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The mean is one of the most commonly used measures of central tendency. It is calculated by adding up all the values in a data set and then dividing by the number of values. This gives you the average. For instance, if you have data points like 2, 4, 6, 8, and 10, the mean would be \( \frac{2 + 4 + 6 + 8 + 10}{5} = 6 \).

A key feature of the mean is that it uses all the data points in its computation. However, this means that it is sensitive to outliers. An outlier is a value that "lies outside" most of the other values in your data set. For example, if your data set is 2, 4, 6, 8, and 50, the mean becomes \( \frac{2 + 4 + 6 + 8 + 50}{5} = 14 \).

As you can see, the presence of 50, which is an outlier, significantly increases the mean, indicating that mean is not resistant to outliers.

The median is another important measure of central tendency, which represents the middle value when a data set is arranged in ascending order. If the number of observations is even, the median is the average of the two middle numbers. For instance, in the data set 3, 5, 7, the median is 5. For an even set, such as 3, 5, 7, 9, the median would be \( \frac{5 + 7}{2} = 6 \).

Unlike the mean, the median is a positional average meaning it does not consider the magnitude of the values. As a result, it is not affected by outliers. If the data set changes to 3, 5, 50, the median remains 5. Therefore, the median is considered a resistant measure of center.

Resistance to outliers is a valuable property for a measure of central tendency, as it prevents the measure from being unduly affected by extreme outliers. Outliers can distort statistical analyses by pulling the measure of center towards them.

• The mean is not resistant as it incorporates all values, including outliers, in its calculation.
• The median is resistant because its calculation is based on the position of data points, not their value.
• The trimmed mean, by discarding a percentage of the highest and lowest data points, gains resistance to outliers.
• The midrange, on the other hand, relies directly on extremes, thus it is not resistant.

Understanding resistance helps in selecting the best measure when summarizing data, especially in the presence of outliers.

The trimmed mean offers a balance between the mean and the median. This measure is calculated by removing a certain percentage of the smallest and largest values in the data set and then determining the mean of the remaining data.

For example, in a data set of 1, 3, 5, 7, and 100, a trimmed mean with 20% trimming would remove 1 and 100, calculating the mean as \( \frac{3 + 5 + 7}{3} = 5 \).

By eliminating the extremes, the trimmed mean reduces the potential distortion from outliers, thus increasing its resistance. It provides a compromise between using all data points and completely ignoring outliers, like when using the median.

The midrange is a measure of central tendency that considers only the smallest and largest observations in a data set. It is calculated by averaging these two extremes. For example, in the data set 2, 3, 10, the midrange would be \( \frac{2 + 10}{2} = 6 \).

Due to its reliance on the most extreme values, the midrange is highly sensitive to outliers. If an outlier is present, it can significantly skew the midrange. For example, if the data set changes to 2, 3, 100, the midrange becomes \( \frac{2 + 100}{2} = 51 \), demonstrating a substantial shift.

This sensitivity makes the midrange a less reliable measure in the presence of outliers, compared to more resistant measures like the median or trimmed mean.