Understanding the Problem: Mean of Observations

The question provides us with a set of observations that have been modified by subtracting a constant value \(k\) from each original observation. We are given the mean of these modified observations, denoted as \(M\). Our goal is to find the mean of the original observations before the constant \(k\) was subtracted.

Let the original observations be \(x_1, x_2, x_3, \dots, x_n\). The modified observations are given as \((x_1 - k), (x_2 - k), (x_3 - k), \dots, (x_n - k)\).

Formulating the Solution: Mean of Original Data

The mean of a set of observations is calculated by summing all the observations and dividing by the total number of observations. We know the mean of the modified observations is \(M\). We can write this mathematically and then use algebraic manipulation to find the mean of the original observations.

Step-by-Step Calculation: Finding the Original Mean

The mean of the modified observations is given by:

Let's simplify the numerator by grouping the \(x_i\) terms and the \(k\) terms:

Since there are \(n\) observations, the term \((k + k + \dots + k)\) repeated \(n\) times is equal to \(nk\). So the numerator becomes:

Now, substitute this back into the equation for \(M\):

We can separate the terms in the numerator:

The term \(\frac{nk}{n}\) simplifies to \(k\):

Let the mean of the original observations \(x_1, x_2, \dots, x_n\) be denoted by \(\bar{x}\). By definition, the mean of the original observations is:

Substitute \(\bar{x}\) into the equation for \(M\):

We want to find the value of the original mean, \(\bar{x}\). To isolate \(\bar{x}\), we add \(k\) to both sides of the equation:

Thus, the mean of the original observations \(x_1, x_2, \dots, x_n\) is \(M + k\).

Result Analysis: Comparing with Options

The calculated mean of the original observations is \(M + k\). Let's compare this with the given options:

• Option 1: \(M\)
• Option 2: \(M + k\)
• Option 3: \(M - k\)
• Option 4: \(kM\)

Our result \(M + k\) matches Option 2.

This demonstrates an important property of the mean: if each observation in a dataset is shifted by a constant value \(k\), the mean of the dataset is also shifted by the same constant value \(k\). If the mean of \(x_i - k\) is \(M\), then the mean of \(x_i\) is \(M + k\).

Revision Table: Key Concepts

Additional Information: Properties of Mean

The mean is a fundamental measure of central tendency in statistics. It has several important properties that are useful when dealing with transformations of data:

• Linearity Property: If \(y_i = a x_i + b\), where \(a\) and \(b\) are constants, and \(\bar{x}\) is the mean of \(x_i\), then the mean of \(y_i\), denoted \(\bar{y}\), is \(\bar{y} = a \bar{x} + b\). In our problem, the transformation is \(y_i = x_i - k\), which can be written as \(y_i = 1 \cdot x_i + (-k)\). Here, \(a=1\) and \(b=-k\). If \(\bar{y}\) is the mean of the modified observations, then \(\bar{y} = 1 \cdot \bar{x} + (-k)\), which means \(M = \bar{x} - k\). Solving for \(\bar{x}\) gives \(\bar{x} = M + k\). This confirms our step-by-step calculation.
• Effect of Outliers: The mean is sensitive to extreme values (outliers). A single very large or very small observation can significantly affect the mean.
• Sum of Deviations: The sum of the deviations of each observation from the mean is always zero: \(\sum_{i=1}^n (x_i - \bar{x}) = 0\).

Understanding how the mean behaves under linear transformations like shifting (adding/subtracting a constant) and scaling (multiplying/dividing by a constant) is crucial for analyzing data in various statistical contexts.