Calculating Mean from Deviations

This problem involves finding the mean of a set of values given the sum of their deviations from two different constant points. The deviation of a value \(x_i\) from a constant 'a' is \(x_i - a\). The algebraic sum of these deviations is \(\sum (x_i - a)\).

Understanding the Properties of Deviations

A key property of deviations is that the sum of deviations from the mean is always zero. However, when deviations are taken from a value other than the mean, the sum is not zero. We can express the sum of deviations from any constant 'a' in terms of the mean (\(\bar{x}\)) and the number of values (n):

\(\sum_{i=1}^{n} (x_i - a) = \sum x_i - \sum a\)

Since \(\sum a = na\) and \(\sum x_i = n\bar{x}\), we get:

\(\sum_{i=1}^{n} (x_i - a) = n\bar{x} - na = n(\bar{x} - a)\)

This formula connects the sum of deviations from a constant 'a' to the mean (\(\bar{x}\)) and the number of observations (n).

Setting Up Equations from Given Information

We are given two pieces of information about the sum of deviations:

1. The algebraic sum of deviations from 100 is -20. Using the formula above with \(a = 100\), we have:

\(\sum_{i=1}^{n} (x_i - 100) = n(\bar{x} - 100) = -20\) (Equation 1)

2. The algebraic sum of deviations from 92 is 140. Using the formula with \(a = 92\), we have:

\(\sum_{i=1}^{n} (x_i - 92) = n(\bar{x} - 92) = 140\) (Equation 2)

Solving for the Mean Value

We now have a system of two equations with two unknowns: the number of values (n) and the mean (\(\bar{x}\)).

Equation 1: \(n(\bar{x} - 100) = -20\)

Equation 2: \(n(\bar{x} - 92) = 140\)

We can solve this system to find the value of the mean, \(\bar{x}\).

One way to solve this is to divide Equation 2 by Equation 1:

\(\frac{n(\bar{x} - 92)}{n(\bar{x} - 100)} = \frac{140}{-20}\)

Assuming \(n \neq 0\) (which must be true for a set of values), we can cancel 'n':

\(\frac{\bar{x} - 92}{\bar{x} - 100} = -7\)

Now, we can solve for \(\bar{x}\) by cross-multiplying:

\(\bar{x} - 92 = -7(\bar{x} - 100)\)

\(\bar{x} - 92 = -7\bar{x} + 700\)

Combine terms involving \(\bar{x}\) and constant terms:

\(\bar{x} + 7\bar{x} = 700 + 92\)

\(8\bar{x} = 792\)

Divide by 8 to find \(\bar{x}\):

\(\bar{x} = \frac{792}{8}\)

Performing the division:

\(\bar{x} = 99\)

The mean of the values is 99.

Verification (Optional but Recommended)

We can find the number of values 'n' and check if the equations hold true.

Using Equation 1: \(n(99 - 100) = -20 \Rightarrow n(-1) = -20 \Rightarrow n = 20\)

Using Equation 2: \(n(99 - 92) = 140 \Rightarrow n(7) = 140 \Rightarrow n = \frac{140}{7} = 20\)

Since both equations give the same value for n (n=20), our calculated mean \(\bar{x} = 99\) is correct.

Revision Table: Key Concepts

Additional Information: Properties of Mean

The mean is a fundamental measure of central tendency. It has several important properties:

• The sum of the deviations of individual values from the mean is always zero: \(\sum (x_i - \bar{x}) = 0\). This property was implicitly used in the formula \(\sum (x_i - a) = n(\bar{x} - a)\), as it arises from \(\sum (x_i - \bar{x} + \bar{x} - a) = \sum (x_i - \bar{x}) + \sum (\bar{x} - a) = 0 + n(\bar{x} - a)\).
• The mean is affected by every value in the dataset.
• The mean is sensitive to extreme values (outliers).
• The sum of the squares of the deviations from the mean is minimum, i.e., \(\sum (x_i - \bar{x})^2\) is less than \(\sum (x_i - a)^2\) for any \(a \neq \bar{x}\).
• If each value in the dataset is increased or decreased by a constant 'c', the mean is also increased or decreased by 'c'.
• If each value in the dataset is multiplied or divided by a constant 'k', the mean is also multiplied or divided by 'k'.