Why Is the Variance Preferred Over the Range as an Index

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Examlex · Accepted Answer

The Answer is The variance is often preferred over the range as an index of variability for several reasons:

1. Sensitivity to all data points: Variance takes into account every value in the data set, which means it reflects the variability of every individual observation. In contrast, the range only considers the two extreme values (the maximum and the minimum) and ignores all the other data points. This makes the variance a more comprehensive measure of variability.

2. Mathematical properties: Variance has desirable mathematical properties that make it useful in statistical analyses. It is the square of the standard deviation, another important measure of dispersion, and it plays a key role in various statistical methods, such as hypothesis testing, confidence intervals, and regression analysis. These properties are not shared by the range.

3. Less sensitive to outliers: Although the variance is affected by outliers (extreme values), it is generally less sensitive to them than the range. The range can be dramatically influenced by a single outlier, which may not be representative of the overall variability in the data set. Variance, by considering all data points, dilutes the impact of outliers.

4. Basis for other statistical measures: Variance is the foundation for other important statistical concepts, such as the standard deviation, which is the square root of the variance and provides a measure of variability that is in the same units as the original data. Variance is also used in the calculation of the coefficient of variation, which is a standardized measure of dispersion.

5. Additivity: Variance has the property of additivity for independent random variables, which means that the variance of the sum of two independent random variables is equal to the sum of their variances. This property is particularly useful in the analysis of variance (ANOVA) and other statistical techniques that involve combining variances from different sources.

In summary, the variance is preferred over the range as an index of variability because it considers all data points, has useful mathematical properties, is less sensitive to outliers, serves as the basis for other statistical measures, and has the property of additivity. These characteristics make it a more reliable and informative measure of variability in a data set.

Why Is the Variance Preferred Over the Range as an Index

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