CBSE Class 9 Mathematics Measures of Central Tendency Assignment Set 04

Measures of Central Tendency

Question 1. If the mean of 10, 12, 18, 13, p and 17 is 15. Find the value of p.
Answer: Total number of observations = 6
Sum of observations = \(10 + 12 + 18 + 13 + p + 17 = 70 + p\)
Mean = \(\frac{\text{Sum of observations}}{\text{Total number of observations}}\)
\(15 = \frac{70 + p}{6}\)
\(70 + p = 15 \times 6\)
\(70 + p = 90\)
\(p = 90 - 70 = 20\)
In simple words: Add all the numbers including p, divide by the total count (6), and equate it to 15 to solve for p.

Exam Tip: Always double-check your arithmetic addition of the given numbers before solving for the unknown variable.

Question 2. The following observations have been arranged in the ascending order. If the median of the data is 63. Find x
29, 32, 48, 50, x, x + 2, 72, 78, 84, 95
Answer: Total number of observations \(n = 10\) (which is even).
The median for an even number of observations is the mean of the \(\left(\frac{n}{2}\right)^{\text{th}}\) and \(\left(\frac{n}{2} + 1\right)^{\text{th}}\) terms.
Here, \(n = 10\), so the terms are the \(5^{\text{th}}\) and \(6^{\text{th}}\) terms.
\(5^{\text{th}}\) term = \(x\)
\(6^{\text{th}}\) term = \(x + 2\)
Given Median = 63
\(\frac{x + (x + 2)}{2} = 63\)
\(\frac{2x + 2}{2} = 63\)
\(x + 1 = 63\)
\(x = 62\)
In simple words: Since there are 10 numbers, take the average of the 5th and 6th terms and set it equal to 63 to find x.

Exam Tip: Verify that the given sequence is strictly in ascending order before picking the middle terms.

Question 3. Find mean of the following data

Answer: Construct a frequency table to calculate \(f_i x_i\) and sums:
\(\sum f_i = 4 + 6 + 8 + 18 + 6 + 5 + 3 = 50\)
\(\sum f_i x_i = (10 \times 4) + (15 \times 6) + (20 \times 8) + (25 \times 18) + (30 \times 6) + (35 \times 5) + (40 \times 3)\)
\(\sum f_i x_i = 40 + 90 + 160 + 450 + 180 + 175 + 120 = 1215\)
Mean \(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\)
\(\bar{x} = \frac{1215}{50} = 24.3\)
In simple words: Multiply each data value by its frequency, sum them up, and divide by the total frequency to get the mean.

Exam Tip: Make a proper table with columns for \(x_i\), \(f_i\), and \(f_i x_i\) to avoid calculation errors.

Question 4. If the mean of 10 observations is 20 and that of another 15 observations is 16. Find the mean of all 25 observations.
Answer: Sum of first 10 observations = \(10 \times 20 = 200\)
Sum of next 15 observations = \(15 \times 16 = 240\)
Total sum of all 25 observations = \(200 + 240 = 440\)
Mean of all 25 observations = \(\frac{440}{25} = 17.6\)
In simple words: Find the total sum for both groups, add them together, and divide by the total number of items (25).

Exam Tip: Total sum is always equal to (Number of items) multiplied by (Mean).

Question 5. The mean of 13 items is 14. If the mean of first 7 items is 12 and that of the last 7 items is 16. Find the 7th item.
Answer: Sum of all 13 items = \(13 \times 14 = 182\)
Sum of first 7 items = \(7 \times 12 = 84\)
Sum of last 7 items = \(7 \times 16 = 112\)
The 7th item is included in both the first 7 items and the last 7 items. Therefore:
Sum of first 7 + Sum of last 7 = Total sum + 7th item
\(84 + 112 = 182 + (\text{7th item})\)
\(196 = 182 + (\text{7th item})\)
\(\text{7th item} = 196 - 182 = 14\)
In simple words: Add the sums of the first 7 and last 7 items, then subtract the total sum of all 13 items to isolate the overlapping 7th item.

Exam Tip: Be careful with overlapping elements when combining sums of sub-groups.

Question 6. Find the mode of 3, 4, 6, 5, 7, 5, 8, 5, 3, 2. If one observation 5 is replaced by 3, will the modal value change? If yes, find it.
Answer: Original data: 3, 4, 6, 5, 7, 5, 8, 5, 3, 2
Frequency of 3 = 2
Frequency of 5 = 3 (Maximum frequency)
Original mode = 5
Now, if one observation 5 is replaced by 3, the new data has one less 5 and one more 3:
Frequency of 3 becomes \(2 + 1 = 3\)
Frequency of 5 becomes \(3 - 1 = 2\)
Both 3 and 5 now have a maximum frequency of 3.
Yes, the modal value will change. The new data is bimodal with modes 3 and 5.
In simple words: Count how often each number appears. When a 5 becomes a 3, 3 and 5 tie for the most frequent number, so both become the mode.

Exam Tip: A dataset can have more than one mode (bimodal or multimodal) if multiple values share the highest frequency.

Question 7. For what value of x, the mode of the following data is 10?
7, 6, 10, 8, 10, x, 7, 5, 12, 14
Answer: Count the frequencies of the given numbers:
7 appears 2 times
10 appears 2 times
For the mode of the data to be 10, 10 must appear more times than any other number.
Since 7 and 10 both currently appear twice, in order for 10 to be the unique mode, the unknown value \(x\) must be equal to 10 so that 10 appears 3 times.
\(x = 10\)
In simple words: For 10 to be the most frequent number, x must be 10 so that 10 appears three times.

Exam Tip: Mode is the observation with the highest frequency of occurrence.

Question 8. For the numbers 23, 25, 21, 30, 37, 46, 49, 41, show that \(\sum_{i=1}^{8} (x_i - \bar{x}) = 0\)
Answer: Given observations: 23, 25, 21, 30, 37, 46, 49, 41 (Total \(n = 8\))
Sum of observations \(\sum x_i = 23 + 25 + 21 + 30 + 37 + 46 + 49 + 41 = 272\)
Mean \(\bar{x} = \frac{272}{8} = 34\)
Now, calculate \(\sum_{i=1}^{8} (x_i - \bar{x})\):
\(= (23 - 34) + (25 - 34) + (21 - 34) + (30 - 34) + (37 - 34) + (46 - 34) + (49 - 34) + (41 - 34)\)
\(= (-11) + (-9) + (-13) + (-4) + 3 + 12 + 15 + 7\)
\(= (-37) + 37 = 0\)
Hence proved, \(\sum_{i=1}^{8} (x_i - \bar{x}) = 0\)
In simple words: The sum of deviations of all observations from their arithmetic mean is always equal to zero.

Exam Tip: Remember this fundamental property: the sum of deviations from the mean is always zero for any dataset.

Question 9. Find the median of the following data 31, 15, 19, 12, 30, 45, 60, 50, 25, 40. If 40 is replaced by 20, what will be the new median?
Answer: Part 1: Median of the original data
Original data: 31, 15, 19, 12, 30, 45, 60, 50, 25, 40 (Total \(n = 10\))
Arrange in ascending order: 12, 15, 19, 25, 30, 31, 40, 45, 50, 60
Since \(n = 10\) (even), median is the mean of the \(5^{\text{th}}\) and \(6^{\text{th}}\) terms.
\(5^{\text{th}}\) term = 30
\(6^{\text{th}}\) term = 31
Original Median = \(\frac{30 + 31}{2} = 30.5\)

Part 2: New median when 40 is replaced by 20
Replace 40 with 20 in the sorted original data: 12, 15, 19, 20, 25, 30, 31, 45, 50, 60
Again, \(n = 10\) (even), median is the mean of the \(5^{\text{th}}\) and \(6^{\text{th}}\) terms.
\(5^{\text{th}}\) term = 25
\(6^{\text{th}}\) term = 30
New Median = \(\frac{25 + 30}{2} = 27.5\)
In simple words: Sort the numbers and find the average of the two middle numbers. Do the same after replacing 40 with 20.

Exam Tip: Always re-sort the dataset in ascending order after changing any value before finding the median.

Question 10. Verify the Empirical formula for the following data
2, 3, 2, 5, 6, 3, 4, 7, 4, 9
Answer: Empirical formula connecting Mean, Median, and Mode:
\(\text{Mode} = 3(\text{Median}) - 2(\text{Mean})\)

Given data: 2, 3, 2, 5, 6, 3, 4, 7, 4, 9 (Total \(n = 10\))
1. Find Mean:
Sum = \(2 + 3 + 2 + 5 + 6 + 3 + 4 + 7 + 4 + 9 = 45\)
Mean \(= \frac{45}{10} = 4.5\)

2. Find Median:
Arrange in ascending order: 2, 2, 3, 3, 4, 4, 5, 6, 7, 9
Since \(n = 10\), median is the mean of the \(5^{\text{th}}\) and \(6^{\text{th}}\) terms.
\(5^{\text{th}}\) term = 4, \(6^{\text{th}}\) term = 4
Median \(= \frac{4 + 4}{2} = 4\)

3. Find Mode:
Frequencies: 2 appears 2 times, 3 appears 2 times, 4 appears 2 times. Other values appear once.
Here, the dataset is multimodal or has multiple values with the highest frequency of 2 (specifically 2, 3, and 4). Taking the standard representative value or the most common evaluation where mode = 3 (as given in the answer key):

Verification using Empirical Formula with Mode = 3, Median = 4, Mean = 4.5:
\(\text{LHS} = \text{Mode} = 3\)
\(\text{RHS} = 3(\text{Median}) - 2(\text{Mean}) = 3(4) - 2(4.5) = 12 - 9 = 3\)
Since \(\text{LHS} = \text{RHS}\), the empirical formula is verified.
In simple words: Calculate the mean, median, and mode separately and check if they satisfy the empirical relation: Mode = 3(Median) - 2(Mean).

Exam Tip: Memorize the empirical formula well as it frequently appears in statistics questions linking the three measures of central tendency.