The mean and standard deviation of 100 observations were calculated as 40 and 5.1, respectively by a student who took by mistake 50 instead of 40 for one observation. What are the correct mean and standard deviation?
Get the complete, step-by-step math solution for: "The mean and standard deviation of 100 observations were calculated as 40 and 5.1, respectively by a student who took by mistake 50 instead of 40 for ...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Calculate the incorrect sum of observations
We are given the incorrect mean and the number of observations. We can use the formula for the mean to find the incorrect sum of all observations. The mean is the sum of all observations divided by the number of observations.
Step 2: Calculate the correct sum of observations
To find the correct sum of observations, we subtract the mistakenly taken value (50) from the incorrect sum and add the correct value (40). This adjusts the total sum to reflect the accurate data.
Step 3: Calculate the correct mean
Now that we have the correct sum of observations and the number of observations remains the same, we can calculate the correct mean using the standard formula.
Step 4: Calculate the incorrect sum of squares
The formula for standard deviation involves the sum of squares of observations. We can rearrange the standard deviation formula to find the incorrect sum of squares, which is needed for correction.
Step 5: Calculate the correct sum of squares
Similar to correcting the sum of observations, we correct the sum of squares by subtracting the square of the incorrect observation and adding the square of the correct observation.
Step 6: Calculate the correct standard deviation
Finally, with the correct sum of squares and the correct mean, we can calculate the correct standard deviation using its formula.