MEAN AND MEDIAN - Questions & Answers
EXERCISE 18(A)
1. Multiple Choice Type :
Choose the correct answer from the options given below.
(a) The mean of x - 2, x, x + 2, x + 4 is :
(i) x + 1 (ii) x (iii) 4x + 4 (iv) x + 2
Step 1: Calculate the sum of the given observations: (x - 2) + x + (x + 2) + (x + 4).
Step 2: Sum = 4x + 4.
Step 3: Identify the total number of observations, which is 4.
Step 4: Mean = Sum of observations / Total number of observations.
Step 5: Mean = (4x + 4) / 4 = x + 1.
Answer: (i) x + 1
Step 2: Sum = 4x + 4.
Step 3: Identify the total number of observations, which is 4.
Step 4: Mean = Sum of observations / Total number of observations.
Step 5: Mean = (4x + 4) / 4 = x + 1.
Answer: (i) x + 1
(b) Mean of 10 observations (numbers) is 20. If one number is included, the mean becomes 21, the included number is :
(i) 431 (ii) 31 (iii) 231 (iv) 131
Step 1: Calculate the total sum of the initial 10 observations = 10 * 20 = 200.
Step 2: When one number is included, total observations become 11, and new mean is 21.
Step 3: Calculate the new total sum of 11 observations = 11 * 21 = 231.
Step 4: Included number = New total sum - Initial total sum.
Step 5: Included number = 231 - 200 = 31.
Answer: (ii) 31
Step 2: When one number is included, total observations become 11, and new mean is 21.
Step 3: Calculate the new total sum of 11 observations = 11 * 21 = 231.
Step 4: Included number = New total sum - Initial total sum.
Step 5: Included number = 231 - 200 = 31.
Answer: (ii) 31
(c) Mean of 20 observations (numbers) is 30. If one number is excluded, the mean of remaining numbers becomes 28. The excluded number is :
(i) 532 (ii) 1132 (iii) 68 (iv) 572
Step 1: Calculate the total sum of the initial 20 observations = 20 * 30 = 600.
Step 2: When one number is excluded, total observations become 19, and new mean is 28.
Step 3: Calculate the new total sum of 19 observations = 19 * 28 = 532.
Step 4: Excluded number = Initial total sum - New total sum.
Step 5: Excluded number = 600 - 532 = 68.
Answer: (iii) 68
Step 2: When one number is excluded, total observations become 19, and new mean is 28.
Step 3: Calculate the new total sum of 19 observations = 19 * 28 = 532.
Step 4: Excluded number = Initial total sum - New total sum.
Step 5: Excluded number = 600 - 532 = 68.
Answer: (iii) 68
(d) If each observation of the data is decreased by 15; then the mean :
(i) remains same (ii) is increased by 15 (iii) is decreased by 15 (iv) is multiplied by 15
Step 1: Recall the property of the mean: if a constant value is subtracted from each observation, the mean decreases by that same constant value.
Step 2: Here, each observation is decreased by 15.
Step 3: Therefore, the overall mean will also decrease by 15.
Answer: (iii) is decreased by 15
Step 2: Here, each observation is decreased by 15.
Step 3: Therefore, the overall mean will also decrease by 15.
Answer: (iii) is decreased by 15
(e) If the mean of x_1 and x_2 is 12.5 and mean of x_1, x_2 and x_3 is 16. The value of x_3 is :
(i) 32 (ii) 3.5 (iii) 285 (iv) 23
Step 1: Calculate the sum of x_1 and x_2 from their mean: 2 * 12.5 = 25.
Step 2: Calculate the sum of x_1, x_2, and x_3 from their mean: 3 * 16 = 48.
Step 3: The value of x_3 = (Sum of all three) - (Sum of first two).
Step 4: x_3 = 48 - 25 = 23.
Answer: (iv) 23
Step 2: Calculate the sum of x_1, x_2, and x_3 from their mean: 3 * 16 = 48.
Step 3: The value of x_3 = (Sum of all three) - (Sum of first two).
Step 4: x_3 = 48 - 25 = 23.
Answer: (iv) 23
2. Find the mean of 43, 51, 50, 57 and 54.
Step 1: Write down the formula for Mean: Mean = (Sum of all observations) / (Total number of observations).
Step 2: Find the sum of the numbers: 43 + 51 + 50 + 57 + 54 = 255.
Step 3: Count the total number of observations, which is 5.
Step 4: Divide the sum by the count: 255 / 5 = 51.
Answer: The mean is 51.
Step 2: Find the sum of the numbers: 43 + 51 + 50 + 57 + 54 = 255.
Step 3: Count the total number of observations, which is 5.
Step 4: Divide the sum by the count: 255 / 5 = 51.
Answer: The mean is 51.
3. Find the mean of first six natural numbers.
Step 1: List the first six natural numbers: 1, 2, 3, 4, 5, 6.
Step 2: Find the sum of these numbers: 1 + 2 + 3 + 4 + 5 + 6 = 21.
Step 3: Count the total number of observations, which is 6.
Step 4: Divide the sum by the count: Mean = 21 / 6 = 3.5.
Answer: The mean is 3.5.
Step 2: Find the sum of these numbers: 1 + 2 + 3 + 4 + 5 + 6 = 21.
Step 3: Count the total number of observations, which is 6.
Step 4: Divide the sum by the count: Mean = 21 / 6 = 3.5.
Answer: The mean is 3.5.
4. Find the mean of first ten odd natural numbers.
Step 1: List the first ten odd natural numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.
Step 2: Find the sum of these numbers = 100.
Step 3: Total number of observations is 10.
Step 4: Calculate Mean = 100 / 10 = 10.
Answer: The mean is 10.
Step 2: Find the sum of these numbers = 100.
Step 3: Total number of observations is 10.
Step 4: Calculate Mean = 100 / 10 = 10.
Answer: The mean is 10.
5. Find the mean of all factors of 10.
Step 1: Find all the factors of 10, which are the numbers that divide 10 evenly.
Step 2: The factors are 1, 2, 5, and 10.
Step 3: Find the sum of these factors: 1 + 2 + 5 + 10 = 18.
Step 4: Total number of factors is 4.
Step 5: Calculate Mean = 18 / 4 = 4.5.
Answer: The mean is 4.5.
Step 2: The factors are 1, 2, 5, and 10.
Step 3: Find the sum of these factors: 1 + 2 + 5 + 10 = 18.
Step 4: Total number of factors is 4.
Step 5: Calculate Mean = 18 / 4 = 4.5.
Answer: The mean is 4.5.
6. Find the mean of x + 3, x + 5, x + 7, x + 9 and x + 11. (BM)
Step 1: Write down all the observations: (x + 3), (x + 5), (x + 7), (x + 9), (x + 11).
Step 2: Add all the observations together: (x+x+x+x+x) + (3+5+7+9+11).
Step 3: Sum = 5x + 35.
Step 4: Total number of observations = 5.
Step 5: Mean = (5x + 35) / 5 = x + 7.
Answer: The mean is x + 7.
Step 2: Add all the observations together: (x+x+x+x+x) + (3+5+7+9+11).
Step 3: Sum = 5x + 35.
Step 4: Total number of observations = 5.
Step 5: Mean = (5x + 35) / 5 = x + 7.
Answer: The mean is x + 7.
7. If different values of variable x are 9.8, 5.4, 3.7, 1.7, 1.8, 2.6, 2.8, 8.6, 10.5 and 11.1; find
(i) the mean x̄
(ii) the value of Σ(x - x̄) (HOTS)
Step 1: For part (i), list the values: 9.8, 5.4, 3.7, 1.7, 1.8, 2.6, 2.8, 8.6, 10.5, 11.1.
Step 2: Total number of observations (n) = 10.
Step 3: Find the sum of these values: 9.8+5.4+3.7+1.7+1.8+2.6+2.8+8.6+10.5+11.1 = 58.0.
Step 4: Mean (x̄) = 58.0 / 10 = 5.8.
Step 5: For part (ii), recall the property of the mean that the sum of deviations of all observations from their mean is always equal to zero.
Step 6: Therefore, Σ(x - x̄) = 0.
Answer: (i) Mean = 5.8 (ii) Σ(x - x̄) = 0
Step 2: Total number of observations (n) = 10.
Step 3: Find the sum of these values: 9.8+5.4+3.7+1.7+1.8+2.6+2.8+8.6+10.5+11.1 = 58.0.
Step 4: Mean (x̄) = 58.0 / 10 = 5.8.
Step 5: For part (ii), recall the property of the mean that the sum of deviations of all observations from their mean is always equal to zero.
Step 6: Therefore, Σ(x - x̄) = 0.
Answer: (i) Mean = 5.8 (ii) Σ(x - x̄) = 0
8. The mean of 15 observations is 32. Find the resulting mean, if each observation is : (BM)
(i) increased by 3 (ii) decreased by 7
(iii) multiplied by 2 (iv) divided by 0.5
(v) increased by 60% (vi) decreased by 20%
Step 1: The original mean is 32. Use the properties of mean for each case.
Step 2: (i) If each observation is increased by 3, the new mean increases by 3: 32 + 3 = 35.
Step 3: (ii) If each observation is decreased by 7, the new mean decreases by 7: 32 - 7 = 25.
Step 4: (iii) If each observation is multiplied by 2, the new mean is multiplied by 2: 32 * 2 = 64.
Step 5: (iv) If each observation is divided by 0.5, the new mean is divided by 0.5: 32 / 0.5 = 64.
Step 6: (v) If each observation is increased by 60%, the mean increases by 60%. New mean = 32 * (1 + 0.60) = 32 * 1.6 = 51.2.
Step 7: (vi) If each observation is decreased by 20%, the mean decreases by 20%. New mean = 32 * (1 - 0.20) = 32 * 0.8 = 25.6.
Answers:
(i) 35
(ii) 25
(iii) 64
(iv) 64
(v) 51.2
(vi) 25.6
Step 2: (i) If each observation is increased by 3, the new mean increases by 3: 32 + 3 = 35.
Step 3: (ii) If each observation is decreased by 7, the new mean decreases by 7: 32 - 7 = 25.
Step 4: (iii) If each observation is multiplied by 2, the new mean is multiplied by 2: 32 * 2 = 64.
Step 5: (iv) If each observation is divided by 0.5, the new mean is divided by 0.5: 32 / 0.5 = 64.
Step 6: (v) If each observation is increased by 60%, the mean increases by 60%. New mean = 32 * (1 + 0.60) = 32 * 1.6 = 51.2.
Step 7: (vi) If each observation is decreased by 20%, the mean decreases by 20%. New mean = 32 * (1 - 0.20) = 32 * 0.8 = 25.6.
Answers:
(i) 35
(ii) 25
(iii) 64
(iv) 64
(v) 51.2
(vi) 25.6
9. The mean of 5 numbers is 18. If one number is excluded, the mean of remaining numbers becomes 16. Find the excluded number.
Step 1: Calculate the total sum of the initial 5 numbers: 5 * 18 = 90.
Step 2: When one number is excluded, 4 numbers remain, and their new mean is 16.
Step 3: Calculate the total sum of the remaining 4 numbers: 4 * 16 = 64.
Step 4: The excluded number = (Sum of 5 numbers) - (Sum of 4 numbers).
Step 5: Excluded number = 90 - 64 = 26.
Answer: The excluded number is 26.
Step 2: When one number is excluded, 4 numbers remain, and their new mean is 16.
Step 3: Calculate the total sum of the remaining 4 numbers: 4 * 16 = 64.
Step 4: The excluded number = (Sum of 5 numbers) - (Sum of 4 numbers).
Step 5: Excluded number = 90 - 64 = 26.
Answer: The excluded number is 26.
10. If the mean of observations x, x + 2, x + 4, x + 6 and x + 8 is 11, find the value of x.
Step 1: Write down the sum of all 5 observations: x + (x + 2) + (x + 4) + (x + 6) + (x + 8).
Step 2: Sum = 5x + 20.
Step 3: Total number of observations = 5.
Step 4: Mean = (5x + 20) / 5 = x + 4.
Step 5: Given that the mean is 11, set up the equation: x + 4 = 11.
Step 6: Solve for x: x = 11 - 4 = 7.
Answer: The value of x is 7.
Step 2: Sum = 5x + 20.
Step 3: Total number of observations = 5.
Step 4: Mean = (5x + 20) / 5 = x + 4.
Step 5: Given that the mean is 11, set up the equation: x + 4 = 11.
Step 6: Solve for x: x = 11 - 4 = 7.
Answer: The value of x is 7.
EXERCISE 18(B)
1. Multiple Choice Type :
Choose the correct answer from the options given below.
(a) 12 observations in a data are written in ascending order. If the last (12th) observation is doubled, the median will increase by :
(i) 12 (ii) 24 (iii) 0 (iv) 6
Step 1: Since there are 12 observations (an even number), the median is the average of the 6th and 7th terms.
Step 2: Doubling the 12th observation only changes the highest value at the end of the array.
Step 3: The values of the 6th and 7th terms remain completely unaffected.
Step 4: Therefore, the median does not change. Its increase is 0.
Answer: (iii) 0
Step 2: Doubling the 12th observation only changes the highest value at the end of the array.
Step 3: The values of the 6th and 7th terms remain completely unaffected.
Step 4: Therefore, the median does not change. Its increase is 0.
Answer: (iii) 0
(b) 12 observations in a data are written in descending order. If the first observation is doubled, the median will :
(i) remain same (ii) increase by 12 (iii) decrease by 12 (iv) change by 12
Step 1: In descending order, the median of 12 terms still depends purely on the two middle terms (6th and 7th).
Step 2: The first observation is the largest value in the set.
Step 3: Doubling the largest value makes it even larger but keeps it in the 1st position.
Step 4: The 6th and 7th terms stay exactly the same, so the median will remain the same.
Answer: (i) remain same
Step 2: The first observation is the largest value in the set.
Step 3: Doubling the largest value makes it even larger but keeps it in the 1st position.
Step 4: The 6th and 7th terms stay exactly the same, so the median will remain the same.
Answer: (i) remain same
(c) Median of numbers 10, 12, 9, 8, 10, 12, 10, 6 and 4 is :
(i) 12 (ii) 10 (iii) 6 (iv) 9
Step 1: Count the numbers: there are 9 observations in total (an odd number).
Step 2: Arrange the data in ascending order: 4, 6, 8, 9, 10, 10, 10, 12, 12.
Step 3: For an odd number of observations (n=9), the median is the ((n+1)/2)th term.
Step 4: Median = ((9+1)/2)th term = 5th term.
Step 5: The 5th term in the sorted array is 10.
Answer: (ii) 10
Step 2: Arrange the data in ascending order: 4, 6, 8, 9, 10, 10, 10, 12, 12.
Step 3: For an odd number of observations (n=9), the median is the ((n+1)/2)th term.
Step 4: Median = ((9+1)/2)th term = 5th term.
Step 5: The 5th term in the sorted array is 10.
Answer: (ii) 10
(d) The median of 80 observations is 60. If each observation is doubled, the resulting median will be :
(i) 60 (ii) 20 (iii) 140 (iv) 120
Step 1: According to the property of central tendencies, if every observation is multiplied by a constant, the median is also multiplied by that same constant.
Step 2: The original median is 60.
Step 3: Since each observation is doubled (multiplied by 2), the new median will be 60 * 2.
Step 4: New median = 120.
Answer: (iv) 120
Step 2: The original median is 60.
Step 3: Since each observation is doubled (multiplied by 2), the new median will be 60 * 2.
Step 4: New median = 120.
Answer: (iv) 120
(e) If each observation in a data is decreased by two, their :
(i) mean decreases by 2, but median remains same.
(ii) mean remains same, but median decreases, by 2.
(iii) mean and median both decrease by 2.
(iv) mean and median both remain the same.
Step 1: If each data point is decreased by a constant value, the entire dataset shifts backward by that amount.
Step 2: Because of this uniform shift, both the arithmetic mean and the middle value (median) will shift backward by the exact same amount.
Step 3: Thus, both the mean and the median will decrease by 2.
Answer: (iii) mean and median both decrease by 2.
Step 2: Because of this uniform shift, both the arithmetic mean and the middle value (median) will shift backward by the exact same amount.
Step 3: Thus, both the mean and the median will decrease by 2.
Answer: (iii) mean and median both decrease by 2.
2. Find the median of :
(i) 25, 16, 26, 16, 32, 31, 19, 28 and 35
(ii) 241, 243, 347, 350, 327, 299, 261, 292, 271, 258 and 257
(iii) 63, 17, 50, 9, 25, 43, 21, 50, 14 and 34
(iv) 233, 173, 189, 208, 194, 204, 194, 185, 200 and 220
For (i):
Step 1: Arrange data in ascending order: 16, 16, 19, 25, 26, 28, 31, 32, 35.
Step 2: Number of terms (n) = 9 (odd).
Step 3: Median is the 5th term.
Answer (i): Median = 26.
For (ii):
Step 1: Arrange in ascending order: 241, 243, 257, 258, 261, 271, 292, 299, 327, 347, 350.
Step 2: Number of terms (n) = 11 (odd).
Step 3: Median is the ((11+1)/2)th term = 6th term.
Answer (ii): Median = 271.
For (iii):
Step 1: Arrange in ascending order: 9, 14, 17, 21, 25, 34, 43, 50, 50, 63.
Step 2: Number of terms (n) = 10 (even).
Step 3: Median is the average of 5th and 6th terms.
Step 4: 5th term = 25, 6th term = 34.
Step 5: Median = (25 + 34) / 2 = 59 / 2 = 29.5.
Answer (iii): Median = 29.5.
For (iv):
Step 1: Arrange in ascending order: 173, 185, 189, 194, 194, 200, 204, 208, 220, 233.
Step 2: Number of terms (n) = 10 (even).
Step 3: Median is the average of 5th and 6th terms.
Step 4: 5th term = 194, 6th term = 200.
Step 5: Median = (194 + 200) / 2 = 394 / 2 = 197.
Answer (iv): Median = 197.
Step 1: Arrange data in ascending order: 16, 16, 19, 25, 26, 28, 31, 32, 35.
Step 2: Number of terms (n) = 9 (odd).
Step 3: Median is the 5th term.
Answer (i): Median = 26.
For (ii):
Step 1: Arrange in ascending order: 241, 243, 257, 258, 261, 271, 292, 299, 327, 347, 350.
Step 2: Number of terms (n) = 11 (odd).
Step 3: Median is the ((11+1)/2)th term = 6th term.
Answer (ii): Median = 271.
For (iii):
Step 1: Arrange in ascending order: 9, 14, 17, 21, 25, 34, 43, 50, 50, 63.
Step 2: Number of terms (n) = 10 (even).
Step 3: Median is the average of 5th and 6th terms.
Step 4: 5th term = 25, 6th term = 34.
Step 5: Median = (25 + 34) / 2 = 59 / 2 = 29.5.
Answer (iii): Median = 29.5.
For (iv):
Step 1: Arrange in ascending order: 173, 185, 189, 194, 194, 200, 204, 208, 220, 233.
Step 2: Number of terms (n) = 10 (even).
Step 3: Median is the average of 5th and 6th terms.
Step 4: 5th term = 194, 6th term = 200.
Step 5: Median = (194 + 200) / 2 = 394 / 2 = 197.
Answer (iv): Median = 197.
3. The following data have been arranged in ascending order. If their median is 63, find the value of x.
34, 37, 53, 55, x, x + 2, 77, 83, 89 and 100.
Step 1: The data is in ascending order and the total number of terms (n) is 10 (even).
Step 2: The median is the average of the 5th and 6th terms.
Step 3: Identify the 5th term (x) and the 6th term (x + 2).
Step 4: Set up the equation for the median: (x + x + 2) / 2 = 63.
Step 5: Simplify the equation: (2x + 2) / 2 = 63.
Step 6: x + 1 = 63.
Step 7: x = 62.
Answer: The value of x is 62.
Step 2: The median is the average of the 5th and 6th terms.
Step 3: Identify the 5th term (x) and the 6th term (x + 2).
Step 4: Set up the equation for the median: (x + x + 2) / 2 = 63.
Step 5: Simplify the equation: (2x + 2) / 2 = 63.
Step 6: x + 1 = 63.
Step 7: x = 62.
Answer: The value of x is 62.
4. In 10 numbers, arranged in increasing order, the 7th number is increased by 8, how much will the median be changed ?
Step 1: There are 10 numbers, which is an even count.
Step 2: The median is solely determined by the average of the 5th and 6th numbers.
Step 3: Increasing the 7th number does not affect the values or positions of the 5th and 6th numbers.
Step 4: Therefore, the median will not change at all.
Answer: The median will be changed by 0 (no change).
Step 2: The median is solely determined by the average of the 5th and 6th numbers.
Step 3: Increasing the 7th number does not affect the values or positions of the 5th and 6th numbers.
Step 4: Therefore, the median will not change at all.
Answer: The median will be changed by 0 (no change).
5. Out of 10 students, who appeared in a test, three secured less than 30 marks and 3 secured more than 75 marks. The marks secured by the remaining 4 students are 35, 48, 66 and 40. Find the median score of the whole group. (LS)
Step 1: Total students = 10. To find the median, we need to imagine arranging all 10 scores in ascending order.
Step 2: The 3 lowest scores (less than 30) will occupy the 1st, 2nd, and 3rd positions.
Step 3: The 3 highest scores (more than 75) will occupy the 8th, 9th, and 10th positions.
Step 4: The remaining 4 middle scores must occupy the 4th, 5th, 6th, and 7th positions.
Step 5: Arrange the 4 middle scores in ascending order: 35, 40, 48, 66.
Step 6: In the full 10-student array, the 5th position is 40 and the 6th position is 48.
Step 7: The median of 10 terms is the average of the 5th and 6th terms.
Step 8: Median = (40 + 48) / 2 = 88 / 2 = 44.
Answer: The median score is 44.
Step 2: The 3 lowest scores (less than 30) will occupy the 1st, 2nd, and 3rd positions.
Step 3: The 3 highest scores (more than 75) will occupy the 8th, 9th, and 10th positions.
Step 4: The remaining 4 middle scores must occupy the 4th, 5th, 6th, and 7th positions.
Step 5: Arrange the 4 middle scores in ascending order: 35, 40, 48, 66.
Step 6: In the full 10-student array, the 5th position is 40 and the 6th position is 48.
Step 7: The median of 10 terms is the average of the 5th and 6th terms.
Step 8: Median = (40 + 48) / 2 = 88 / 2 = 44.
Answer: The median score is 44.
6. The median of observations 10, 11, 13, 17, x + 5, 20, 22, 24 and 53 (arranged in ascending order) is 18; find the value of x.
Step 1: Count the total number of observations (n). There are 9 terms (odd).
Step 2: The median is the ((9+1)/2)th term, which is the 5th term.
Step 3: Identify the 5th term from the given ascending array, which is (x + 5).
Step 4: Set the 5th term equal to the given median: x + 5 = 18.
Step 5: Solve for x: x = 18 - 5 = 13.
Answer: The value of x is 13.
Step 2: The median is the ((9+1)/2)th term, which is the 5th term.
Step 3: Identify the 5th term from the given ascending array, which is (x + 5).
Step 4: Set the 5th term equal to the given median: x + 5 = 18.
Step 5: Solve for x: x = 18 - 5 = 13.
Answer: The value of x is 13.
TEST YOURSELF
1. Multiple Choice Type :
Choose the correct answer from the options given below.
(a) If each observation of a given set of data is increased by 5, their mean :
(i) remains the same (ii) becomes five times (iii) is decreased by 5 (iv) is increased by 5
Step 1: Use the basic properties of the mean.
Step 2: If a constant value 'k' is added to every observation in a dataset, the new mean is equal to the old mean plus 'k'.
Step 3: Since each observation is increased by 5, the overall mean is also increased by 5.
Answer: (iv) is increased by 5
Step 2: If a constant value 'k' is added to every observation in a dataset, the new mean is equal to the old mean plus 'k'.
Step 3: Since each observation is increased by 5, the overall mean is also increased by 5.
Answer: (iv) is increased by 5
(b) The median of observations (written in ascending order) 26, 29, 42, 53, x, x + 2, 70, 75, 82, 83 and 100 is 65; then :
(i) x = 65 (ii) x + 2 = 65 (iii) (x+(x+2))/2 = 6 (iv) none of these
Step 1: Count the number of observations in the list. There are 11 terms in total (odd number).
Step 2: The median is the ((11+1)/2)th term, which is the 6th term.
Step 3: Find the 6th term in the list, which is (x + 2).
Step 4: Given that the median is 65, we write: x + 2 = 65.
Answer: (ii) x + 2 = 65
Step 2: The median is the ((11+1)/2)th term, which is the 6th term.
Step 3: Find the 6th term in the list, which is (x + 2).
Step 4: Given that the median is 65, we write: x + 2 = 65.
Answer: (ii) x + 2 = 65
(c) The mean of x - 5, x - 3, x - 1 and x + 1 is :
(i) 4x - 8 (ii) (4x - 8) / 4 (iii) (x - 3 + x - 1) / 2 (iv) none of these
Step 1: Find the sum of the four observations: (x - 5) + (x - 3) + (x - 1) + (x + 1).
Step 2: Sum = 4x - 8.
Step 3: Number of observations = 4.
Step 4: Mean = Sum / 4 = (4x - 8) / 4.
Answer: (ii) (4x - 8) / 4
Step 2: Sum = 4x - 8.
Step 3: Number of observations = 4.
Step 4: Mean = Sum / 4 = (4x - 8) / 4.
Answer: (ii) (4x - 8) / 4
(d) 26, x + 4, x + 2 and 18 are in descending order of their values and their median is 20, then the value of x is :
(i) 19 (ii) 17 (iii) 15 (iv) 13
Step 1: There are 4 terms (even) given in descending order.
Step 2: The median is the average of the 2 middle terms (the 2nd and 3rd terms).
Step 3: The 2nd term is x + 4 and the 3rd term is x + 2.
Step 4: Set up the equation for the median: ((x + 4) + (x + 2)) / 2 = 20.
Step 5: (2x + 6) / 2 = 20.
Step 6: x + 3 = 20, which means x = 17.
Answer: (ii) 17
Step 2: The median is the average of the 2 middle terms (the 2nd and 3rd terms).
Step 3: The 2nd term is x + 4 and the 3rd term is x + 2.
Step 4: Set up the equation for the median: ((x + 4) + (x + 2)) / 2 = 20.
Step 5: (2x + 6) / 2 = 20.
Step 6: x + 3 = 20, which means x = 17.
Answer: (ii) 17
(e) Statement (1) : For n number of data in a set, median = (n/2)th term
Statement (2) : If n is even, median = 1/2 [ (n/2)th term + (n/2 + 1)th term ]
(i) Both the statements are true.
(ii) Both the statements are false.
(iii) Statement 1 is true, and statement 2 is false.
(iv) Statement 1 is false, and statement 2 is true.
Step 1: Evaluate Statement (1): The median formula depends on whether n is odd or even. If n is odd, it's ((n+1)/2)th term. If even, it's an average. The simple statement median = (n/2)th term is mathematically incorrect. Thus, Statement 1 is false.
Step 2: Evaluate Statement (2): This is the exact correct standard formula for finding the median when the number of observations (n) is even. Thus, Statement 2 is true.
Answer: (iv) Statement 1 is false, and statement 2 is true.
Step 2: Evaluate Statement (2): This is the exact correct standard formula for finding the median when the number of observations (n) is even. Thus, Statement 2 is true.
Answer: (iv) Statement 1 is false, and statement 2 is true.
(f) Statement (1) : The mean of 100 observations is 50 and one of these observations is increased by 150, the sum of resulting observations is 100 x 50 + 150.
Statement (2) : The sum of resulting observations = 100 x 50 + 100.
(i) Both the statements are true.
(ii) Both the statements are false.
(iii) Statement 1 is true, and statement 2 is false.
(iv) Statement 1 is false, and statement 2 is true.
Step 1: Calculate the initial total sum of 100 observations: 100 * 50.
Step 2: If a single observation is increased by 150, the total sum of the entire data set simply increases by that exact same 150.
Step 3: Resulting sum = Initial sum + 150 = (100 * 50) + 150. This matches Statement (1).
Step 4: Statement (2) suggests the sum increases by 100, which is incorrect.
Answer: (iii) Statement 1 is true, and statement 2 is false.
Step 2: If a single observation is increased by 150, the total sum of the entire data set simply increases by that exact same 150.
Step 3: Resulting sum = Initial sum + 150 = (100 * 50) + 150. This matches Statement (1).
Step 4: Statement (2) suggests the sum increases by 100, which is incorrect.
Answer: (iii) Statement 1 is true, and statement 2 is false.
(g) Assertion (A) : The mean of x_1, x_2 and x_3 is m. Then the value of (x_1 - m) + (x_2 - m) + (x_3 - m) = 0.
Reason (R) : x_1 + x_2 + x_3 = 3m implies (x_1 - m) + (x_2 - m) + (x_3 - m) = (x_1 + x_2 + x_3) - 3m
(i) A is true, R is false.
(ii) A is false, R is true.
(iii) Both A and R are true and R is the correct reason for A.
(iv) Both A and R are true and R is the incorrect reason for A.
Step 1: Analyze Assertion (A). It states the sum of deviations of all observations from their mean is zero. This is a fundamental property of the mean and is always mathematically true.
Step 2: Analyze Reason (R). It shows the algebraic breakdown: Since mean is m, sum of 3 terms is 3m. Expanding the deviations gives (x_1 + x_2 + x_3) - 3m, which becomes 3m - 3m = 0.
Step 3: The reason accurately proves why the assertion is true.
Answer: (iii) Both A and R are true and R is the correct reason for A.
Step 2: Analyze Reason (R). It shows the algebraic breakdown: Since mean is m, sum of 3 terms is 3m. Expanding the deviations gives (x_1 + x_2 + x_3) - 3m, which becomes 3m - 3m = 0.
Step 3: The reason accurately proves why the assertion is true.
Answer: (iii) Both A and R are true and R is the correct reason for A.
(h) Assertion (A) : Mean of n observations is x and mean of another set of n observations is y, the combined mean of all the observations is (x + y) / 2.
Reason (R) : Total of all the observations = nx + ny. Therefore Mean of all the observations = (nx + ny) / 2n.
(i) A is true, R is false.
(ii) A is false, R is true.
(iii) Both A and R are true and R is the correct reason for A.
(iv) Both A and R are true and R is the incorrect reason for A.
Step 1: Look at Reason (R). The sum of the first set is n*x and the second is n*y. Total sum = nx + ny. Total number of observations is n + n = 2n. The mean is (nx + ny) / 2n. Factoring out n gives n(x+y)/2n = (x+y)/2. The reason is completely accurate.
Step 2: Look at Assertion (A). It states the combined mean is (x+y)/2, which perfectly matches the conclusion drawn from the algebraic steps in the Reason.
Answer: (iii) Both A and R are true and R is the correct reason for A.
Step 2: Look at Assertion (A). It states the combined mean is (x+y)/2, which perfectly matches the conclusion drawn from the algebraic steps in the Reason.
Answer: (iii) Both A and R are true and R is the correct reason for A.
2. The mean of 100 observations is 40. It is found that an observation 53 was misread as 83. Find the correct mean.
Step 1: Calculate the incorrect total sum of the 100 observations: 100 * 40 = 4000.
Step 2: Identify the error: The incorrect value added was 83, and the correct value that should have been added is 53.
Step 3: Find the correct total sum by subtracting the wrong value and adding the right value: Correct sum = 4000 - 83 + 53.
Step 4: Correct sum = 4000 - 30 = 3970.
Step 5: Calculate the correct mean = Correct sum / Total observations = 3970 / 100.
Step 6: Correct mean = 39.7.
Answer: The correct mean is 39.7.
Step 2: Identify the error: The incorrect value added was 83, and the correct value that should have been added is 53.
Step 3: Find the correct total sum by subtracting the wrong value and adding the right value: Correct sum = 4000 - 83 + 53.
Step 4: Correct sum = 4000 - 30 = 3970.
Step 5: Calculate the correct mean = Correct sum / Total observations = 3970 / 100.
Step 6: Correct mean = 39.7.
Answer: The correct mean is 39.7.
3. The mean of 200 items was 50. Later on, it was discovered that two items were misread as 92 and 8 instead of 192 and 88. Find the correct mean.
Step 1: Calculate the incorrect total sum of 200 items: 200 * 50 = 10000.
Step 2: Identify the errors: Wrong values added were 92 and 8. Correct values that should have been added are 192 and 88.
Step 3: Calculate the correct total sum: Correct sum = 10000 - (92 + 8) + (192 + 88).
Step 4: Correct sum = 10000 - 100 + 280 = 10180.
Step 5: Calculate the correct mean = 10180 / 200.
Step 6: Correct mean = 50.9.
Answer: The correct mean is 50.9.
Step 2: Identify the errors: Wrong values added were 92 and 8. Correct values that should have been added are 192 and 88.
Step 3: Calculate the correct total sum: Correct sum = 10000 - (92 + 8) + (192 + 88).
Step 4: Correct sum = 10000 - 100 + 280 = 10180.
Step 5: Calculate the correct mean = 10180 / 200.
Step 6: Correct mean = 50.9.
Answer: The correct mean is 50.9.
4. Find the mean of 75 numbers, if the mean of 45 of them is 18 and the mean of the remaining ones is 13.
Step 1: Calculate the sum of the first 45 numbers: 45 * 18 = 810.
Step 2: Find the count of the remaining numbers: 75 - 45 = 30 numbers.
Step 3: Calculate the sum of these remaining 30 numbers: 30 * 13 = 390.
Step 4: Find the total sum of all 75 numbers: 810 + 390 = 1200.
Step 5: Calculate the final overall mean: 1200 / 75.
Step 6: Overall mean = 16.
Answer: The mean is 16.
Step 2: Find the count of the remaining numbers: 75 - 45 = 30 numbers.
Step 3: Calculate the sum of these remaining 30 numbers: 30 * 13 = 390.
Step 4: Find the total sum of all 75 numbers: 810 + 390 = 1200.
Step 5: Calculate the final overall mean: 1200 / 75.
Step 6: Overall mean = 16.
Answer: The mean is 16.
5. The mean weight of 120 students of a school is 52.75 kg. If the mean weight of 50 of them is 51 kg, find the mean weight of the remaining students.
Step 1: Calculate the total weight of all 120 students: 120 * 52.75 = 6330 kg.
Step 2: Calculate the total weight of the given 50 students: 50 * 51 = 2550 kg.
Step 3: Find the number of remaining students: 120 - 50 = 70 students.
Step 4: Calculate the total weight of these remaining 70 students: 6330 - 2550 = 3780 kg.
Step 5: Find the mean weight of the remaining students: 3780 / 70.
Step 6: Mean weight = 54 kg.
Answer: The mean weight is 54 kg.
Step 2: Calculate the total weight of the given 50 students: 50 * 51 = 2550 kg.
Step 3: Find the number of remaining students: 120 - 50 = 70 students.
Step 4: Calculate the total weight of these remaining 70 students: 6330 - 2550 = 3780 kg.
Step 5: Find the mean weight of the remaining students: 3780 / 70.
Step 6: Mean weight = 54 kg.
Answer: The mean weight is 54 kg.
6. The mean marks (out of 100) of boys and girls in an examination are 70 and 73 respectively. If the mean marks of all the students in that examination is 72.25, find the ratio of the number of boys to the number of girls.
Step 1: Let the number of boys be 'b' and the number of girls be 'g'.
Step 2: Total marks of all boys = 70 * b = 70b.
Step 3: Total marks of all girls = 73 * g = 73g.
Step 4: Total marks of all students together = 72.25 * (b + g).
Step 5: Set up the equation: 70b + 73g = 72.25b + 72.25g.
Step 6: Group the 'b' terms and 'g' terms on opposite sides: 73g - 72.25g = 72.25b - 70b.
Step 7: 0.75g = 2.25b.
Step 8: Find the ratio b/g: b/g = 0.75 / 2.25.
Step 9: Simplify the decimal ratio: b/g = 75 / 225 = 1 / 3.
Answer: The ratio of boys to girls is 1:3.
Step 2: Total marks of all boys = 70 * b = 70b.
Step 3: Total marks of all girls = 73 * g = 73g.
Step 4: Total marks of all students together = 72.25 * (b + g).
Step 5: Set up the equation: 70b + 73g = 72.25b + 72.25g.
Step 6: Group the 'b' terms and 'g' terms on opposite sides: 73g - 72.25g = 72.25b - 70b.
Step 7: 0.75g = 2.25b.
Step 8: Find the ratio b/g: b/g = 0.75 / 2.25.
Step 9: Simplify the decimal ratio: b/g = 75 / 225 = 1 / 3.
Answer: The ratio of boys to girls is 1:3.
7. Find x, if 9, x, 14, 18, x, x, 8, 10 and 4 have a mean of 11.
Step 1: Count the total number of observations (n), which is 9.
Step 2: Calculate the sum of all observations: 9 + x + 14 + 18 + x + x + 8 + 10 + 4.
Step 3: Sum = 3x + 63.
Step 4: Set up the formula for mean: (3x + 63) / 9 = 11.
Step 5: Multiply both sides by 9: 3x + 63 = 99.
Step 6: Subtract 63 from both sides: 3x = 36.
Step 7: Solve for x: x = 12.
Answer: The value of x is 12.
Step 2: Calculate the sum of all observations: 9 + x + 14 + 18 + x + x + 8 + 10 + 4.
Step 3: Sum = 3x + 63.
Step 4: Set up the formula for mean: (3x + 63) / 9 = 11.
Step 5: Multiply both sides by 9: 3x + 63 = 99.
Step 6: Subtract 63 from both sides: 3x = 36.
Step 7: Solve for x: x = 12.
Answer: The value of x is 12.
8. In a series of tests, A appeared for 8 tests. Each test was marked out of 30 and averages 25. However, while checking his files, A could only find 7 of the 8 tests. For these he scored 29, 26, 18, 20, 27, 24 and 29. Determine how many marks he scored for the eighth test. (LS)
Step 1: Calculate the total marks A scored across all 8 tests: 8 * 25 = 200.
Step 2: Calculate the sum of the marks for the 7 tests he found: 29 + 26 + 18 + 20 + 27 + 24 + 29.
Step 3: Sum of 7 tests = 173.
Step 4: The score of the 8th test is the difference between total marks and the sum of 7 tests.
Step 5: Score of 8th test = 200 - 173 = 27.
Answer: He scored 27 marks for the eighth test.
Step 2: Calculate the sum of the marks for the 7 tests he found: 29 + 26 + 18 + 20 + 27 + 24 + 29.
Step 3: Sum of 7 tests = 173.
Step 4: The score of the 8th test is the difference between total marks and the sum of 7 tests.
Step 5: Score of 8th test = 200 - 173 = 27.
Answer: He scored 27 marks for the eighth test.
9. Find the mean of 8, 12, 16, 22, 10 and 4. Find the resulting mean, if each of the observations, given above, be :
(i) multiplied by 3. (ii) divided by 2.
(iii) multiplied by 3 and then divided by 2.
(iv) increased by 25%.
(v) decreased by 40%.
Step 1: Find original mean. Sum = 8 + 12 + 16 + 22 + 10 + 4 = 72. Number of terms = 6. Original Mean = 72 / 6 = 12.
Step 2: Apply properties of mean for each scenario. Whatever mathematical operation is applied uniformly to all data points is also applied to the mean.
Step 3: (i) If multiplied by 3, new mean = 12 * 3 = 36.
Step 4: (ii) If divided by 2, new mean = 12 / 2 = 6.
Step 5: (iii) If multiplied by 3 then divided by 2, new mean = (12 * 3) / 2 = 36 / 2 = 18.
Step 6: (iv) If increased by 25%, new mean = 12 * (1 + 0.25) = 12 * 1.25 = 15.
Step 7: (v) If decreased by 40%, new mean = 12 * (1 - 0.40) = 12 * 0.60 = 7.2.
Answers:
Initial mean = 12
(i) 36
(ii) 6
(iii) 18
(iv) 15
(v) 7.2
Step 2: Apply properties of mean for each scenario. Whatever mathematical operation is applied uniformly to all data points is also applied to the mean.
Step 3: (i) If multiplied by 3, new mean = 12 * 3 = 36.
Step 4: (ii) If divided by 2, new mean = 12 / 2 = 6.
Step 5: (iii) If multiplied by 3 then divided by 2, new mean = (12 * 3) / 2 = 36 / 2 = 18.
Step 6: (iv) If increased by 25%, new mean = 12 * (1 + 0.25) = 12 * 1.25 = 15.
Step 7: (v) If decreased by 40%, new mean = 12 * (1 - 0.40) = 12 * 0.60 = 7.2.
Answers:
Initial mean = 12
(i) 36
(ii) 6
(iii) 18
(iv) 15
(v) 7.2
10. The mean of 18, 24, 15, 2x + 1 and 12 is 21. Find the value of x.
Step 1: Count the total number of observations, which is 5.
Step 2: Find the sum of all observations: 18 + 24 + 15 + (2x + 1) + 12.
Step 3: Sum = 2x + 70.
Step 4: Set up the mean equation: (2x + 70) / 5 = 21.
Step 5: Multiply both sides by 5: 2x + 70 = 105.
Step 6: Subtract 70 from both sides: 2x = 35.
Step 7: Divide by 2: x = 17.5.
Answer: The value of x is 17.5.
Step 2: Find the sum of all observations: 18 + 24 + 15 + (2x + 1) + 12.
Step 3: Sum = 2x + 70.
Step 4: Set up the mean equation: (2x + 70) / 5 = 21.
Step 5: Multiply both sides by 5: 2x + 70 = 105.
Step 6: Subtract 70 from both sides: 2x = 35.
Step 7: Divide by 2: x = 17.5.
Answer: The value of x is 17.5.
11. The mean of 6 numbers is 42. If one number is excluded, the mean of remaining numbers is 45. Find the excluded number.
Step 1: Calculate the total sum of the initial 6 numbers: 6 * 42 = 252.
Step 2: If one number is excluded, 5 numbers remain, and their new mean is 45.
Step 3: Calculate the total sum of these remaining 5 numbers: 5 * 45 = 225.
Step 4: The excluded number is the difference between the initial sum and the new sum.
Step 5: Excluded number = 252 - 225 = 27.
Answer: The excluded number is 27.
Step 2: If one number is excluded, 5 numbers remain, and their new mean is 45.
Step 3: Calculate the total sum of these remaining 5 numbers: 5 * 45 = 225.
Step 4: The excluded number is the difference between the initial sum and the new sum.
Step 5: Excluded number = 252 - 225 = 27.
Answer: The excluded number is 27.
12. The mean of 10 numbers is 24. If one more number is included, the new mean is 25. Find the included number.
Step 1: Calculate the total sum of the initial 10 numbers: 10 * 24 = 240.
Step 2: When one number is included, there are 11 numbers in total, with a new mean of 25.
Step 3: Calculate the new total sum of 11 numbers: 11 * 25 = 275.
Step 4: The included number is the difference between the new sum and the initial sum.
Step 5: Included number = 275 - 240 = 35.
Answer: The included number is 35.
Step 2: When one number is included, there are 11 numbers in total, with a new mean of 25.
Step 3: Calculate the new total sum of 11 numbers: 11 * 25 = 275.
Step 4: The included number is the difference between the new sum and the initial sum.
Step 5: Included number = 275 - 240 = 35.
Answer: The included number is 35.
13. The following observations have been arranged in ascending order. If the median of the data is 78, find the value of x.
44, 47, 63, 65, x+13, 87, 93, 99, 110.
Step 1: Count the total number of observations (n), which is 9 (odd).
Step 2: The median is the ((9+1)/2)th term, which is the 5th term.
Step 3: Identify the 5th term in the array, which is (x + 13).
Step 4: Set the 5th term equal to the given median: x + 13 = 78.
Step 5: Solve for x: x = 78 - 13 = 65.
Answer: The value of x is 65.
Step 2: The median is the ((9+1)/2)th term, which is the 5th term.
Step 3: Identify the 5th term in the array, which is (x + 13).
Step 4: Set the 5th term equal to the given median: x + 13 = 78.
Step 5: Solve for x: x = 78 - 13 = 65.
Answer: The value of x is 65.
14. The following observations have been arranged in ascending order. If the median of these observations is 58, find the value of x.
24, 27, 43, 48, x-1, x+3, 68, 73, 80, 90.
Step 1: Count the total number of observations (n), which is 10 (even).
Step 2: The median is the average of the 5th and 6th terms.
Step 3: Identify the 5th term (x - 1) and the 6th term (x + 3).
Step 4: Set up the equation for the median: ((x - 1) + (x + 3)) / 2 = 58.
Step 5: Simplify numerator: (2x + 2) / 2 = 58.
Step 6: x + 1 = 58.
Step 7: x = 57.
Answer: The value of x is 57.
Step 2: The median is the average of the 5th and 6th terms.
Step 3: Identify the 5th term (x - 1) and the 6th term (x + 3).
Step 4: Set up the equation for the median: ((x - 1) + (x + 3)) / 2 = 58.
Step 5: Simplify numerator: (2x + 2) / 2 = 58.
Step 6: x + 1 = 58.
Step 7: x = 57.
Answer: The value of x is 57.
15. Find the mean of the following data : (HOTS)
30, 32, 24, 34, 26, 28, 30, 35, 33, 25
(i) Show that the sum of the deviations of all the given observations from the mean is zero.
(ii) Find the median of the given data.
Step 1: First, calculate the mean. Sum = 30 + 32 + 24 + 34 + 26 + 28 + 30 + 35 + 33 + 25 = 297.
Step 2: Total observations = 10.
Step 3: Mean = 297 / 10 = 29.7.
Step 4: For part (i), find the deviation of each observation from 29.7 by subtracting 29.7 from each number.
Step 5: Deviations: (30 - 29.7)=0.3, (32 - 29.7)=2.3, (24 - 29.7)=-5.7, (34 - 29.7)=4.3, (26 - 29.7)=-3.7, (28 - 29.7)=-1.7, (30 - 29.7)=0.3, (35 - 29.7)=5.3, (33 - 29.7)=3.3, (25 - 29.7)=-4.7.
Step 6: Sum of positive deviations = 0.3 + 2.3 + 4.3 + 0.3 + 5.3 + 3.3 = 15.8.
Step 7: Sum of negative deviations = -5.7 - 3.7 - 1.7 - 4.7 = -15.8.
Step 8: Total sum of deviations = 15.8 + (-15.8) = 0. This shows the property holds true.
Step 9: For part (ii), arrange the data in ascending order: 24, 25, 26, 28, 30, 30, 32, 33, 34, 35.
Step 10: Since n = 10, median is the average of 5th and 6th terms: (30 + 30) / 2 = 30.
Answers:
Mean = 29.7
(i) Sum of deviations is shown to be zero.
(ii) Median = 30
Step 2: Total observations = 10.
Step 3: Mean = 297 / 10 = 29.7.
Step 4: For part (i), find the deviation of each observation from 29.7 by subtracting 29.7 from each number.
Step 5: Deviations: (30 - 29.7)=0.3, (32 - 29.7)=2.3, (24 - 29.7)=-5.7, (34 - 29.7)=4.3, (26 - 29.7)=-3.7, (28 - 29.7)=-1.7, (30 - 29.7)=0.3, (35 - 29.7)=5.3, (33 - 29.7)=3.3, (25 - 29.7)=-4.7.
Step 6: Sum of positive deviations = 0.3 + 2.3 + 4.3 + 0.3 + 5.3 + 3.3 = 15.8.
Step 7: Sum of negative deviations = -5.7 - 3.7 - 1.7 - 4.7 = -15.8.
Step 8: Total sum of deviations = 15.8 + (-15.8) = 0. This shows the property holds true.
Step 9: For part (ii), arrange the data in ascending order: 24, 25, 26, 28, 30, 30, 32, 33, 34, 35.
Step 10: Since n = 10, median is the average of 5th and 6th terms: (30 + 30) / 2 = 30.
Answers:
Mean = 29.7
(i) Sum of deviations is shown to be zero.
(ii) Median = 30
16. Find the mean and median of the data : (HOTS)
35, 48, 92, 76, 64, 52, 51, 63 and 71.
If 51 is replaced by 66, what will be the new median ?
Step 1: Calculate the mean. Find the sum: 35 + 48 + 92 + 76 + 64 + 52 + 51 + 63 + 71 = 552.
Step 2: Total number of observations (n) = 9.
Step 3: Mean = 552 / 9 = 61.33.
Step 4: Find original median by arranging in ascending order: 35, 48, 51, 52, 63, 64, 71, 76, 92.
Step 5: Median is the 5th term, which is 63.
Step 6: If 51 is replaced by 66, insert 66 in correct sorted position and remove 51.
Step 7: New ascending order: 35, 48, 52, 63, 64, 66, 71, 76, 92.
Step 8: The new median is the 5th term of this new list, which is 64.
Answers:
Mean = 61.33
Original Median = 63
New Median = 64
Step 2: Total number of observations (n) = 9.
Step 3: Mean = 552 / 9 = 61.33.
Step 4: Find original median by arranging in ascending order: 35, 48, 51, 52, 63, 64, 71, 76, 92.
Step 5: Median is the 5th term, which is 63.
Step 6: If 51 is replaced by 66, insert 66 in correct sorted position and remove 51.
Step 7: New ascending order: 35, 48, 52, 63, 64, 66, 71, 76, 92.
Step 8: The new median is the 5th term of this new list, which is 64.
Answers:
Mean = 61.33
Original Median = 63
New Median = 64
17. The mean of x, x+2, x+4, x+6 and x+8 is 11, find the mean of the first three observations.
Step 1: Set up the equation for the mean of the 5 observations: (x + (x+2) + (x+4) + (x+6) + (x+8)) / 5 = 11.
Step 2: Sum the terms in the numerator: 5x + 20.
Step 3: 5x + 20 = 55.
Step 4: 5x = 35, so x = 7.
Step 5: List the first three observations: x, x+2, x+4.
Step 6: Substitute x = 7: 7, 9, 11.
Step 7: Calculate the mean of these first three observations: (7 + 9 + 11) / 3.
Step 8: Mean = 27 / 3 = 9.
Answer: The mean of the first three observations is 9.
Step 2: Sum the terms in the numerator: 5x + 20.
Step 3: 5x + 20 = 55.
Step 4: 5x = 35, so x = 7.
Step 5: List the first three observations: x, x+2, x+4.
Step 6: Substitute x = 7: 7, 9, 11.
Step 7: Calculate the mean of these first three observations: (7 + 9 + 11) / 3.
Step 8: Mean = 27 / 3 = 9.
Answer: The mean of the first three observations is 9.
18. Find the mean and median of all the positive factors of 72.
Step 1: List all positive factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Step 2: Total number of factors (n) = 12.
Step 3: Find the sum of all these factors: 1+2+3+4+6+8+9+12+18+24+36+72 = 195.
Step 4: Calculate Mean = 195 / 12 = 16.25.
Step 5: Since there are 12 factors (even), the median is the average of the 6th and 7th terms.
Step 6: 6th term is 8 and 7th term is 9.
Step 7: Calculate Median = (8 + 9) / 2 = 17 / 2 = 8.5.
Answer: Mean = 16.25, Median = 8.5
Step 2: Total number of factors (n) = 12.
Step 3: Find the sum of all these factors: 1+2+3+4+6+8+9+12+18+24+36+72 = 195.
Step 4: Calculate Mean = 195 / 12 = 16.25.
Step 5: Since there are 12 factors (even), the median is the average of the 6th and 7th terms.
Step 6: 6th term is 8 and 7th term is 9.
Step 7: Calculate Median = (8 + 9) / 2 = 17 / 2 = 8.5.
Answer: Mean = 16.25, Median = 8.5
19. The mean weight of 60 students in a class is 40 kg. The mean weight of boys is 50 kg while that of girls is 30 kg. Find the number of boys and girls in the class.
Step 1: Let the number of boys be 'b'. Since total students are 60, number of girls is '60 - b'.
Step 2: Total weight of all 60 students = 60 * 40 = 2400 kg.
Step 3: Total weight of boys = 50 * b = 50b.
Step 4: Total weight of girls = 30 * (60 - b) = 1800 - 30b.
Step 5: The sum of weights of boys and girls must equal total weight: 50b + 1800 - 30b = 2400.
Step 6: Simplify the equation: 20b + 1800 = 2400.
Step 7: 20b = 600, so b = 30.
Step 8: The number of boys is 30. Therefore, number of girls is 60 - 30 = 30.
Answer: The number of boys is 30 and the number of girls is 30.
Step 2: Total weight of all 60 students = 60 * 40 = 2400 kg.
Step 3: Total weight of boys = 50 * b = 50b.
Step 4: Total weight of girls = 30 * (60 - b) = 1800 - 30b.
Step 5: The sum of weights of boys and girls must equal total weight: 50b + 1800 - 30b = 2400.
Step 6: Simplify the equation: 20b + 1800 = 2400.
Step 7: 20b = 600, so b = 30.
Step 8: The number of boys is 30. Therefore, number of girls is 60 - 30 = 30.
Answer: The number of boys is 30 and the number of girls is 30.
20. The average of n numbers x_1, x_2, x_3 ....... x_n is A. If x_1 is replaced by (x + a)x_1, x_2 is replaced by (x + a)x_2 and so on. Find the new average.
Step 1: The original sum of the n numbers is x_1 + x_2 + ... + x_n = n * A.
Step 2: Write down the sum of the new numbers: (x + a)x_1 + (x + a)x_2 + ... + (x + a)x_n.
Step 3: Factor out the common term (x + a): (x + a) * (x_1 + x_2 + ... + x_n).
Step 4: Substitute the original sum (nA) into the expression: (x + a) * (nA).
Step 5: The new average is the new sum divided by n: (x + a) * nA / n.
Step 6: Cancel out 'n' to find the final new average = A * (x + a).
Answer: The new average is A(x + a).
Step 2: Write down the sum of the new numbers: (x + a)x_1 + (x + a)x_2 + ... + (x + a)x_n.
Step 3: Factor out the common term (x + a): (x + a) * (x_1 + x_2 + ... + x_n).
Step 4: Substitute the original sum (nA) into the expression: (x + a) * (nA).
Step 5: The new average is the new sum divided by n: (x + a) * nA / n.
Step 6: Cancel out 'n' to find the final new average = A * (x + a).
Answer: The new average is A(x + a).
21. The heights (in cm) of the volley-ball players from team A and team B were recorded as :
Team A : 180, 178, 176, 181, 190, 175, 187
Team B : 174, 175, 190, 179, 178, 185, 177
Which team had the greater average height ?
Find the median of team A and team B.
Step 1: Calculate the mean of Team A. Sum = 180+178+176+181+190+175+187 = 1267.
Step 2: Team A Mean = 1267 / 7 = 181 cm.
Step 3: Calculate the mean of Team B. Sum = 174+175+190+179+178+185+177 = 1258.
Step 4: Team B Mean = 1258 / 7 = 179.71 cm.
Step 5: Team A has the greater average height.
Step 6: Find Team A median by arranging in order: 175, 176, 178, 180, 181, 187, 190.
Step 7: Team A Median = 4th term = 180 cm.
Step 8: Find Team B median by arranging in order: 174, 175, 177, 178, 179, 185, 190.
Step 9: Team B Median = 4th term = 178 cm.
Answers:
Team A has a greater average height.
Team A median = 180 cm.
Team B median = 178 cm.
Step 2: Team A Mean = 1267 / 7 = 181 cm.
Step 3: Calculate the mean of Team B. Sum = 174+175+190+179+178+185+177 = 1258.
Step 4: Team B Mean = 1258 / 7 = 179.71 cm.
Step 5: Team A has the greater average height.
Step 6: Find Team A median by arranging in order: 175, 176, 178, 180, 181, 187, 190.
Step 7: Team A Median = 4th term = 180 cm.
Step 8: Find Team B median by arranging in order: 174, 175, 177, 178, 179, 185, 190.
Step 9: Team B Median = 4th term = 178 cm.
Answers:
Team A has a greater average height.
Team A median = 180 cm.
Team B median = 178 cm.
22. The Mean of the following arrayed data 5, 8, (3x - 1), (4x + 1), (3x + 7) is equal to its Median. Find the value of x.
Step 1: Arrayed data implies it is already in ascending order. Number of terms (n) = 5.
Step 2: The Median is the 3rd term, which is (3x - 1).
Step 3: Calculate the sum for the Mean: 5 + 8 + (3x - 1) + (4x + 1) + (3x + 7) = 10x + 20.
Step 4: The Mean = (10x + 20) / 5 = 2x + 4.
Step 5: Since Mean = Median, equate the two expressions: 2x + 4 = 3x - 1.
Step 6: Solve the equation: 4 + 1 = 3x - 2x.
Step 7: x = 5.
Answer: The value of x is 5.
Step 2: The Median is the 3rd term, which is (3x - 1).
Step 3: Calculate the sum for the Mean: 5 + 8 + (3x - 1) + (4x + 1) + (3x + 7) = 10x + 20.
Step 4: The Mean = (10x + 20) / 5 = 2x + 4.
Step 5: Since Mean = Median, equate the two expressions: 2x + 4 = 3x - 1.
Step 6: Solve the equation: 4 + 1 = 3x - 2x.
Step 7: x = 5.
Answer: The value of x is 5.
Case-Study Based Question
1. The data representing the number of people in municipalities in different towns is shown by the bar graph given below.
(i) What is the number of people living in municipality F ?
(ii) What is the total number of people living in municipalities A to C ?
(iii) What is the total population of all the municipalities under consideration ?
(iv) What the average (mean) of people living in municipalities B to E ?
Step 1: Carefully read the bar graph values on the y-axis for each municipality on the x-axis.
Step 2: Read values: A=1600, B=1400, C=1600, D=800, E=1200, F=800, G=400.
Step 3: For (i), read the value for F directly, which is 800.
Step 4: For (ii), sum the values for A, B, and C: 1600 + 1400 + 1600 = 4600.
Step 5: For (iii), sum all values from A to G: 1600+1400+1600+800+1200+800+400 = 7800.
Step 6: For (iv), sum the values from B to E (B, C, D, E): 1400 + 1600 + 800 + 1200 = 5000.
Step 7: Divide by the 4 municipalities to find the mean: 5000 / 4 = 1250.
Answers:
(i) 800
(ii) 4600
(iii) 7800
(iv) 1250
Step 2: Read values: A=1600, B=1400, C=1600, D=800, E=1200, F=800, G=400.
Step 3: For (i), read the value for F directly, which is 800.
Step 4: For (ii), sum the values for A, B, and C: 1600 + 1400 + 1600 = 4600.
Step 5: For (iii), sum all values from A to G: 1600+1400+1600+800+1200+800+400 = 7800.
Step 6: For (iv), sum the values from B to E (B, C, D, E): 1400 + 1600 + 800 + 1200 = 5000.
Step 7: Divide by the 4 municipalities to find the mean: 5000 / 4 = 1250.
Answers:
(i) 800
(ii) 4600
(iii) 7800
(iv) 1250
2. The Frequency Polygon shown depicts the monthly rainfall in the Kathmandu (Nepal).
Answer the following:
(i) Which month had the highest rainfall?
(ii) How many months had rainfall above 200 mm?
(iii) How many months had rainfall below 100 mm?
(iv) Calculate the Mean rainfall in the months from June to September.
Step 1: Carefully analyze the peaks and values on the line graph (Frequency Polygon) for each month.
Step 2: Approximate read values: Jan:0, Feb:10, Mar:25, Apr:50, May:125, Jun:250, Jul:350, Aug:300, Sep:150, Oct:50, Nov:10, Dec:0.
Step 3: For (i), the highest peak is in July (350 mm).
Step 4: For (ii), count the months with values over 200 mm: June(250), July(350), August(300). There are 3 months.
Step 5: For (iii), count the months strictly below 100 mm: Jan, Feb, Mar, Apr, Oct, Nov, Dec. Total is 7 months.
Step 6: For (iv), calculate mean for Jun, Jul, Aug, Sep. Sum = 250 + 350 + 300 + 150 = 1050.
Step 7: Mean = 1050 / 4 months = 262.5 mm.
Answers:
(i) July
(ii) 3 months
(iii) 7 months
(iv) 262.5 mm
Step 2: Approximate read values: Jan:0, Feb:10, Mar:25, Apr:50, May:125, Jun:250, Jul:350, Aug:300, Sep:150, Oct:50, Nov:10, Dec:0.
Step 3: For (i), the highest peak is in July (350 mm).
Step 4: For (ii), count the months with values over 200 mm: June(250), July(350), August(300). There are 3 months.
Step 5: For (iii), count the months strictly below 100 mm: Jan, Feb, Mar, Apr, Oct, Nov, Dec. Total is 7 months.
Step 6: For (iv), calculate mean for Jun, Jul, Aug, Sep. Sum = 250 + 350 + 300 + 150 = 1050.
Step 7: Mean = 1050 / 4 months = 262.5 mm.
Answers:
(i) July
(ii) 3 months
(iii) 7 months
(iv) 262.5 mm
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Quick Review Flashcards - Click to flip and test your knowledge!
Question
In statistics, what is the specific term used to denote the arithmetic average of a set of observations?
Answer
Mean
Question
Which mathematical symbol is generally used to denote the mean of a set of observations $x$?
Answer
$\bar{x}$
Question
What is the general formula for the mean $\bar{x}$ of $n$ observations $x_1, x_2, x_3, \dots, x_n$?
Answer
$\bar{x} = \frac{x_1 + x_2 + x_3 + \dots + x_n}{n}$
Question
How is the mean $\bar{x}$ expressed using summation notation $\sum$ for $n$ observations $x_i$?
Answer
$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$
Question
The symbol $\sum_{i=1}^{n} x_i$ is a shorthand for which mathematical operation performed on observations $x_1$ to $x_n$?
Answer
Addition (or finding the sum)
Question
To find the mean of any set of observations, the sum of those observations must be divided by the _____.
Answer
total number of observations
Question
According to the properties of the mean, what is the sum of the deviations of all observations $x$ from their mean $\bar{x}$?
Answer
Zero (i.e., $\sum(x - \bar{x}) = 0$)
Question
If each observation in a data set is increased by a quantity $a$, how is the resulting mean affected?
Answer
The mean is also increased by the same quantity $a$.
Question
If each observation $x_i$ in a set with mean $\bar{x}$ is decreased by a quantity $a$, what is the new mean?
Answer
$\bar{x} - a$
Question
If each observation in a data set is multiplied by a constant $a$, by what factor does the mean change?
Answer
$a$
Question
If $\bar{x}$ is the mean of $n$ observations, what is the resulting mean if every observation is divided by a non-zero quantity $a$?
Answer
$\frac{\bar{x}}{a}$
Question
How is the total sum of observations ($\sum x$) calculated if the mean $\bar{x}$ and the number of observations $n$ are known?
Answer
$\sum x = n \cdot \bar{x}$
Question
When correcting a mean where one value was wrongly recorded, what is the first step after finding the incorrect total sum?
Answer
Subtract the incorrect observation and add the correct observation to find the correct total sum.
Question
If one observation is excluded from a set of $n$ observations, the new total number of observations used to calculate the mean becomes _____.
Answer
$n - 1$
Question
If a group of $n$ observations has mean $x$ and a second group of $n$ observations has mean $y$, what is the combined mean of both groups?
Answer
$\frac{x + y}{2}$
Question
In a combined group where $n_1$ items have mean $\bar{x}_1$ and $n_2$ items have mean $\bar{x}_2$, what is the formula for the total sum of all items?
Answer
$n_1 \bar{x}_1 + n_2 \bar{x}_2$
Question
What statistical term represents the value of the middle term in a given set of data?
Answer
Median
Question
What mandatory step must be performed on a raw data set before calculating the median?
Answer
Arrange the data in ascending or descending order of magnitude.
Question
If the total number of observations $n$ is an odd number, which term in the ordered sequence is the median?
Answer
$(\frac{n+1}{2})^{th}$ term
Question
If the total number of observations $n$ is an even number, the median is the average of which two terms?
Answer
The $(\frac{n}{2})^{th}$ term and the $(\frac{n}{2} + 1)^{th}$ term
Question
Which formula is used to calculate the median when the number of observations $n$ is even?
Answer
$\frac{1}{2} [(\frac{n}{2})^{th} \text{ term} + (\frac{n}{2} + 1)^{th} \text{ term}]$
Question
If a data set is already arranged in ascending order, how does doubling the value of the very last observation affect the median?
Answer
The median remains the same.
Question
If a data set is arranged in descending order and the first observation is doubled, how is the median affected?
Answer
The median remains the same.
Question
If every observation in a data set is decreased by 2, how do the mean and median change?
Answer
Both the mean and median decrease by 2.
Question
If every observation in a set is doubled, what happens to the resulting median?
Answer
The median is doubled.
Question
In a set of 80 observations with a median of 60, what is the new median if every observation is doubled?
Answer
120
Question
When a missing value $x$ is part of an ordered sequence and the median is known, what equation is used if $n$ is even?
Answer
$\frac{1}{2} [(\text{lower middle term}) + (\text{upper middle term})] = \text{Median}$
Question
If the mean of $x, x+2, x+4, x+6$ and $x+8$ is 11, what is the value of $x$?
Answer
7
Question
If the mean of 10 observations is 20 and one number is included to make the new mean 21, what is the value of the included number?
Answer
31
Question
What is the effect on the sum of observations if one observation is increased by a value $k$?
Answer
The total sum increases by $k$.
Question
If the mean of 75 numbers is known, and the mean of 45 of those numbers is also known, how is the sum of the remaining 30 numbers found?
Answer
Subtract the sum of the 45 numbers from the total sum of the 75 numbers.
Question
In a frequency polygon, what does the horizontal axis ($x$-axis) typically represent?
Answer
The variable or category under consideration (e.g., Months, Municipalities).
Question
In a bar graph representing population, what does the height of each bar typically indicate?
Answer
The frequency or number of people in that category.
Question
To calculate the average rainfall from June to September using a graph, which monthly values must be summed before dividing by 4?
Answer
The rainfall values for June, July, August, and September.
Question
If the median of an ordered set of 10 observations is 78, which two terms are averaged to get this value?
Answer
The $5^{th}$ and $6^{th}$ terms
Question
If the mean of 6 numbers is 42 and one number is excluded, leaving a mean of 45, what is the excluded number?
Answer
27
Question
What is the median of the first six natural numbers (1, 2, 3, 4, 5, 6)?
Answer
3.5
Question
What is the mean of the first five even natural numbers (2, 4, 6, 8, 10)?
Answer
6
Question
Which measure of central tendency is most affected by rearranging the data into ascending order?
Answer
Median
Question
If $\sum x = 251$ for 7 observations, what is the mean expressed as a mixed fraction?
Answer
$35 \frac{6}{7}$
Question
Concept: Ungrouped Data
Answer
Definition: Data that has not been organised into groups or classes; typically presented as a simple list of observations.
Question
When calculating the mean of prime numbers between 20 and 50, which numbers are included in the sum?
Answer
23, 29, 31, 37, 41, 43, and 47
Question
If the mean of 40 observations is 160, what is the total sum of these observations?
Answer
6,400
Question
In an ordered set, if the $7^{th}$ number is increased by 8 and it remains the $7^{th}$ largest number, how is the median of 10 observations affected?
Answer
The median remains unchanged (as the median is the average of the $5^{th}$ and $6^{th}$ terms).
Question
What is the mean of the first ten odd natural numbers?
Answer
10
Question
What is the median of the set of numbers: 10, 12, 9, 8, 10, 12, 10, 6 and 4?
Answer
10
Question
How does increasing every observation in a set by $25\%$ affect the mean?
Answer
The mean also increases by $25\%$.
Question
If the mean of $x, x+2, x+4, x+6$ and $x+8$ is 11, what is the mean of the first three observations?
Answer
9
Question
True or False: The mean, arithmetic mean, and average are considered the same in basic statistics.
Answer
True
Question
If the mean of 100 observations is 30, and two observations (32 and 12) were wrongly taken instead of 23 and 11, what is the correct mean?
Answer
29.9
Question
If the mean of $n$ numbers is $A$, and each observation $x_i$ is replaced by $(x+a)x_i$, what is the new mean?
Answer
$(x+a)A$
Question
What is the median of the following five observations arranged in increasing order: 35, 48, 66, 40, and 35?
Answer
40 (after re-ordering to 35, 35, 40, 48, 66)
Question
In the formula for median when $n$ is odd, what does the superscript 'th' signify?
Answer
The position of the term in the ordered sequence.
Question
If the mean weight of 60 students is 40 kg, what is the total weight of the class?
Answer
2,400 kg
Question
If a student needs to find the ratio of boys to girls in a class given their individual mean weights and the class mean weight, which formula is used?
Answer
The combined mean formula: $\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}$
Question
Property: If each observation is decreased by $40\%$, the mean is _____.
Answer
decreased by $40\%$
Question
When finding the median of 14 observations, which two terms must be averaged?
Answer
The $7^{th}$ and $8^{th}$ terms
Question
If the mean of $x-5, x-3, x-1$ and $x+1$ is calculated, what is the resulting expression?
Answer
$\frac{4x-8}{4}$ (or $x-2$)
Question
Formula: The median for an even number of data points $n$ is defined as _____.
Answer
$\frac{1}{2} [(\frac{n}{2})^{th} \text{ term} + (\frac{n}{2} + 1)^{th} \text{ term}]$
Question
If a data set has an odd number of observations, does the median have to be one of the original values in the set?
Answer
Yes