Calculating the Mean of Observations

The question asks us to find the mean of the first three observations given the mean of five observations which are in an arithmetic progression.

Let the five observations be \(x\), \(x + 2\), \(x + 4\), \(x + 6\), and \(x + 8\).

Step-by-Step Mean Calculation

First, let's find the mean of the five observations. The mean (\(m\)) is calculated by summing all observations and dividing by the number of observations.

Sum of the five observations:

\(x + (x + 2) + (x + 4) + (x + 6) + (x + 8)\)

\(= x + x + 2 + x + 4 + x + 6 + x + 8\)

\(= 5x + (2 + 4 + 6 + 8)\)

\(= 5x + 20\)

Number of observations = 5

The mean (\(m\)) is given by:

\(m = \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(m = \frac{5x + 20}{5}\)

We can simplify this expression by dividing each term in the numerator by 5:

\(m = \frac{5x}{5} + \frac{20}{5}\)

\(m = x + 4\)

So, we have the relationship \(m = x + 4\). We can rearrange this to express \(x\) in terms of \(m\):

\(x = m - 4\)

Finding the Mean of the First Three Observations

Now, let's consider the first three observations: \(x\), \(x + 2\), and \(x + 4\).

Sum of the first three observations:

\(x + (x + 2) + (x + 4)\)

\(= x + x + 2 + x + 4\)

\(= 3x + (2 + 4)\)

\(= 3x + 6\)

Number of first three observations = 3

The mean of the first three observations is:

\(\frac{\text{Sum of the first three observations}}{\text{Number of first three observations}}\)

\(= \frac{3x + 6}{3}\)

Simplifying this expression:

\(= \frac{3x}{3} + \frac{6}{3}\)

\(= x + 2\)

So, the mean of the first three observations is \(x + 2\).

Expressing the Mean in terms of m

We found that the mean of the first three observations is \(x + 2\).

We also established the relationship \(x = m - 4\).

Now, substitute the expression for \(x\) into the mean of the first three observations:

Mean of first three observations \(= (m - 4) + 2\)

\(= m - 4 + 2\)

\(= m - 2\)

Therefore, the mean of the first three observations is \(m - 2\).

Since the mean of the first three observations is \(x+2\) and \(x = m-4\), the mean is \((m-4) + 2 = m-2\).

Revision Table: Key Concepts

Additional Information: Properties of Arithmetic Progression

The given observations \(x, x+2, x+4, x+6, x+8\) form an arithmetic progression with the first term \(a = x\) and common difference \(d = 2\).

• For an arithmetic progression with an odd number of terms, the mean is equal to the middle term. In the case of five terms, the middle term is the 3rd term, which is \(x + 4\). This confirms our calculation that \(m = x + 4\).
• For the first three terms \(x, x+2, x+4\), which are also in AP, the mean is the middle term, which is \(x+2\). This confirms our calculation for the mean of the first three terms.

This property of the mean being the middle term in an AP with an odd number of terms can often be a quick way to find the mean or relate terms if you recognize the pattern.