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A normal value is the single value that occurs most often in a data set. Data Set A has a normal value of 4, Data Set B has a normal value of 3, and Data Set C has a normal value of 1.
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Standard Deviations
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By definition, one standard deviation is the amount of variance within a set of measurements that encompasses approximately the top 68 percent of all measurements in the data set; in other words, knowing the standard deviation of your data set tells you how densely the data points are clustered around the mean. Simply put, the smaller the standard deviation, the more consistent the data. To illustrate, the standard deviation of Data Set A is approximately 1.5, the standard deviation of Data Set B is approximately 6.0, and the standard deviation of Data Set C is approximately 2.6. A common rule in this case is: Data with a standard deviation greater than half of its mean should be treated as suspect. If the data is accurate, the phenomenon the data represents is not displaying a normal distribution pattern. Applying this rule, Data Set A is likely to be a reasonable example of a normal distribution; Data Set B may or may not be a reasonable representation of a normal distribution; and Data Set C is undoubtedly not a reasonable representation of a normal distribution.
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Uniform distributions sometimes known as linear distributions represent a collection of data that is roughly equivalent to a set of random numbers evenly spaced between the upper and lower bounds. In a uniform distribution, every number in the data set is represented approximately the same number of times. Uniform distributions are frequently used when modeling user delays, but are not common in response time results data. In fact, uniformly distributed results in response time data may be an indication of suspect results.
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Figure 15.5 Uniform Distributions
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Also known as bell curves, normal distributions are data sets whose member data are weighted toward the center (or median value). When graphed, the shape of the bell of normally distributed data can vary from tall and narrow to short and squat, depending on the standard deviation of the data set. The smaller the standard deviation, the taller and more narrow the bell. Statistically speaking, most measurements of human variance result in data sets that are normally distributed. As it turns out, end-user response times for Web applications are also frequently normally distributed.
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Figure 15.6 Normal Distribution
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Statistical Significance
Mathematically calculating statistical significance, or reliability, based on sample size is a task that is too arduous and complex for most commercially driven softwaredevelopment projects. Fortunately, there is a commonsense approach that is both efficient and accurate enough to identify the most significant concerns related to statistical significance. Unless you have a good reason to use a mathematically rigorous calculation for statistical significance, a commonsense approximation is generally sufficient. In support of the commonsense approach described below, consider this excerpt from a StatSoft, Inc. (http://www.statsoftinc.com) discussion on the topic:
There is no way to avoid arbitrariness in the final decision as to what level of significance will be treated as really significant. That is, the selection of some level of significance, up to which the results will be rejected as invalid, is arbitrary.
Typically, it is fairly easy to add iterations to performance tests to increase the total number of measurements collected; the best way to ensure statistical significance is simply to collect additional data if there is any doubt about whether or not the collected data represents reality. Whenever possible, ensure that you obtain a sample size of at least 100 measurements from at least two independent tests. Although there is no strict rule about how to decide which results are statistically similar without complex equations that call for huge volumes of data that commercially driven software projects rarely have the time or resources to collect, the following is a reasonable approach to apply if there is doubt about the significance or reliability of data after evaluating two test executions where the data was expected to be similar. Compare results from at least five test executions and apply the rules of thumb below to determine whether or not test results are similar enough to be considered reliable: 1. If more than 20 percent (or one out of five) of the test-execution results appear not to be similar to the others, something is generally wrong with the test environment, the application, or the test itself. 2. If a 90th percentile value for any test execution is greater than the maximum or less than the minimum value for any of the other test executions, that data set is probably not statistically similar. 3. If measurements from a test are noticeably higher or lower, when charted side-byside, than the results of the other test executions, it is probably not statistically similar.
Figure 15.7 Result Comparison 4. If one data set for a particular item (e.g., the response time for a single page) in a test is noticeably higher or lower, but the results for the data sets of the remaining items appear similar, the test itself is probably statistically similar (even though it is probably worth the time to investigate the reasons for the difference of the one dissimilar data set.
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