Standard Deviation - Relationship Between Standard Deviation and Mean

Relationship Between Standard Deviation and Mean

The mean and the standard deviation of a set of data are usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point. The precise statement is the following: suppose x1, ..., xn are real numbers and define the function:

Using calculus or by completing the square, it is possible to show that σ(r) has a unique minimum at the mean:

Variability can also be measured by the coefficient of variation, which is the ratio of the standard deviation to the mean. It is a dimensionless number.

Often, we want some information about the precision of the mean we obtained. We can obtain this by determining the standard deviation of the sampled mean. The standard deviation of the mean is related to the standard deviation of the distribution by:

where N is the number of observations in the sample used to estimate the mean. This can easily be proven with:

\begin{align} \operatorname{var}(X) &\equiv \sigma^2_X\\ \operatorname{var}(X_1+X_2) &\equiv \operatorname{var}(X_1) + \operatorname{var}(X_2)\\ \operatorname{var}(cX_1) &\equiv c^2 \, \operatorname{var}(X_1) \end{align}

hence

\begin{align} \operatorname{var}(\text{mean}) &= \operatorname{var}\left (\frac{1}{N} \sum_{i=1}^N X_i \right) = \frac{1}{N^2}\operatorname{var}\left (\sum_{i=1}^N X_i \right ) \\ &= \frac{1}{N^2}\sum_{i=1}^N \operatorname{var}(X_i) = \frac{N}{N^2} \operatorname{var}(X) = \frac{1}{N} \operatorname{var} (X). \end{align}

Resulting in:

Read more about this topic:  Standard Deviation

Famous quotes containing the words relationship between, relationship and/or standard:

    “It would be a fallacy to deduce that the slow writer necessarily comes up with superior work. There seems to be scant relationship between prolificness and quality.”
    —Fannie Hurst (1889–1968)

    “When any relationship is characterized by difference, particularly a disparity in power, there remains a tendency to model it on the parent-child-relationship. Even protectiveness and benevolence toward the poor, toward minorities, and especially toward women have involved equating them with children.”
    —Mary Catherine Bateson (20th century)

    “If the Revolution has the right to destroy bridges and art monuments whenever necessary, it will stop still less from laying its hand on any tendency in art which, no matter how great its achievement in form, threatens to disintegrate the revolutionary environment or to arouse the internal forces of the Revolution, that is, the proletariat, the peasantry and the intelligentsia, to a hostile opposition to one another. Our standard is, clearly, political, imperative and intolerant.”
    —Leon Trotsky (1879–1940)