Example : Find the median absolute deviation of climatological monthly precipitation in South America for January to December Select the "Datasets by Catagory" link in the blue banner on the Data Library page. CHECK The above command computes the median value over the monthly climatologies at each grid point in the field. Enter the command: [T] medianover Click the OK button. Similar to the IQR example, the Amazon Basin exhibits high intraannual precipitation variability, while areas to the north and south exhibit lower precipitation variability.
Trimmed Variance Similar to variance, except that a proportion of the largest and smallest values in the dataset are ommitted before it is calculated.
Sometimes multiplied by an adjustment factor to make it more consistant with the ordinary sample variance. Wilks, Daniel S. Statisical Methods in the Atmospheric Sciences. Analogous to the trimmed mean. Click on the "Cloud Characteristics and Radiation Budget" link. Click on the "monthly" link Select the "outgoing longwave radiation" link under the Datasets and Variables subheading.
Select the Average over "XY" link. The result is located under the Expert Mode text box in bold: Make a note of this value. In the source bar, click on the [X Y] average box. This operation undoes the replacebypercentile command. Return to Expert Mode. Enter the following command under the text already there: [T]. The result should be Click on the "Filters" link in the function bar.
The value of the root mean square is 7. Calculate the trimmed variance by squaring the value above. To see the results of this operation, choose the viewer window with land drawn in black. Maximum Observed Sea Surface Temperatures. Return to the dataset page by clicking on the right-most link on the blue source bar. To see your results, choose the viewer with land shaded in black. Select the "cloud cover" link under the Datasets and Variables subheading.
Click on the "Data Selection" link in the function bar. Enter the following text below the text already there: dataflag [T]sum dup 1. The dataflag [T]sum commands determines, for each grid point, the number of non-missing elements in the time series. The next step is to take n and divide it by n The following five lines of code reference the dataset being used, including the temporal and spatial ranges. And what better number to represent a minimum than 0?
Does the measure also have an absolute maximum? Not really. An arbitrary collection can be arbitrarily large and the numbers can be as different from each other as you want. Another property this measure must have is to increase when the numbers get more different from each other and decrease when they get more similar. The range of a collection of numbers is defined as the difference between the maximum and the minimum value in the collection.
In our example collection, the largest number is 10 and the smallest is 1. First, if all numbers in the collection were the same for example, [1, 1, 1, 1, 1] or [10, 10, 10, 10, 10], the minimum and the maximum would also be the same both 1 or both And subtracting a number from itself always results in 0. This property holds. What about the second property?
If you think about it, the only way to change the range of a collection is if you change its minimum or maximum values. As the numbers become more dispersed, the range increases, and vice versa. Which is what we would expect from a measure of dispersion. The mean absolute difference of a collection of numbers is the arithmetic mean of the absolute differences between all pairs of numbers in the collection. From some perspective, you can see the mean absolute difference as a generalization of the range.
Well, instead of only looking at the difference between the most extreme values, here you take into account the differences between all numbers in the collection. And now this measure is also going to be sensitive to changes in all values, not just the minimum and the maximum. If you need a refresher on combinatorics, you will find my post on the topic useful. But what if we came up with a measure that compares all numbers to some unique and special number?
And, as you know, the most common measures of central tendency are the mean, the mode, and the median. Here M stands for any measure of central tendency. The median of [1, 4, 4, 9, 10] is the middle value, which is 4.
Here are the absolute differences:. In this case, the mode the most common value happens to be equal to the median, so the calculations will be exactly the same. Hence, the mean absolute deviation around the mode for [1, 4, 4, 9, 10] is also equal to 2. As you can see, they all measure the dispersion in the collection somewhat differently. But which one is the most accurate? Which ones should you trust more and under what circumstances? The variance is arguably the most commonly used measure of dispersion.
First, similar to mean absolute deviation, the variance also measures deviations from one particular central tendency. Namely, the mean of the collection. Therefore, we will again take the differences between the mean and each number. Did you wonder why the mean absolute deviation takes the absolute value of the differences? Why not simply sum the positive and negative differences together? For example, consider this collection:. Then, the total deviation is. So, we got a measure of 0, which is exactly what we would have gotten if we had the collection [2, 2, 2].
It completely disregards the second required property for any measure of dispersion. Namely, as the numbers get more different from each other, the measure should increase in value. Outliers are single observations which, if excluded from the calculations, have noticeable influence on the results. For example, if we had entered '21' instead of '2. It does not necessarily follow, however, that outliers should be excluded from the final data summary, or that they always result from an erroneous measurement.
The median has the advantage that it is not affected by outliers, so for example the median in the example would be unaffected by replacing '2. However, it is not statistically efficient, as it does not make use of all the individual data values. A third measure of location is the mode. This is the value that occurs most frequently, or, if the data are grouped, the grouping with the highest frequency. It is not used much in statistical analysis, since its value depends on the accuracy with which the data are measured; although it may be useful for categorical data to describe the most frequent category.
The expression 'bimodal' distribution is used to describe a distribution with two peaks in it. This can be caused by mixing populations. For example, height might appear bimodal if one had men and women on the population.
Some illnesses may raise a biochemical measure, so in a population containing healthy and ill people one might expect a bimodal distribution.
However, some illnesses are defined by the measure e. Measures of dispersion describe the spread of the data. They include the range, interquartile range, standard deviation and variance. The range is given as the smallest and largest observations. This is the simplest measure of variability. Note in statistics unlike physics a range is given by two numbers, not the difference between the smallest and largest. For some data it is very useful, because one would want to know these numbers, for example knowing in a sample the ages of youngest and oldest participant.
If outliers are present it may give a distorted impression of the variability of the data, since only two observations are included in the estimate. The quartiles, namely the lower quartile, the median and the upper quartile, divide the data into four equal parts; that is there will be approximately equal numbers of observations in the four sections and exactly equal if the sample size is divisible by four and the measures are all distinct.
Note that there are in fact only three quartiles and these are points not proportions. However, the meaning of the first statement is clear and so the distinction is really only useful to display a superior knowledge of statistics!
The quartiles are calculated in a similar way to the median; first arrange the data in size order and determine the median, using the method described above. Now split the data in two the lower half and upper half, based on the median. The first quartile is the middle observation of the lower half, and the third quartile is the middle observation of the upper half.
This process is demonstrated in Example 2, below. The interquartile range is a useful measure of variability and is given by the lower and upper quartiles. The median is the average of the 9th and 10th observations 2.
The first half of the data has 9 observations so the first quartile is the 5th observation, namely 1. Similarly the 3rd quartile would be the 5th observation in the upper half of the data, or the 14th observation, namely 2. Hence the interquartile range is 1. Next add each of the n squared differences.
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