Understanding the Problem: Calculating the Mean After Omitting Observations

The problem asks us to find the new mean of a set of observations after certain incorrect values are removed. We start with the original mean and number of observations, use this to find the original sum, then adjust the sum and the count of observations based on the incorrect data to be omitted.

Initial Data Analysis

• Original number of observations (\(n_{\text{original}}\)): 100
• Original mean (\(\bar{x}_{\text{original}}\)): 20
• Incorrect observations to be omitted: 21, 21, 18, 20

Step-by-Step Solution for Correcting the Mean

Step 1: Calculate the Original Sum of Observations

The mean is calculated as the sum of observations divided by the number of observations. We can use this formula to find the total sum from the original data:

Formula: \(\text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}}\)

So, \(\text{Sum of Observations} = \text{Mean} \times \text{Number of Observations}\)

Original Sum (\(\Sigma x_{\text{original}}\)) = \(n_{\text{original}} \times \bar{x}_{\text{original}}\)

\(\Sigma x_{\text{original}} = 100 \times 20\)

\(\Sigma x_{\text{original}} = 2000\)

The original sum of the 100 observations was 2000.

Step 2: Identify and Sum the Incorrect Observations to be Omitted

The problem states that four observations were incorrect and should be omitted. These observations are 21, 21, 18, and 20.

Sum of Incorrect Observations (\(\Sigma x_{\text{incorrect}}\)) = \(21 + 21 + 18 + 20\)

\(\Sigma x_{\text{incorrect}} = 80\)

Step 3: Calculate the New Sum of Observations

Since the incorrect observations are to be omitted, we subtract their sum from the original total sum to get the sum of the remaining (correct) observations.

New Sum (\(\Sigma x_{\text{new}}\)) = Original Sum (\(\Sigma x_{\text{original}}\)) - Sum of Incorrect Observations (\(\Sigma x_{\text{incorrect}}\))

\(\Sigma x_{\text{new}} = 2000 - 80\)

\(\Sigma x_{\text{new}} = 1920\)

Step 4: Calculate the New Number of Observations

When observations are omitted, the total number of observations decreases by the number of observations removed.

New Number of Observations (\(n_{\text{new}}\)) = Original Number of Observations (\(n_{\text{original}}\)) - Number of Incorrect Observations

\(n_{\text{new}} = 100 - 4\)

\(n_{\text{new}} = 96\)

Step 5: Calculate the New Mean

Now we have the new sum of observations and the new number of observations. We can calculate the new mean using the standard mean formula.

New Mean (\(\bar{x}_{\text{new}}\)) = \(\frac{\text{New Sum of Observations}}{\text{New Number of Observations}}\)

\(\bar{x}_{\text{new}} = \frac{1920}{96}\)

To simplify the division: \(1920 \div 96\)

\(1920 \div 96 = (192 \times 10) \div 96\)

Since \(192 = 2 \times 96\):

\((2 \times 96 \times 10) \div 96 = 2 \times 10\)

\(\bar{x}_{\text{new}} = 20\)

The mean after omitting the incorrect observations is 20.

Summary of Calculations for Correcting Mean

Conclusion on Correcting the Mean

By correctly adjusting both the total sum of the observations and the count of observations, we found that the new mean remains 20 when the specified incorrect observations are omitted.

Revision Table: Correcting Mean Calculations

When correcting a mean due to incorrect data, the approach depends on whether the incorrect values are to be replaced or omitted.

• If incorrect values are to be replaced:
  • Subtract the sum of incorrect values.
  • Add the sum of correct values.
  • The number of observations remains the same.
  • Calculate the new mean with the adjusted sum and original count.
• If incorrect values are to be omitted (as in this problem):
  • Subtract the sum of incorrect values.
  • Subtract the number of incorrect values from the total count.
  • Calculate the new mean with the adjusted sum and adjusted count.

Additional Information: Importance of Accurate Mean Calculation

The mean is a fundamental measure of central tendency. An accurate mean is crucial for many statistical analyses and interpretations. Errors in original data or transcription errors can significantly skew the mean. Identifying and handling such incorrect observations, whether by omission or correction, is a vital step in ensuring the reliability of statistical results. This problem demonstrates a common scenario in data analysis where data cleaning is necessary before computing summary statistics like the mean.

Measures of central tendency include:

• Mean: The average value. Affected by extreme values.
• Median: The middle value when data is ordered. Less affected by extreme values.
• Mode: The most frequently occurring value.

Understanding how to recalculate these measures when data changes is an important statistical skill.