For a statistical data $x_1, x_2, \ldots, x_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum_{i=1}^{10} x_i^2 = 371$. He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is

Step 1: Calculate the incorrect sum of observations

The mean of a dataset is calculated by dividing the sum of all observations by the number of observations. We are given the incorrect mean and the number of observations, so we can find the incorrect sum of observations.

Step 2: Calculate the correct sum of observations

To find the correct sum of observations, we subtract the incorrectly noted values from the incorrect sum and add the correct values. The incorrect values were 4 and 5, and the correct values are 6 and 8.

Step 3: Calculate the correct sum of squares

Similarly, to find the correct sum of squares, we subtract the squares of the incorrectly noted values from the incorrect sum of squares and add the squares of the correct values. The incorrect sum of squares was given as 371.

Step 4: Calculate the correct mean

Now that we have the correct sum of observations, we can calculate the correct mean by dividing it by the total number of observations, which is 10.

Step 5: Calculate the variance of the corrected data

The variance is calculated using the formula: sum of squares divided by the number of observations, minus the square of the mean. We use the corrected sum of squares and the corrected mean for this calculation.

More from this chapter

• Calculate the mean and variance of the first n natural numbers.
• 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 observ
• 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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