Calculating Correct Mean After Input Errors
The problem involves finding the correct mean of a set of numbers when some values used in the initial calculation were incorrect.
Initial Calculation Summary
- Number of observations ($n$): 80
- Incorrect Mean ($\bar{x}_{incorrect}$): 49
Step 1: Calculate the Initial Sum
The sum of the numbers initially calculated is the product of the incorrect mean and the number of observations.
Initial Sum = Incorrect Mean $\times$ Number of observations
Sum$_{initial}$ = $\sum_{i=1}^{80} x_{incorrect, i} = \bar{x}_{incorrect} \times n = 49 \times 80 = 3920$
Step 2: Identify Sum of Incorrect and Correct Values
The values that were incorrectly used and the correct values are given.
- Incorrect numbers: 69 and 74
- Sum of incorrect numbers = $69 + 74 = 143$
- Correct numbers: 92 and 83
- Sum of correct numbers = $92 + 83 = 175$
Step 3: Calculate the Correct Sum
To find the correct sum, subtract the sum of the incorrect numbers from the initial sum and add the sum of the correct numbers.
Correct Sum = Initial Sum - Sum of incorrect numbers + Sum of correct numbers
Sum$_{correct}$ = $3920 - 143 + 175$
Sum$_{correct}$ = $3920 + (175 - 143)$
Sum$_{correct}$ = $3920 + 32 = 3952$
Step 4: Calculate the Correct Mean
The correct mean is calculated by dividing the correct sum by the total number of observations.
Correct Mean = Correct Sum / Number of observations
$\bar{x}_{correct} = \frac{\text{Sum}_{correct}}{n} = \frac{3952}{80}$
$\bar{x}_{correct} = 49.4$
Therefore, the correct mean is 49.4.
Important Questions from Average
The average of 28 numbers is 77. The average of first 14 numbers is 74 and the average of last 15 numbers is 84. If the 14 th number is excluded, then what is the average of remaining numbers? (correct to one decimal places)
24 students collected money for donation. The average contribution was Rs. 50. Later on, their teacher also contributed some money. Now the average contribution is Rs. 56. The teacher’s contribution is:
Out of 6 numbers, the sum of the first 5 numbers is 7 times the 6 th number. If their average is 136, then the 6 th number is:
The average of five numbers is 30. If one number is excluded, then average becomes 31. What is the excluded number?
The average weight of 49 students in a class is 39 kg. Seven of them whose average weight is 40 kg leave the class and other seven students whose average weight is 54 kg join the class. What is the new average weight (in kg) of the class?
