Why is the median resistant, but the mean is not?

Verified step by step guidance

1. Step 1: Understand the definitions of mean and median. The mean is the arithmetic average of all data points, calculated as \(\text{mean} = \frac{\sum x_i}{n}\), where \(x_i\) are the data values and \(n\) is the number of observations. The median is the middle value when the data are ordered from smallest to largest.
2. Step 2: Recognize what it means for a statistic to be resistant. A resistant statistic is one that is not affected much by extreme values or outliers in the data set.
3. Step 3: Consider how the mean responds to extreme values. Since the mean involves summing all data points, an extremely large or small value can significantly change the total sum, thus shifting the mean noticeably.
4. Step 4: Consider how the median responds to extreme values. Because the median depends only on the middle position in the ordered data, extreme values at the ends do not affect which value is in the middle, so the median remains stable.
5. Step 5: Summarize that the median is resistant because it depends on the position within ordered data, not the magnitude of values, while the mean is non-resistant because it depends on the magnitude of all values, making it sensitive to outliers.

Key Concepts

Median

The median is the middle value in an ordered data set, dividing it into two equal halves. It is resistant because it depends only on the position of values, not their magnitude, so extreme values (outliers) have little effect on it.

Mean

The mean is the arithmetic average of all data points, calculated by summing values and dividing by the count. It is sensitive to extreme values because every data point contributes to the total sum, so outliers can significantly shift the mean.

Resistance to Outliers

Resistance refers to a statistic's ability to remain unaffected by extreme or unusual data points. The median is resistant because it relies on data order, while the mean is not because it incorporates all values equally, making it vulnerable to skewed data.