The Median in Statistics

Explanation

By definition, there are as many values above the median as below it. Each sample has a unique median. (II) and (III) hold for the mean, not for the median.

Let's analyze each statement and determine which ones are true:

I. There are as many values above the median as below it. This statement is true for a symmetrical distribution. In a symmetrical distribution, the median divides the data into two equal halves, with half the values above and half below it. Therefore, statement I is true.

II. The sum of the differences between the observations in a sample and the median of the sample equals zero. This statement is true. By definition, the median is the value that divides the data into two equal halves. Therefore, the sum of the differences between each observation and the median will be zero because the positive and negative differences cancel each other out. For example, if you have a dataset with three numbers: 1, 3, and 5, the median is 3. The differences between the observations and the median are -2, 0, and 2, and their sum is zero. Therefore, statement II is true.

III. The median is greatly affected by "outliers." This statement is false. The median is a measure of central tendency that is less affected by outliers compared to the mean. In fact, the median is often preferred over the mean when dealing with data that has extreme values or outliers. Outliers have a smaller impact on the median because it only considers the position of the middle value(s) in the dataset, rather than the actual values themselves. Therefore, statement III is false.