Understanding the Sum of Deviations from the Mean

The question asks for the sum of deviations of a given set of variate values from their mean. This involves two main steps: first, calculating the mean of the given values, and second, finding the difference (deviation) between each value and the mean, and then summing up these differences.

Calculating the Mean of Variate Values

The given variate values are 73, 85, 92, 105, and 120.

The mean (average) of a set of values is calculated by summing all the values and dividing by the total number of values.

Let the variate values be denoted by \(x_i\). The sum of the values is:

\(\sum x_i = 73 + 85 + 92 + 105 + 120\)

\(\sum x_i = 475\)

The number of variate values is \(n = 5\).

The mean, denoted by \(\bar{x}\), is calculated as:

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

\(\bar{x} = \frac{475}{5}\)

\(\bar{x} = 95\)

So, the mean of the given variate values is 95.

Calculating the Deviations from the Mean

The deviation of a variate value \(x_i\) from the mean \(\bar{x}\) is given by \(x_i - \bar{x}\).

We calculate the deviation for each given value:

• Deviation for 73: \(73 - 95 = -22\)
• Deviation for 85: \(85 - 95 = -10\)
• Deviation for 92: \(92 - 95 = -3\)
• Deviation for 105: \(105 - 95 = 10\)
• Deviation for 120: \(120 - 95 = 25\)

Summing the Deviations

The sum of the deviations is the sum of all the individual deviations calculated above:

Sum of deviations = \((-22) + (-10) + (-3) + 10 + 25\)

Sum of deviations = \(-22 - 10 - 3 + 10 + 25\)

Sum of deviations = \(-35 + 35\)

Sum of deviations = \(0\)

Alternatively, we can show this in a table format:

The sum of the deviations of the variate values from their mean is 0.

Property of the Arithmetic Mean

This result demonstrates a fundamental property of the arithmetic mean: the sum of the deviations of a set of values from their arithmetic mean is always zero. Mathematically, for any set of values \(x_1, x_2, \dots, x_n\) and their mean \(\bar{x}\), the following holds true:

\(\sum_{i=1}^{n} (x_i - \bar{x}) = 0\)

This property arises because the mean is the balancing point of the data set. The sum of positive deviations (values above the mean) is exactly equal in magnitude to the sum of negative deviations (values below the mean).

Revision Table: Key Concepts

Additional Information: Why Sum of Deviations is Zero

Let's quickly look at the mathematical proof for why the sum of deviations from the mean is always zero:

Given a set of values \(x_1, x_2, \dots, x_n\).

The mean is \(\bar{x} = \frac{\sum x_i}{n}\).

This implies \(n\bar{x} = \sum x_i\).

Now consider the sum of deviations:

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

Using the property of summation that allows distributing the sum:

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

We know that \(\sum_{i=1}^{n} x_i\) is simply the sum of all the values.

The term \(\sum_{i=1}^{n} \bar{x}\) means adding the constant value \(\bar{x}\) \(n\) times. This is equal to \(n \times \bar{x}\).

So, the sum of deviations becomes:

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

Since we established that \(\sum x_i = n\bar{x}\) from the definition of the mean:

\(\sum x_i - n\bar{x} = n\bar{x} - n\bar{x} = 0\)

Thus, the sum of deviations from the arithmetic mean is always zero.

This property is unique to the arithmetic mean and is a key characteristic that distinguishes it from other measures of central tendency like the median or mode.