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?

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.

More from this chapter

• Calculate the mean and variance of the first n natural numbers.
• For a statistical data x_1, x_2,, x_10 of 10 values, a student obtained the mean as 5.5 and Σ_i=1^(10) x_i² = 371. He later found that he had noted two values i
• Let the mean and variance of five observations x_1=1, x_2=3, x_3=a, x_4=7 and x_5=b, with a>b, be 5 and 10 respectively. Then the variance of the observations n
• The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.
• If each of the observation x_1, x_2,..., x_n is increased by ' a ', where a is a negative or positive number, show that the variance remains unchanged.
• The variance of the numbers 8, 21, 34, 47,, 320 is:
• The variance of 20 observations is 5. If each observation is multiplied by 2, find the new variance of the resulting observations.
• Find the mean deviation about the median for the following data: x_i: 3, 6, 9, 12, 13, 15, 21, 22 f_i: 3, 4, 5, 2, 4, 5, 4, 3.

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