UP Board Solutions Class 9 Maths Chapter 20 Statistics Ex 20.7

Balaji Class 9 Maths Solutions Chapter 20 Statistics Ex 20.7 सांख्यिकी

Ex 20.7 Statistics अतिलघु उत्तरीय प्रश्न (Very Short Answer Type Questions)

Question 1. संख्याओं 84, 78, 54, 56, 68, 22, 34, 45, 39, 54 का माध्यक ज्ञात कीजिए। हलः संख्याओं को आरोही क्रम में रखने पर 22, 34, 39,45, 54, 54, 56, 68,78,84 n = 10 (सम संख्या) माध्यक या माध्यिका = \( \frac{\left( \frac{n}{2} \right)\text{वाँ पद}+\left( \frac{n}{2}+1 \right)\text{वाँ पद}}{2} \)
\( \frac{\text{5वाँ पद} + \text{6वाँ पद}}{2} = \frac{54+54}{2} = \frac{108}{2} = 54 \)
Answer: 54
In simple words: To find the median for an even number of data points, arrange them in ascending order and take the average of the two middle terms. Here, the 5th and 6th terms are both 54, so their average is 54.

🎯 Exam Tip: Always arrange the data in ascending or descending order before calculating the median. For an even number of observations, the median is the average of the two middle terms.

Question 2. संख्याओं 4, 4, 5, 7, 6, 7, 7, 12, 3 का माध्यक ज्ञात कीजिए। हलः आरोही क्रम में रखने पर = 3,4,4,5,6,7,7,7,12 n = 9 (विषम संख्या)
Answer: माध्यक या माध्यिका = 6
In simple words: When there's an odd number of data points, arrange them from smallest to largest. The median is simply the middle value in that ordered list. Here, the 5th term out of 9 is 6.

🎯 Exam Tip: For an odd number of observations, the median is the \((n+1)/2\)th observation in the sorted list. Ensure careful sorting.

Question 3. 133, 73, 89, 108, 94, 140, 99, 85, 100, 120 का माध्यक ज्ञात कीजिए। हलः आरोही क्रम = 73,85,89,94,99,100, 108,120,133,140 n = 10 (सम संख्या)
Answer: माध्यक या माध्यिका = \( \frac{\text{5वाँ पद} + \text{6वाँ पद}}{2} = \frac{99+100}{2} = \frac{199}{2} = 99.5 \)
In simple words: First, arrange the numbers in increasing order. Since there are 10 numbers (an even count), the median is the average of the 5th (99) and 6th (100) numbers, which is 99.5.

🎯 Exam Tip: Double-check the ascending order of data, especially for larger datasets. Any mistake in sorting will lead to an incorrect median.

Question 4. प्रथम 10 प्राकृतिक संख्याओं का माध्यक ज्ञात कीजिए। हलः प्रथम 10 प्राकृतिक संख्याएँ = 1,2,3,4,5,6,7,8,9,10 n = 10 (सम संख्या)
Answer: माध्यक या माध्यिका = \( \frac{\text{5वाँ पद} + \text{6वाँ पद}}{2} = \frac{5+6}{2} = \frac{11}{2} = 5.5 \)
In simple words: The first 10 natural numbers are 1 through 10. Since this is an even set, the median is the average of the two middle numbers, which are 5 and 6. The average of 5 and 6 is 5.5.

🎯 Exam Tip: Remember that natural numbers start from 1. For a sequence of consecutive numbers, the median calculation is straightforward once the count (even/odd) is determined.

Question 5. प्रथम 9 पूर्ण संख्याओं का माध्यक ज्ञात कीजिए। हलः प्रथम 9 पूर्ण संख्या = 0,1,2,3,4,5,6,7,8 n = 9 (विषम संख्या)
Answer: माध्यक या माध्यिका = 4
In simple words: The first 9 whole numbers start from 0 and go up to 8. Since there are 9 numbers (an odd count), the median is the middle number, which is the 5th number in the sequence, 4.

🎯 Exam Tip: Be careful with the definition of "whole numbers" (starting from 0) versus "natural numbers" (starting from 1). This affects the data set and thus the median.

Ex 20.7 Statistics लघु उत्तरीय प्रश्न (Short Answer Type Questions)

Question 6. निम्नलिखित आँकड़े आरोही क्रम में व्यवस्थित हैं, 59, 62, 65, x, x +2, 72, 85, 99 यदि इन आँकड़ों का माध्यक 67 है, तो x का मान ज्ञात कीजिए । हल: n = 8 (सम संख्या) माध्यिका = \( \frac{\left( \frac{8}{2} \right)\text{वाँ पद} + \left( \frac{8}{2}+1 \right)\text{वाँ पद}}{2} = \frac{\text{4वाँ पद} + \text{5वाँ पद}}{2} \)
\( 67 = \frac{x+(x+2)}{2} \)
\( 134 = 2x + 2 \)

\( \implies 2x = 132 \)

\( \implies x = \frac{132}{2} \)

\( \implies x = 66 \)
Answer: x = 66
In simple words: The data is already sorted and has 8 terms, so the median is the average of the 4th and 5th terms. Given the median is 67 and the terms are x and x+2, we set up an equation (x + x + 2)/2 = 67 and solve for x, finding x = 66.

🎯 Exam Tip: When the median is given for a sorted dataset with an unknown variable, set up the median formula (average of middle terms for even 'n', or middle term for odd 'n') equal to the given median and solve the resulting algebraic equation.

Question 7. निम्नलिखित आँकड़ों का माध्यक ज्ञात कीजिए : (i) 83, 37, 70, 29, 45, 63, 41, 70, 34, 51 (ii) 41, 43, 127, 99, 71, 92, 71, 58, 57 हल:
(i) आरोही क्रम में = 29,34, 37,41, 45, 51, 63,70,70,83 माध्यिका = \( \frac{\left( \frac{10}{2} \right)\text{वाँ पद} + \left( \frac{10}{2}+1 \right)\text{वाँ पद}}{2} \)
\( = \frac{\text{5वाँ पद} + \text{6वाँ पद}}{2} = \frac{45+51}{2} = \frac{96}{2} = 48 \)
(ii) आरोही क्रम में = 41, 43, 57, 58, 71, 71, 92, 99, 127 n = 9 (विषम संख्या) माध्यिका \( = \frac{9+1}{2} = \frac{10}{2} \) वाँ पद = 5 वाँ पद = 71
Answer: (i) 48
(ii) 71
In simple words: For part (i), sort the 10 numbers to get an even count; the median is the average of the 5th (45) and 6th (51) terms, which is 48. For part (ii), sort the 9 numbers to get an odd count; the median is the 5th term (71) in the sorted list.

🎯 Exam Tip: Pay close attention to whether the number of observations (n) is odd or even, as this dictates the correct median formula. Always double-check the sorting process.

Question 8. प्रथम 12 अभाज्य संख्याओं का माध्यक ज्ञात कीजिए । हलः प्रथम 12 अभाज्य संख्याओं का माध्यक = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 n = 12 (सम संख्या)
Answer: माध्यक या माध्यिका = \( \frac{\text{6वाँ पद} + \text{7वाँ पद}}{2} = \frac{13+17}{2} = \frac{30}{2} = 15 \)
In simple words: List the first 12 prime numbers in order. Since there are 12 numbers (an even count), the median is the average of the 6th prime (13) and the 7th prime (17), which is 15.

🎯 Exam Tip: Accurately identifying prime numbers is crucial. Remember that 2 is the only even prime number, and 1 is not a prime number. Once listed, apply the standard median formula for an even dataset.

Question 9. कुल 10 प्रेक्षणों के एक अवरोही क्रम में व्यवस्थित 5वाँ तथा 6वाँ प्रेक्षण क्रमशः 13 तथा 11 है। सभी 10 प्रेक्षणों का माध्यक मान क्या है? हलः
Answer: माध्यक या माध्यिका = \( \frac{\text{5वाँ पद} + \text{6वाँ पद}}{2} = \frac{13+11}{2} = \frac{24}{2} = 12 \)
In simple words: The data is already sorted in descending order, and we have 10 observations (an even number). The median is the average of the 5th and 6th observations, which are given as 13 and 11, respectively. Their average is (13+11)/2 = 12.

🎯 Exam Tip: Even if data is in descending order, the median calculation remains the same: average of the two middle terms for an even count, or the single middle term for an odd count. The order itself doesn't change the median value, only which terms are "middle".

Ex 20.7 Statistics दीर्घ उत्तरीय प्रश्न (Long Answer Type Questions)

Question 10. निम्नलिखित आँकड़ों का माध्यक ज्ञात कीजिए : 19, 25, 59, 48, 35, 31, 30, 32, 51 यदि 25 को 52 में बदल दिया जाये तो नया माध्यक क्या होगा? हलः यदि 25 को 52 में बदल दिया जाए तो आरोही क्रम = 19, 30, 31, 32, 35, 48, 51, 52,59 माध्यिका = 5 वाँ पद =35
Answer:Initial data: 19, 25, 59, 48, 35, 31, 30, 32, 51. n = 9 (विषम संख्या) Sorted initial data: 19, 25, 30, 31, 32, 35, 48, 51, 59. Initial Median (5th term) = 32.
After changing 25 to 52, the new sorted data: 19, 30, 31, 32, 35, 48, 51, 52, 59. n = 9 (विषम संख्या) New Median (5th term) = 35.
In simple words: First, sort the original 9 numbers to find the median (the 5th term), which is 32. Then, replace 25 with 52 and re-sort the numbers. The new median (the 5th term) becomes 35.

🎯 Exam Tip: For problems involving changes to a dataset, always re-sort the entire set after the change before recalculating the median. This ensures the correct order for identifying the middle term(s).

Question 11. निम्नलिखित आँकड़ों का माध्यक ज्ञात कीजिए : 46, 64, 87, 41, 58, 77, 55, 33, 92 यदि प्रेक्षण 92 को 19 से बदल दिया जाये तो नया माध्यक निकालिए। हलः आरोही क्रम = 33,41,46,55,58,64,77,87,92 n = 9 (विषम संख्या) यदि प्रेक्षण 92 को 19 में बदल दिया जाय तो आरोही क्रम = 19, 33, 41, 46, 55, 58,64,77,87 माध्यिका = 5 वाँ पद = 55
Answer:Initial data: 46, 64, 87, 41, 58, 77, 55, 33, 92. n = 9 (विषम संख्या) Sorted initial data: 33, 41, 46, 55, 58, 64, 77, 87, 92. Initial Median (5th term) = 58.
After changing 92 to 19, the new sorted data: 19, 33, 41, 46, 55, 58, 64, 77, 87. n = 9 (विषम संख्या) New Median (5th term) = 55.
In simple words: For the original set of 9 numbers, sorting them gives a median of 58. After changing 92 to 19 and re-sorting, the new list has 19 as the first term, and the middle (5th) term becomes 55.

🎯 Exam Tip: When a value is changed, its position in the sorted list might shift significantly, especially if the new value is much smaller or larger than the original. Always perform a full re-sort.

Question 12. निम्नलिखित प्रेक्षणों का माध्यक ज्ञात कीजिए : 46, 64,87, 41, 58, 77, 35, 90, 55, 92, 33 यदि 92 को 99 से तथा 41 को 43 से बदल दिया जाये तो नया माध्यक ज्ञात कीजिए। हलः आरोही क्रम = 33, 35,41,46, 55, 58, 64,77,87,90, 92 n = 11 (विषम सँख्या) यदि 92 को 99 से तथा 41 को 43 से बदले दे तब आरोही क्रम - 33, 35, 43, 46, 55, 58,64,77,87,90, 99 n = 11 (विषम संख्या)
Answer:Initial data: 46, 64, 87, 41, 58, 77, 35, 90, 55, 92, 33. n = 11 (विषम संख्या) Sorted initial data: 33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92. Initial Median (6th term) = 58.
After changing 92 to 99 and 41 to 43, the new data: 46, 64, 87, 43, 58, 77, 35, 90, 55, 99, 33. New sorted data: 33, 35, 43, 46, 55, 58, 64, 77, 87, 90, 99. n = 11 (विषम संख्या) New Median (6th term) = 58.
In simple words: First, sort the original 11 numbers; the median is the 6th term, 58. Then, update two values (92 to 99, 41 to 43) and re-sort the list. The new median is again the 6th term, which is still 58.

🎯 Exam Tip: Multiple changes to data points require meticulous re-sorting and identification of the middle term(s). Even if the median doesn't change, show all steps of re-sorting and calculation clearly.

Question 13. 15 विद्यार्थियों का भार (किग्रा में) निम्नलिखित है : 31, 35, 27, 29, 32, 43, 37, 41, 34, 28, 36, 44, 45, 42, 30 माध्यक ज्ञात कीजिए। यदि भार 44 किग्रा को 46 किग्रा से बदल दिया जाये तथा 27 किग्रा को 25 किग्रा से बदल दिया जाये तो नया माध्यक ज्ञात कीजिए । हलः आरोही क्रम = 27, 28, 29, 30, 31, 32, 34, 35, 36, 37,41, 42, 43, 44, 45 n = 15 (विषम संख्या) 44 किग्रा को 46 में तथा 27 किग्रा को 25 किग्रा से बदल दिया जाए। तो आरोही क्रम = 25, 28, 29, 30, 31, 32, 34, 35, 36, 37,41, 42, 43, 45, 46 ∴ n = 15 (विषम संख्या)
Answer:Initial data: 31, 35, 27, 29, 32, 43, 37, 41, 34, 28, 36, 44, 45, 42, 30. n = 15 (विषम संख्या) Sorted initial data: 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 42, 43, 44, 45. Initial Median (8th term) = 35.
After changing 44 kg to 46 kg and 27 kg to 25 kg, the new data: 31, 35, 25, 29, 32, 43, 37, 41, 34, 28, 36, 46, 45, 42, 30. New sorted data: 25, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 42, 43, 45, 46. n = 15 (विषम संख्या) New Median (8th term) = 35.
In simple words: First, sort the 15 weights to find the original median, which is the 8th term (35 kg). Then, modify the two specified weights (44 to 46, 27 to 25) and re-sort the entire list. The new median is still the 8th term, which remains 35 kg.

🎯 Exam Tip: When dealing with a large dataset and multiple changes, carefully update the values in your sorted list and then re-sort. A single transcription error can lead to an incorrect median.

Question 14. आरोही क्रम में व्यवस्थित निम्नलिखित प्रेक्षणों का माध्यक 22 है x ज्ञात कीजिए। 8, 11, 13, 15, x + 1, x + 3, 30, 35, 40, 43 हलः n = 10 (सम संख्या ) माध्यक = \( \frac{\left( \frac{10}{2} \right)\text{वाँ पद} + \left( \frac{10}{2}+1 \right)\text{वाँ पद}}{2} \)
\( 22 = \frac{\text{5वाँ पद} + \text{6वाँ पद}}{2} \)
\( 22 = \frac{x+1+x+3}{2} \)

\( \implies 44 = 2x + 4 \)

\( \implies 44-4 = 2x \)

\( \implies 40 = 2x \)

\( \implies x = 20 \)
Answer: x = 20
In simple words: The data is already sorted and has 10 terms (an even number), so the median is the average of the 5th and 6th terms. Given the median is 22 and the terms are (x+1) and (x+3), we solve the equation (x+1 + x+3)/2 = 22 to find x = 20.

🎯 Exam Tip: Always correctly identify the middle terms for even 'n' and set up the algebraic equation with precision. Carefully perform the algebraic manipulations to solve for the unknown variable.

Balaji Publications Mathematics Class 9 Solutions