the mean of 30 observations was 17 with sd2. later it was found that two of the observations were incorrect, which were recorded as 19vand 21.find the correct sd if the incorrect values are deleted and correct values 22 and 15 are included

To find the correct standard deviation after adjusting for the incorrect observations, we need to follow a systematic approach. Let’s break it down step by step.

Initial Data Overview

We started with 30 observations that had a mean of 17 and a variance (which is the square of the standard deviation) of 2. This means the standard deviation (SD) is the square root of 2, which is approximately 1.41. The two incorrect observations were recorded as 19 and 21, and we need to replace them with the correct values of 22 and 15.

Step 1: Calculate the Total of the Original Observations

The mean is calculated as the total sum of observations divided by the number of observations. Therefore, we can find the total sum of the original observations:

• Total = Mean × Number of Observations
• Total = 17 × 30 = 510

Step 2: Adjust the Total for Incorrect Observations

Next, we need to adjust this total by removing the incorrect values and adding the correct ones:

• Remove the incorrect values: 510 - 19 - 21 = 470
• Add the correct values: 470 + 22 + 15 = 507

Step 3: Calculate the New Mean

Now that we have the new total, we can calculate the new mean with the updated observations:

• New Mean = New Total / Number of Observations
• New Mean = 507 / 30 = 16.9

Step 4: Calculate the New Variance

To find the new standard deviation, we first need to calculate the new variance. The formula for variance is:

• Variance = (Sum of Squares of Observations) / N - (Mean^2)

We need to calculate the sum of squares of the original observations. We can derive this from the variance:

• Variance = SD² = 2
• Sum of Squares = Variance × N + (Mean² × N)
• Sum of Squares = 2 × 30 + (17² × 30) = 60 + 8670 = 8730

Now, we need to adjust this sum of squares for the incorrect observations:

• Remove the squares of the incorrect values: 8730 - (19² + 21²) = 8730 - (361 + 441) = 8730 - 802 = 7928
• Add the squares of the correct values: 7928 + (22² + 15²) = 7928 + (484 + 225) = 7928 + 709 = 8637

Step 5: Calculate the New Variance and Standard Deviation

Now we can calculate the new variance:

• New Variance = Sum of Squares / N - (New Mean²)
• New Variance = 8637 / 30 - (16.9²) = 287.9 - 285.61 = 2.29

Finally, the new standard deviation is the square root of the new variance:

• New SD = √2.29 ≈ 1.51

Final Result

The corrected standard deviation, after replacing the incorrect observations with the correct ones, is approximately 1.51. This process illustrates how adjustments in data can significantly impact statistical measures like the mean and standard deviation.