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3.7: Median and Mean

  • Page ID
    2300
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    Learning Objectives

    • State whether it is the mean or median that minimizes the mean absolute deviation
    • State whether it is the mean or median that is the balance point on a balance scale

    In the section "What is central tendency," we saw that the center of a distribution could be defined three ways:

    1. the point on which a distribution would balance
    2. the value whose average absolute deviation from all the other values is minimized
    3. the value whose average squared difference from all the other values is minimized

    From the simulation in this chapter, you discovered (we hope) that the mean is the point on which a distribution would balance, the median is the value that minimizes the sum of absolute deviations, and the mean is the value that minimizes the sum of the squared deviations.

    Table \(\PageIndex{1}\) shows the absolute and squared deviations of the numbers \(2, 3, 4, 9\) and \(16\) from their median of \(4\) and their mean of \(6.8\). You can see that the sum of absolute deviations from the median (\(20\)) is smaller than the sum of absolute deviations from the mean (\(22.8\)). On the other hand, the sum of squared deviations from the median (\(174\)) is larger than the sum of squared deviations from the mean (\(134.8\)).

    Table \(\PageIndex{1}\): Absolute and squared deviations from the median of 4 and the mean of 6.8
    Value Absolute Deviation from Median Absolute Deviation from Mean Squared Deviation from Median Squared Deviation from Mean
    2 2 4.8 4 23.04
    3 1 3.8 1 14.44
    4 0 2.8 0 7.84
    9 5 2.2 25 4.84
    16 12 9.2 144 84.64
    Total 20 22.8 174 134.8

    Figure \(\PageIndex{1}\) shows that the distribution balances at the mean of \(6.8\) and not at the median of \(4\). The relative advantages and disadvantages of the mean and median are discussed in the section "Comparing Measures" later in this chapter.

    balance1.jpg
    Figure \(\PageIndex{1}\): The distribution balances at the mean of \(6.8\) and not at the median of \(4.0\).

    When a distribution is symmetric, then the mean and the median are the same. Consider the following distribution: \(1, 3, 4, 5, 6, 7, 9\). The mean and median are both \(5\). The mean, median, and mode are identical in the bell-shaped normal distribution.


    This page titled 3.7: Median and Mean is shared under a Public Domain license and was authored, remixed, and/or curated by David Lane via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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