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The range is a simple yet telling measure of the spread in a dataset. It's calculated by subtracting the smallest value from the largest value. For example, if you have a dataset of {3, 7, 8, 10, 15}, the minimum value is 3 and the maximum is 15. Thus, the range is 15 - 3 = 12. This measure gives us a quick glimpse at how spread out the values are.
However, because it's solely dependent on the two extreme values, the range is incredibly sensitive to outliers. An outlier, by definition, is an unusually large or small data point compared to others in the dataset. Adding an outlier to our previous dataset, say 100, would make the range 100 - 3 = 97. This shows how one single outlier can drastically alter the range, even though it might not affect the other data points in the same way.

The mean, or average, offers a different perspective. Calculated by summing all values in the dataset and dividing by the count of values, it provides a central tendency. For instance, using the dataset {3, 7, 8, 10, 15}, the mean is \((3 + 7 + 8 + 10 + 15) \div 5 = 8.6\).
Unlike the range, the mean considers all values, making it more robust against individual extreme outliers, especially in larger datasets. When an outlier is included, such as a 100 in our dataset, the new mean becomes \((3 + 7 + 8 + 10 + 15 + 100) \div 6 = 23.83\). As the dataset size increases, each value, including the outlier, has a smaller relative influence on the mean, thus reducing its impact.

Standard deviation measures how spread out the data points are from the mean. It's a bit more complex to compute but provides significant insight into data variability. In essence, the standard deviation represents the square root of the variance, which is the average of the squared differences from the mean.
For a dataset, this means calculating the mean, finding each data point's deviation from the mean, squaring those deviations, averaging them, and taking the square root of that average. With an outlier present, the standard deviation does increase, but like the mean, its sensitivity reduces as the number of data points increases.

• In smaller datasets, an outlier significantly affects the standard deviation, increasing perceived variability.
• However, in larger datasets, the relative impact of a single outlier diminishes, making the standard deviation a somewhat resistant measure to such anomalies.

Overall, while affected by outliers, standard deviation and mean react less dramatically than the range, especially as a dataset grows.