Understanding the Problem: Calculating the New Mean

The question asks us to find the new mean of a series of numbers ($x_1, x_2, \dots, x_n$) after one of the numbers, specifically $x_2$, is replaced by a new value, $\lambda$. We are given the original mean of the series as $\bar{X}$.

Original Mean and Sum

The mean of a series of $n$ numbers is the sum of the numbers divided by the total count of numbers. For the original series $x_1, x_2, \dots, x_n$, the mean $\bar{X}$ is given by the formula:

\(\bar{X} = \frac{x_1 + x_2 + \dots + x_n}{n}\)

We can write the sum of the original numbers as:

\(\text{Original Sum} = x_1 + x_2 + \dots + x_n = \sum_{i=1}^{n} x_i\)

From the mean formula, we can express the original sum in terms of the original mean and the number of terms:

\(\text{Original Sum} = n \times \bar{X}\)

So, \(\sum_{i=1}^{n} x_i = n\bar{X}\).

Calculating the New Sum After Replacement

The problem states that $x_2$ is replaced by $\lambda$. This means we remove $x_2$ from the sum and add $\lambda$ to the sum. The number of elements in the series remains $n$.

The new sum will be:

\(\text{New Sum} = (\text{Original Sum}) - x_2 + \lambda\)

Substituting the expression for the original sum ($n\bar{X}$):

\(\text{New Sum} = n\bar{X} - x_2 + \lambda\)

Finding the New Mean

The new mean is the new sum divided by the number of elements ($n$).

\(\text{New Mean} = \frac{\text{New Sum}}{n}\)

Substitute the expression for the new sum:

\(\text{New Mean} = \frac{n\bar{X} - x_2 + \lambda}{n}\)

Conclusion

The new mean of the series after replacing $x_2$ with $\lambda$ is \(\frac{n\bar{X} - x_2 + \lambda}{n}\). This matches one of the given options.

Comparing with Options

Let's look at the provided options and compare them with our derived formula:

• Option 1: \(\bar{X} - x_2+ \lambda\) - This formula is incorrect as it does not account for the number of terms $n$.
• Option 2: \(\frac{{\bar X - {x_2} - \lambda }}{n}\) - This formula has the wrong signs for $x_2$ and $\lambda$ and incorrectly uses \(\bar{X}\) instead of \(n\bar{X}\) for the sum.
• Option 3: \(\frac{{\bar X - {x_2} + \lambda }}{n}\) - This formula has the correct structure for the change but incorrectly uses \(\bar{X}\) instead of \(n\bar{X}\) for the original sum.
• Option 4: \(\frac{{n\bar X - {x_2} + \lambda }}{n}\) - This formula exactly matches our derived formula for the new mean.

Therefore, the new mean is \(\frac{n\bar{X} - x_2 + \lambda}{n}\).

Revision Table: Mean Calculation

Additional Information: Properties of Mean

The mean is a fundamental measure of central tendency. Here are some key points about the mean:

• Affected by Outliers: The mean is sensitive to extreme values (outliers).
• Unique Value: For a given set of data, there is only one mean.
• Sum of Deviations: The sum of the deviations of each observation from the mean is always zero, i.e., \(\sum_{i=1}^{n} (x_i - \bar{X}) = 0\).
• Effect of Adding/Subtracting a Constant: If each observation is increased or decreased by a constant value $c$, the new mean will be the original mean plus or minus $c$. New Mean = \(\bar{X} \pm c\).
• Effect of Multiplying/Dividing by a Constant: If each observation is multiplied or divided by a non-zero constant value $k$, the new mean will be the original mean multiplied or divided by $k$. New Mean = \(k\bar{X}\) or \(\frac{\bar{X}}{k}\).
• Effect of Replacement: As shown in this problem, replacing a value changes the total sum, which in turn changes the mean. The formula \(\frac{n\bar{X} - x_{\text{old}} + x_{\text{new}}}{n}\) is a general way to calculate the new mean after one value is replaced.