}{2}\right)\)-th term
Median = Value of the \(\left(\frac{7+1}{2}\right)\)-th = 4th term
4th term = 10
In simple words: Sort the numbers from smallest to largest and pick the exact middle number.

Exam Tip: Never forget to arrange data in ascending or descending order before finding the median.

Question 5. The mode of 5, 7, 9, 4, 7, 8, 7, 3 is
(i) 7
(ii) 5
(iii) 3
(iv) 9
Answer: (i) 7
Frequency of each number:
3 appears 1 time
4 appears 1 time
5 appears 1 time
7 appears 3 times
8 appears 1 time
9 appears 1 time
Since 7 has the highest frequency, Mode = 7.
In simple words: The mode is the number that repeats most often.

Question 6. If the mean of 10, 12, 18, 13, \(p\), 17 is 15, the value of \(p\) is
(i) 21
(ii) 20
(iii) 19
(iv) 18
Answer: (ii) 20
Number of observations = 6
Mean = 15
Total sum = \(15 \times 6 = 90\)
Sum of given numbers = \(10 + 12 + 18 + 13 + p + 17 = 70 + p\)
\(70 + p = 90\)
\(p = 90 - 70 = 20\)
In simple words: Find the total sum expected from the mean, subtract the known numbers, and the remainder is \(p\).

Exam Tip: Double-check your addition of the known numbers before subtracting.

Question 7. The following data is arranged in ascending order: 59, 62, 65, \(x\), \(x+2\), 72, 80, 90. If the median of the data is 69. The value of \(x\) is
(i) 68
(ii) 69
(iii) 70
(iv) 71
Answer: (i) 68
Total number of observations \(n = 8\) (even)
Median = Mean of the \(\left(\frac{n}{2}\right)\)-th and \(\left(\frac{n}{2} + 1\right)\)-th terms
Median = Mean of the 4th and 5th terms
4th term = \(x\), 5th term = \(x+2\)
Median = \(\frac{x + (x+2)}{2} = \frac{2x+2}{2} = x + 1\)
Given Median = 69
\(x + 1 = 69\)
\(x = 68\)
In simple words: For an even number of items, average the two middle values to find the median and solve for \(x\).

Exam Tip: When \(n\) is even, remember to divide the sum of the two middle terms by 2.

Question 8. The mode of the following data is: 15, 14, 19, 20, 14, 15, 16, 14, 15, 18, 14, 15, 19, 15, 17
(i) 18
(iii) 17
(iii) 16
(iv) 15
Answer: (iv) 15
Counting the frequencies:
14 appears 4 times
15 appears 5 times
16 appears 1 time
17 appears 1 time
18 appears 1 time
19 appears 2 times
20 appears 1 time
Since 15 has the maximum frequency (5 times), Mode = 15.
In simple words: Count how many times each number appears; the one with the highest count is the mode.

Exam Tip: Grouping or tallying numbers in ascending order helps prevent counting mistakes.

Question 9. If the mean of 10 observations is 20 and that of another 15 observations is 16, the mean of all 25 observations is
(i) 17.0
(ii) 17.4
(iii) 17.6
(iv) 17.9
Answer: (iii) 17.6
Sum of first 10 observations = \(10 \times 20 = 200\)
Sum of other 15 observations = \(15 \times 16 = 240\)
Total sum of all 25 observations = \(200 + 240 = 440\)
Combined mean = \(\frac{440}{25} = \frac{1760}{100} = 17.6\)
In simple words: Find the sum of both groups, add them together, and divide by the total count of 25 observations.

Exam Tip: Combined mean formula: \(\bar{x}_{12} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}\).

Question 10. The demand of different shirt sizes are given below

Find the modal shirt size
(i) 44
(ii) 39
(iii) 42
(iv) 15
Answer: (ii) 39
The highest frequency (number of shirts demanded) is 39.
The corresponding shirt size for frequency 39 is 39.
Therefore, the modal shirt size is 39.
In simple words: Look for the largest number in the frequency row and find the shirt size right above it.

Exam Tip: The mode is the value of the variable, not the maximum frequency itself.