OP Malhotra Class 9 Maths Solutions Chapter 15 Mean Median and Frequency Polygon Exercise 15 (B)

S Chand Class 9 ICSE Maths Solutions Chapter 15 Mean, Median and Frequency Polygon Ex 15(B)

Find the median of the following :

Question 1. 2, 3, 5, 7, 9
Answer: The given data set is 2, 3, 5, 7, 9. The total number of terms (n) is 5, which is an odd number. For an odd number of terms, the median is the middle term. We find the position of the median using the formula \( \frac{n+1}{2} \).
\( \implies \) So, it is the \( \frac{5+1}{2} = \frac{6}{2} = 3 \)rd term. The 3rd term in the given data is 5. Therefore, the median is 5. The median divides the data set into two equal halves.
In simple words: The numbers are 2, 3, 5, 7, 9. There are 5 numbers, which is an odd count. The middle number is found by \( (5+1)/2 = 3 \). The 3rd number is 5, so 5 is the median.

🎯 Exam Tip: Always count the number of data points first to determine if 'n' is odd or even, as this affects the median formula.

Question 2. 4, 8, 12, 16, 20, 24, 28, 32
Answer: The given data set is 4, 8, 12, 16, 20, 24, 28, 32. The number of terms (n) is 8, which is an even number. For an even number of terms, the median is the average of the two middle terms. We find the two middle terms, which are the \( \frac{n}{2} \)th term and the \( (\frac{n}{2}+1) \)th term.
\( \implies \) These are the \( \frac{8}{2} = 4 \)th term and the \( (\frac{8}{2}+1) = 5 \)th term. From the given data, the 4th term is 16 and the 5th term is 20. The median is the average of these two terms: \( \frac{1}{2} (16 + 20) = \frac{1}{2} (36) = 18 \). So, the median is 18. It is important that the data is arranged in order before finding the median.
In simple words: There are 8 numbers (an even count). The two middle numbers are the 4th (16) and 5th (20) terms. To find the median, we add them up and divide by 2: \( (16 + 20) / 2 = 36 / 2 = 18 \).

🎯 Exam Tip: For an even number of data points, ensure you correctly identify *both* middle terms before calculating their average.

Question 3. 60, 33, 63, 61, 44, 48, 51
Answer: First, we arrange the given numbers in ascending order: 33, 44, 48, 51, 60, 61, 63. The total number of terms (n) is 7, which is an odd number. For an odd number of terms, the median is the middle term. The position of the median is given by the \( \frac{n+1}{2} \)th term.
\( \implies \) So, it is the \( \frac{7+1}{2} = \frac{8}{2} = 4 \)th term. In the ordered list, the 4th term is 51. Therefore, the median is 51. Arranging data helps to correctly identify the middle value.
In simple words: First, put the numbers in order: 33, 44, 48, 51, 60, 61, 63. There are 7 numbers (odd). The middle number is the \( (7+1)/2 = 4 \)th number, which is 51. So, 51 is the median.

🎯 Exam Tip: Always arrange the data in either ascending or descending order before attempting to find the median, otherwise your answer will be incorrect.

Question 4. 13, 22, 25, 8, 11, 19, 17, 31, 16, 10
Answer: First, arrange the numbers in ascending order: 8, 10, 11, 13, 16, 17, 19, 22, 25, 31. The total number of terms (n) is 10, which is an even number. For an even number of terms, the median is the average of the two middle terms. The median is calculated as the average of the \( \frac{n}{2} \)th term and the \( (\frac{n}{2}+1) \)th term.
\( \implies \) For n=10, these are the \( \frac{10}{2} = 5 \)th term and the \( (\frac{10}{2}+1) = 6 \)th term. From the ordered list, the 5th term is 16 and the 6th term is 17. So, the median is \( \frac{1}{2} (16 + 17) = \frac{33}{2} = 16.5 \). The median of the data is 16.5. When data is even, the median can sometimes be a decimal, even if the data points are whole numbers.
In simple words: First, put the numbers in order: 8, 10, 11, 13, 16, 17, 19, 22, 25, 31. There are 10 numbers (even). The middle numbers are the 5th (16) and 6th (17) terms. We add them and divide by 2: \( (16 + 17) / 2 = 33 / 2 = 16.5 \). So, the median is 16.5.

🎯 Exam Tip: Remember to express the median as a decimal if the average of the two middle terms is not a whole number.

Question 5. First 10 prime numbers.
Answer: The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The total count of these numbers (n) is 10, which is an even number. The median is the average of the \( \frac{n}{2} \)th term and the \( (\frac{n}{2}+1) \)th term.
\( \implies \) For n=10, these are the \( \frac{10}{2} = 5 \)th term and the \( (\frac{10}{2}+1) = 6 \)th term. From the list of prime numbers, the 5th term is 11 and the 6th term is 13. So, the median is \( \frac{1}{2} (11 + 13) = \frac{24}{2} = 12 \). Thus, the median of the first 10 prime numbers is 12. Prime numbers are numbers greater than 1 that have only two factors: 1 and themselves.
In simple words: The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. There are 10 numbers (even). The middle numbers are the 5th (11) and 6th (13) terms. Add them and divide by 2: \( (11 + 13) / 2 = 24 / 2 = 12 \). So, the median is 12.

🎯 Exam Tip: Be careful to list the prime numbers correctly; missing or adding one will lead to an incorrect median.

Question 6. 15 students secured the following marks in a test in Statistics. Find the median marks : 35, 28, 13, 17, 20, 30, 19, 29, 11, 10, 29, 23, 18, 25, 17
Answer: First, arrange the marks in ascending order: 10, 11, 13, 17, 17, 18, 19, 20, 23, 25, 28, 29, 29, 30, 35. The number of scores (n) is 15, which is an odd number. Since n is odd, the median is the \( \frac{n+1}{2} \)th term, which is the \( \frac{15+1}{2} = \frac{16}{2} = 8 \)th term. The 8th term in the sorted list is 20. So, the median marks are 20. The median is a good measure for central tendency when there are outliers in the data, as it is not affected by extreme values.
In simple words: First, put all the marks in order from smallest to largest. There are 15 marks (odd). The middle mark is the \( (15+1)/2 = 8 \)th mark. Looking at the ordered list, the 8th mark is 20. So, the median mark is 20.

🎯 Exam Tip: When sorting a large dataset, double-check your ordering and count to ensure no numbers are missed or duplicated. Using tally marks can help.

Question 7. Calculate the median from the following data:
Roll No. 1, 2, 3, 4, 5, 6, 7
Marks 25, 55, 5, 45, 15, 35, 60
Answer: The given marks are 25, 55, 5, 45, 15, 35, 60. The number of observations (n) is 7, which is an odd number. First, arrange the marks in ascending order: 5, 15, 25, 35, 45, 55, 60. For an odd number of terms, the median is the \( \frac{n+1}{2} \)th term, which is the \( \frac{7+1}{2} = \frac{8}{2} = 4 \)th term. In the ordered list, the 4th term is 35. Therefore, the median marks are 35. A median is helpful because it is less affected by very high or very low scores compared to the mean.
In simple words: The marks are 25, 55, 5, 45, 15, 35, 60. First, put them in order: 5, 15, 25, 35, 45, 55, 60. There are 7 marks (odd). The middle mark is the \( (7+1)/2 = 4 \)th mark. The 4th mark is 35. So, the median marks are 35.

🎯 Exam Tip: For data presented in tables, identify the relevant variable (e.g., "Marks") and treat it as a single list for median calculation.

Question 8. Out of total of 16 observations, arranged in ascending order, the 8th and 9th observations are 25 and 27. What is the median?
Answer: The total number of observations (n) is 16, which is an even number. For an even number of terms, the median is the average of the \( \frac{n}{2} \)th term and the \( (\frac{n}{2}+1) \)th term.
\( \implies \) Here, these are the \( \frac{16}{2} = 8 \)th term and the \( (\frac{16}{2}+1) = 9 \)th term. We are given that the 8th observation is 25 and the 9th observation is 27. So, the median is \( \frac{1}{2} (25 + 27) = \frac{1}{2} (52) = 26 \). The median of these observations is 26. The median is always found by either picking the middle number or averaging the two middle numbers from an ordered list.
In simple words: There are 16 observations (even). The 8th observation is 25 and the 9th is 27. To find the median, we take the average of these two: \( (25 + 27) / 2 = 52 / 2 = 26 \). So, the median is 26.

🎯 Exam Tip: If the data is already sorted and the middle terms are provided, simply apply the formula for even 'n' directly without re-sorting.

Question 9. The median of the following observations arranged in ascending order 8, 9, 12, 18, (x + 2), (x + 4), 30, 31, 34, 39 is 24. Find x.
Answer: The observations are already given in ascending order: 8, 9, 12, 18, \( (x+2) \), \( (x+4) \), 30, 31, 34, 39. The total number of observations (n) is 10, which is an even number. The median for an even number of terms is the average of the \( \frac{n}{2} \)th term and the \( (\frac{n}{2}+1) \)th term.
\( \implies \) For n=10, these are the \( \frac{10}{2} = 5 \)th term and the \( (\frac{10}{2}+1) = 6 \)th term. From the given data, the 5th term is \( (x+2) \) and the 6th term is \( (x+4) \). So, the median is \( \frac{1}{2} [(x+2) + (x+4)] \). This simplifies to \( \frac{1}{2} [2x + 6] = x + 3 \). We are told that the median is 24. Therefore, we can set up the equation: \( x+3 = 24 \).
\( \implies \) \( x = 24 - 3 \).
\( \implies \) \( x = 21 \). The value of x is 21. Algebra is often used in statistics to find unknown values when other properties like the median are known.
In simple words: The numbers are given in order. There are 10 numbers (even). The middle numbers are the 5th (\( x+2 \)) and 6th (\( x+4 \)). The median is the average of these two: \( ((x+2) + (x+4)) / 2 = (2x+6) / 2 = x+3 \). We know the median is 24, so \( x+3 = 24 \).
\( \implies \) \( x = 24 - 3 = 21 \). So, x is 21.

🎯 Exam Tip: When solving for an unknown variable like 'x' in median problems, set up the median formula as an equation and solve carefully.

Question 10. The following data have been in ascending order 18, 20, 25, 26, 30, x, 37, 38, 39, 48. If the median of the data is 35, find x. In the above data, if 48 is replaced by 28, find the new median.
Answer:
(i) The data is already arranged in ascending order: 18, 20, 25, 26, 30, x, 37, 38, 39, 48. There are 10 terms in the data, which is an even number. For n=10, the median is the average of the \( \frac{10}{2} = 5 \)th term and the \( (\frac{10}{2}+1) = 6 \)th term. The 5th term is 30 and the 6th term is x. So, the median is \( \frac{1}{2} (30 + x) = \frac{30+x}{2} \). We are given that the median is 35. So, \( \frac{30+x}{2} = 35 \).
\( \implies \) \( 30 + x = 35 \times 2 \).
\( \implies \) \( 30 + x = 70 \).
\( \implies \) \( x = 70 - 30 \).
\( \implies \) \( x = 40 \). Knowing the median helps us fill in missing data points if the data is already sorted.
(ii) In the original data (18, 20, 25, 26, 30, x, 37, 38, 39, 48), we found \( x=40 \). So the data was: 18, 20, 25, 26, 30, 40, 37, 38, 39, 48. Now, if we replace 48 with 28, the new data set, arranged in ascending order, becomes: 18, 20, 25, 26, 28, 30, 37, 38, 39, 40. There are still 10 terms (even). The new median is the average of the 5th and 6th terms. In the new ordered list, the 5th term is 28 and the 6th term is 30. So, the new median is \( \frac{1}{2} (28 + 30) = \frac{58}{2} = 29 \). Therefore, the new median is 29. Changing even one data point can affect the median if it alters the order or the middle values.
In simple words:
(i) The numbers are given in order, with 'x' as a missing value. There are 10 numbers (even). The median is the average of the 5th (30) and 6th (x) terms, which is \( (30+x)/2 \). We are told the median is 35. So, \( (30+x)/2 = 35 \). Multiply both sides by 2: \( 30+x = 70 \).
\( \implies \) \( x = 70 - 30 = 40 \).
(ii) Using \( x=40 \), the original data (in order) was 18, 20, 25, 26, 30, 40, 37, 38, 39, 48. If 48 is changed to 28, and we put the new list in order: 18, 20, 25, 26, 28, 30, 37, 38, 39, 40. There are 10 numbers (even). The middle numbers are the 5th (28) and 6th (30) terms. The new median is \( (28 + 30) / 2 = 58 / 2 = 29 \).

🎯 Exam Tip: For multi-part questions, ensure you use the result from the first part (like the value of x) correctly in the subsequent parts of the problem.