Calculating Mean from Algebraic Sum of Deviations

The question asks us to find the mean of 10 observations. We are given that the algebraic sum of the deviations of these observations from 15 is 7. This type of problem involves using the concept of deviations to find the mean.

Let the observations be denoted by \(x_1, x_2, \dots, x_{10}\). The number of observations is \(n = 10\).

The deviations of these observations from 15 are \(x_i - 15\) for \(i = 1, 2, \dots, 10\).

The algebraic sum of these deviations is given as 7. So, we have:

\(\sum_{i=1}^{10} (x_i - 15) = 7\)

Understanding the Formula for Mean Using Deviations

The mean (\(\bar{x}\)) of a set of observations can be calculated using a reference point (often called an assumed mean or arbitrary origin), denoted by \(A\). The formula relates the mean, the reference point, and the sum of the deviations from that reference point.

The formula is:

\(\bar{x} = A + \frac{\sum (x_i - A)}{n}\)

Where:

• \(\bar{x}\) is the mean.
• \(A\) is the reference point from which deviations are measured.
• \(\sum (x_i - A)\) is the algebraic sum of the deviations.
• \(n\) is the number of observations.

In this specific problem:

• The number of observations, \(n = 10\).
• The reference point (from which deviations are measured) is 15, so \(A = 15\).
• The algebraic sum of the deviations from 15 is given as 7, so \(\sum_{i=1}^{10} (x_i - 15) = 7\).

Step-by-Step Mean Calculation

Now, we can substitute the given values into the formula to calculate the mean.

Given:

• \(n = 10\)
• \(A = 15\)
• \(\sum (x_i - A) = 7\)

Using the formula:

\(\bar{x} = A + \frac{\sum (x_i - A)}{n}\)

Substitute the values:

\(\bar{x} = 15 + \frac{7}{10}\)

Perform the division:

\(\bar{x} = 15 + 0.7\)

Perform the addition:

\(\bar{x} = 15.7\)

So, the mean of the 10 observations is 15.7. This method is useful for calculating the mean, especially with large numbers, as it simplifies calculations using deviations from a convenient point.

Summary of Mean Calculation from Deviations

To calculate the mean using the algebraic sum of deviations from a point \(A\), we add the reference point \(A\) to the average of the deviations. The average of the deviations is found by dividing the algebraic sum of the deviations by the total number of observations. This allows us to efficiently find the mean.

In this case, with 10 observations and an algebraic sum of deviations from 15 being 7, the Mean is found to be 15.7.