Lecture Number 3 Data Description 41 99 Measures of Variability

Lecture number 3 data description 41 99 measures of

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Lecture Number 3 Data Description October 19, 2016 41 / 99
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Measures of Variability - Variance and Standard Deviation The horizontal distances between each dot (measurement) and the mean ¯ x will help you to measure the variability. If the distances are large, the data are more spread out or variable than if the distances are small. If x i is a particular dot (measurement), then the deviation of that measurement from the mean is ( x i - ¯ x ). Measurements to the right of the mean produce positive deviations, and those to the left produce negative deviations. The values of x and the deviations for our example are listed in the first and second columns of the table on the next slide. Lecture Number 3 Data Description October 19, 2016 42 / 99
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Measures of Variability - Variance and Standard Deviation Because the deviations in the second column of the table contain information on variability, one way to combine the five deviations into one numerical measure is to average them. Unfortunately, the average will not work because some of the deviations are positive, some are negative, and the sum is always zero (unless round-off errors have been introduced into the calculations). Lecture Number 3 Data Description October 19, 2016 43 / 99
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Measures of Variability - Variance and Standard Deviation Another possibility might be to disregard the signs of the deviations and calculate the average of their absolute values. This procedure has been used as a measure of variability in some statistical methods. The standard procedure, which leads to calculation of variance and standard deviation, involves working with the sum of squared deviations, i.e. each deviation is squared and then a sum is obtained for all the deviations as shown in column 3 of the table. From the sum of squared deviations, a single measure called the variance is calculated. To distinguish between the variance of a population and the variance of a sample, we use the symbol σ 2 ( σ is the Greek lower case sigma) for a population variance and s 2 for a sample variance. The variance will be relatively large for highly variable data and relatively small for less variable data. Lecture Number 3 Data Description October 19, 2016 44 / 99
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Measures of Variability - Variance and Standard Deviation We now define the variance for a population and a sample. Definition of Population Variance The variance of a population of N measurements is the average of the squares of the deviations of the measurements about their mean μ . The population variance is denoted by σ 2 and is given by the formula: σ 2 = N i =1 ( x i - μ ) 2 N (6) In practice, we don’t normally deal with populations but rather samples which are subsets of populations. Definition of Sample Variance The variance of a sample of n measurements is the sum of the squared deviations of the measurements about their mean ¯ x divided by ( n - 1).
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