Calculating Measures of Location and Dispersion for the given set of Data
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We subtract this from each of the observations. Note the mean of this column is zero.
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This will always be the case: A convenient method for removing the negative signs is squaring the deviations, which is given in the next column. These values are then summed to get a value of 0. We need to find the average squared deviation. Common-sense would suggest dividing by n , but it turns out that this actually gives an estimate of the population variance, which is too small.
Measures of dispersion
It can be shown that it is better to divide by the degrees of freedom, which is n minus the number of estimated parameters, in this case n An intuitive way of looking at this is to suppose one had n telephone poles each meters apart. How much wire would one need to link them? As with variation, here we are not interested in where the telegraph poles are, but simply how far apart they are. A moment's thought should convince one that n -1 lengths of wire are required to link n telegraph poles.
From the results calculated thus far, we can determine the variance and standard deviation, as follows:. It is this characteristic of the standard deviation which makes it so useful. It holds for a large number of measurements commonly made in medicine. In particular, it holds for data that follow a Normal distribution. Standard deviations should not be used for highly skewed data, such as counts or bounded data, since they do not illustrate a meaningful measure of variation, and instead an IQR or range should be used.
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In particular, if the standard deviation is of a similar size to the mean, then the SD is not an informative summary measure, save to indicate that the data are skewed. Skip to main content. Create new account Request new password. You are here 1b - Statistical Methods. Median The median is defined as the middle point of the ordered data.
Measures of dispersion
Example 1 Calculation of mean and median Consider the following 5 birth weights, in kilograms, recorded to 1 decimal place: Advantages and disadvantages of the mean and median The major advantage of the mean is that it uses all the data values, and is, in a statistical sense, efficient. Mode A third measure of location is the mode.
Measures of Dispersion or Variability Measures of dispersion describe the spread of the data.
Range and Interquartile Range The range is given as the smallest and largest observations. Quartiles and Interquartile Range 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.
Example 2 Calculation of the quartiles Suppose we had 18 birth weights arranged in increasing order. Standard Deviation and Variance The standard deviation of a sample s is calculated as follows: Example 3 Calculation of the standard deviation Consider the data from example 1. Standard deviation is often abbreviated to SD in the medical literature. The other advantage of SD is that along with mean it can be used to detect skewness.
Why is it important to measure the spread of data?
The disadvantage of SD is that it is an inappropriate measure of dispersion for skewed data. SD is used as a measure of dispersion when mean is used as measure of central tendency ie, for symmetric numerical data. National Center for Biotechnology Information , U. Journal List J Pharmacol Pharmacother v. Author information Copyright and License information Disclaimer. This is an open-access article distributed under the terms of the Creative Commons Attribution-Noncommercial-Share Alike 3. This article has been cited by other articles in PMC. For ordinal data or skewed numerical data, median and interquartile range are used.
Footnotes Source of Support: Nil Conflict of Interest: Viva Books Private Limited; Statistics at square one. These help you to know the spread of scores within a bunch of scores. Are the scores really close together or are they really far apart? For example, if you were describing the heights of students in your class to a friend, they might want to know how much the heights vary.
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Are all the men about 5 feet 11 inches within a few centimeters or so? Or is there a lot of variation where some men are 5 feet and others are 6 foot 5 inches?
Measures of dispersion like the range, variance and standard deviation tell you about the spread of scores in a data set. Like central tendency, they help you summarize a bunch of numbers with one or just a few numbers. Home Copyright Tell a Friend.