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Range In statistics, range is defined simply as the difference between the maximum and minimum observations. It is intuitively obvious why we define range in statistics this way – range should suggest how diversely spreads out the values are, and by computing the difference between the maximum and minimum values, we can get an estimate of the spread of the data. Example For example, suppose an experiment involves finding out the weight of lab rats and the values in grams are 320, 367, 423, 471 and 480. In this case, the range is simply computed as 480-320 = 160 grams. Source: http://www.experimentresources.com/range-instatistics.html#ixzz22LkgVypq The Range: Range is defined as the difference between the maximum and the minimum observation of the given data. If denotes the maximum observation denotes the minimum observation then the range is defined as Range Source: http://www.emathzone.com/tutorials/basicstatistics/range-and-coefficient-of-range.html Formula for Range The calculation of the range is very straightforward. All we need to do is find the difference between the largest data value in our set and the smallest data value. Stated succinctly we have the following formula: Range = Maximum Value – Minimum Value. For example, the data set 4,6,10, 15, 18 has maximum of 18, minimum of 4 and range of 18 – 4 = 14. Limitations of Range The range is a very crude measurement of the spread of data because it is extremely sensitive to outliers. A single data value can greatly affect the value of the range. For example, consider the set of data 1, 2, 3, 4, 6, 7, 7, 8. The maximum value is 8, the minimum is 1 and the range is 7. Now consider the same set of data, only with the value 100 included. The range now becomes 100 – 1 = 99. The addition of a single extra data point greatly affected the value of the range. The standard deviation is another measure of spread that is less susceptible to outliers. The drawback is that the calculation of the standard deviation is much more complicated. Source: http://statistics.about.com/od/DescriptiveStatistics/a/What-Is-The-Range-In-Statistics.htm Variance Statistical variance gives a measure of how the data distributes itself about the mean or expected value. Unlike range that only looks at the extremes, the variance looks at all the data point and then determines their distribution. Usage The concept of variance can be extended to continuous data sets too. In that case, instead of summing up the individual differences from the mean, we need to integrate them. This approach is also useful when the number of data points is very large, like the population of a country. Variance is extensively used in probability theory, wherein from a given smaller sample set, more generalized conclusions need to be drawn. This is because variance gives us an idea about the distribution of data around the mean, and thus from this distribution, we can work out where we can expect an unknown data point. Source: http://www.experimentresources.com/statisticalvariance.html#ixzz22LvEJVE5 Source: http://www.quickmba.com/stats/standard-deviation/ Coefficients of Variations Variance and Standard Deviation While the range is useful for describing the "borders" of a data set, it tells us almost nothing about the points that fall between the two extremes. In most cases, we are interested in knowing how far each of the data points is from the mean or median of the data set, as this would enable us to see the spread in all of the data points. Since the median doesn't take the magnitude of each data point into account and the mean does, the mean is the better option for a central point. Source: http://esa21.kennesaw.edu/activities/stats/stats.pdf A coefficient of variation (CV) can be calculated and interpreted in two different settings: analyzing a single variable and interpreting a model. The standard formulation of the CV, the ratio of the standard deviation to the mean, applies in the single variable setting. In the modeling setting, the CV is calculated as the ratio of the root mean squared error (RMSE) to the mean of the dependent variable. In both settings, the CV is often presented as the given ratio multiplied by 100. The CV for a single variable aims to describe the dispersion of the variable in a way that does not depend on the variable's measurement unit. The higher the CV, the greater the dispersion in the variable. The CV for a model aims to describe the model fit in terms of the relative sizes of the squared residuals and outcome values. The lower the CV, the smaller the residuals relative to the predicted value. This is suggestive of a good model fit. Source:http://www.ats.ucla.edu/stat/mult_pkg/faq/ge neral/coefficient_of_variation.htm Coefficient of Variation The coefficient of variation (abbreviated CV) is a way to quantify scatter. It is defined as the standard deviation of a group of values divided by their mean. Often that ratio is multiplied by 100 to express the coefficient of variation as a percent (abbreviated %CV). The CV is useful for comparing scatter of variables measured in different units. You could ask, for example, whether the variation in pulse rate is greater or less than the variation in the concentration of serum sodium. The pulse rate and sodium are measured in completely different units, so comparing their standard deviation would be nonsense. Comparing their coefficients of variation might prove useful to some physiological investigations of homeostasis. Source: http://www.graphpad.com/support/faqid/1088/ Coefficient of Variation (CV) If you know nothing about the data other than the mean, one way to interpret the relative magnitude of the standard deviation is to divide it by the mean. This is called the coefficient of variation. For example, if the mean is 80 and standard deviation is 12, the cv = 12/80 = .15 or 15%. If the standard deviation is .20 and the mean is .50, then the cv = .20/.50 = .4 or 40%. So knowing nothing else about the data, the CV helps us see that even a lower standard deviation doesn't mean less variable data.