As a statistical expert with a deep understanding of the nuances of various measures of central tendency and dispersion, I am well-equipped to discuss the robustness of statistical measures such as the standard deviation.
The standard deviation is a widely used measure of the amount of variation or dispersion in a set of values. It is defined as the square root of the variance, which is the average of the squared differences from the mean. The standard deviation is sensitive to outliers because it involves squaring the differences from the mean, which can amplify the effect of extreme values.

In contrast, the interquartile range (IQR) is a measure of statistical dispersion that is less sensitive to outliers. The IQR is the difference between the first quartile (25th percentile) and the third quartile (75th percentile) in a data set. Since the IQR is based on the middle 50% of the data, it is considered a more resistant measure to extreme values or outliers.

The term resistant in statistics refers to the property of a statistical measure to be unaffected by small changes in the extreme values of the distribution. A resistant measure is desirable because it ensures that the measure remains stable even when the data contains outliers or extreme values.

Now, let's delve into the standard deviation's resistance to outliers. While the standard deviation is a powerful tool for summarizing the spread of a distribution, its sensitivity to outliers is a notable drawback. When a data set contains outliers, the standard deviation can be significantly skewed, leading to a misleading representation of the true variability within the data set. This is because outliers can disproportionately influence the calculation of the mean and subsequently the variance and standard deviation.

To illustrate, consider a data set with the following values: 1, 2, 3, 4, 100. The mean of this data set is (1+2+3+4+100)/5 = 22, and the standard deviation, calculated as the square root of [(1-22)² + (2-22)² + (3-22)² + (4-22)² + (100-22)²]/(5-1), is approximately 57.38. Notice how the single outlier (100) has a substantial impact on the standard deviation, making it much larger than it would be without the outlier.

On the other hand, the IQR for the same data set, which is the range between the 25th percentile (2) and the 75th percentile (4), is 2. This value is unaffected by the outlier at 100, demonstrating the IQR's resistance to the influence of extreme values.

In summary, while the standard deviation is a valuable measure of dispersion, it is not resistant to outliers. The IQR, however, is a more robust measure that maintains its stability in the presence of outliers. When choosing a measure of variability, it is essential to consider the nature of the data and the potential for outliers. If the data is expected to contain outliers, a resistant measure like the IQR may be more appropriate. Conversely, if the data is relatively clean and free of outliers, the standard deviation can provide a comprehensive and accurate depiction of the spread of the data.

read more >>

Notice that only a few numbers are needed to determine the IQR and those numbers are not the extreme observations that may be outliers. The IQR is a type of resistant measure. The second measure of spread or variation is called the standard deviation (SD).read more >>