Question

which of the following is true with regard to the degree of freedom?
the sum of all the differences between the data value and the sample mean is always zero.
the sum of all of the differences between the data value and the sample mean can be any number.
its value is the same as the sample size.
its value is always one more than the sample size.

Explanation:

Step1: Recall the property of the sum of deviations from the mean

Let \(x_1,x_2,\cdots,x_n\) be a sample and \(\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i\) be the sample mean.
The sum of differences (deviations) is \(\sum_{i=1}^{n}(x_i-\bar{x})\).

Step2: Expand the sum

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Since \(\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i\), then \(n\bar{x}=\sum_{i=1}^{n}x_i\).
So \(\sum_{i=1}^{n}(x_i - \bar{x})=\sum_{i=1}^{n}x_i-\sum_{i=1}^{n}x_i=0\).

Step3: Analyze the degree - of - freedom formula

The degree of freedom for sample variance \(s^2=\frac{1}{n - 1}\sum_{i=1}^{n}(x_i-\bar{x})^2\) is \(n-1\), where \(n\) is the sample size. So the degree of freedom is not equal to the sample size \(n\) and not \(n + 1\).

Answer:

The sum of all the differences between the data value and the sample mean is always zero.