02/11/2020
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The simplest measure of variation is the range, which measures the width of the data values.
Suppose we have samples \( x_1,\cdots, x_n \) and
\[ \begin{align} x_\text{max} = \max_i(x_i) & & x_\text{min} = \min_i(x_i) \end{align} \]
Then the range is computed as
\[ \text{Range} = x_\text{max} - x_\text{min} \]
For example, if our samples are \( 22, 22, 26, \) and \( 24 \) then
\[ \text{Range} = 26 - 22 = 4.0 \]
Discuss with a neighbor: is the range resistant to outliers? Why or why not?
For example, suppose we have the sample values \( 0, 0, 0, 0, 0, 0, 0, 0, 0, 1000 \). The range will be 1000 due the outlier;
Note, we could have considered a different way to measure the variation than standard deviation.
Consider, if we want to measure the total deviation we could instead write this as,
\[ \sum_{i=1}^n \vert x_i - \overline{x}\vert \]
We could then divide this by the total number of observations, which gives
\[ \text{Mean absolute deviation} = \frac{\sum_{i=1}^n \vert x_i - \overline{x}\vert}{n} \]
This is a possible choice for a similar measure of the variation, but the main issue lies in that the absolute value is not an “algebraic operation”.
If we want to make calculations or inferences based on the formula above, this will become very difficult and there are few tools that work well with this statistic.
For this reason, using the square as in the sample standard deviation
\[ s = \sqrt{\frac{\sum_{i=1}^n\left(x_i - \overline{x}\right)^2}{n-1}} \]
we get a similar result, but one that is mathematically easier to manipulate.
We will not focus on calculating the sample standard deviation manually in this course;
Suppose we have the samples \( 22, 22, 26 \) and \( 24 \).
We wish to compute how much deviation there is in the data from the sample mean, so we will begin by computing this value
\[ \overline{x} = \frac{22 + 22 + 26 + 24 }{4} = \frac{94}{4}=23.5 \]
We now compute the raw deviation of each sample from the sample mean:
\[ \begin{align} x_1 - \overline{x} =& 22 - 23.5 = -1.5\\ x_2 - \overline{x} =& 22 - 23.5 = -1.5\\ x_3 - \overline{x} =& 26 - 23.5 = 2.5\\ x_4 - \overline{x} =& 24 - 23.5 = 0.5\\ \end{align} \]
Squaring each value, we obtain \( 2.25, 2.25, 6.25, 0.25 \), so that
\[ s = \sqrt{\frac{\sum_{i=1}^4 \left(x_i - \overline{x}\right)^2}{3}} = \sqrt{\frac{11}{3}}\approx 1.9 \]
This shows how the sample standard deviation can be computed, but we will want a few ways to interpret the value.
Courtesy of Mario Triola, Essentials of Statistics, 6th edition
Courtesy of Mario Triola, Essentials of Statistics, 6th edition
As a very rough estimate, we can approximate the sample standard deviation with the range rule.
The only time we should consider using this approximation is when we have no computer or calculator on hand, and need a quick “back-of-an-envelope” calculation.
The range rule of thumb for estimating the standard deviation is given as
\[ s \approx \frac{\text{Range}}{4} \]
Suppose we have the samples \( 22, 22, 26 \) and \( 24 \) once again.
The sample standard deviation of the data is \( \approx 1.9 \).
Discuss with a neighbor: what is the range rule of thumb estimate for the sample standard deviation? Is this very accurate in this case?
The range rule of thumb gives \( \frac{26 - 22}{4} = \frac{4}{4}=1 \), which is not that accurate.
This shows that we should only consider this as a very loose approximation, and in practice we should compute the sample standard deviation directly whenever possible.
When we describe the amount of variation in data, it is commonly described as the dispersion or spread in the data.
The word variance also has a specific meaning in statistics and is another tool for describing the variation / dispersion / spread of the data.
Suppose that the data has a population standard deviation of \( \sigma \) and a sample standard deviation of \( s \).
Then, the data has a population variance of \( \sigma^2 \).
Likewise, the data has a sample variance of \( s^2 \).
Therefore, for either a population parameter or a sample statistic, the variance is the square of the standard deviation.
For example, measuring the heights of students in inches, the standard deviation is in the units inches.
Courtesy of Melikamp CC via Wikimedia Commons
Courtesy of Melikamp CC via Wikimedia Commons
The proportion (or fraction) of any set of data lying within \( K \) standard deviations of of the mean is always at least \( 1-\frac{1}{K^2} \) where \( K>1 \).