Two Interpretations of the Mean (arithmetic average) of a data set: Fair Share and Balance Point

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# Two Interpretations of the Mean (arithmetic average) of a data set: Fair Share and Balance Point - PowerPoint PPT Presentation

Two Interpretations of the Mean (arithmetic average) of a data set: Fair Share and Balance Point. Portions of this presentation are adapted from the Virginia State Dept. of Education, 2010. Mean as “Fair Share”.

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Two Interpretations of the Mean (arithmetic average) of a data set: Fair Share and Balance Point

Portions of this presentation are adapted from the Virginia State Dept. of Education, 2010.

Mean as “Fair Share”

Themean of a data set represents a fair share concept (equal redistribution) of the data. Imagine redistributing the sum of n data values (x1+x2+ …+xn) among all n cases so that each case contributes the same amount to the total.

That amount would be equal to 1/nth of the sum:

(x1+x2+ …+xn)/n is called the “fair share” of the data set.

The 18 members of a book club answered this survey question “how many books did you read within the last month?”

Mean as “Fair Share”
Mean as “Fair Share”

Notice that the number of books read varies across members.

But the entire group read a total of 11+2x12+2x13+5x15+5x16 = 258 books.

If each member had read the same number of books to arrive at this same total for the group, that number would be 1/18thof the total number read: it would be (11+2x12+2x13+5x15+5x16)/18 = 258/18 = 14.333 books

Mean as Fair Share

The “Fair share” conception of the mean enables us to understand that the mean of a set of data values “evens out” or neutralizes the diversity/variability among the values.

Reporting the mean of a data set is reporting the hypothetical equal value that each data case would need to assume in order to preserve the sum of the original values.

Mean as “Balance Point”

Mean can be defined as the point on a number line where the data distribution is balanced.

This means that the sum of the distances between the mean and all the data values above the mean is equal to the sum of the distances between the mean and all the data values below the mean.

Where is the balance point for this

data set?

3 is the

Balance Point

X

X

X

X

X

X

Where is the balance point for this

data set?

MEAN

Sum of the distances below the mean

1+1+1+2 = 5

Sum of the distances above the mean

2 + 3 = 5

X

X

X

X

X

X