The Sample Median

In the box below, please enter the sample median of the data given above, then click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

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You got the correct answer of 21.5. Congratulations!

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Here is the mathematical formula for the median. Note that the notation is rather complicated. This is due to the fact that data are discrete.

In this formula, there are several symbols to know:

Thus, this formula can be read as “The median is the set of all values \( x \) such that at least half of the data are at most \( x \) and at least half of the data are at least \( x \).”

This formula says that the sample median is a value that splits the data into two equal halves once it is sorted from smallest to largest. By its mathematical definition, there may be an infinite number of medians in a data set. If the sample size is even and the middle two values (after sorting the data) are different, then any number between the two (including the two) are medians. Statisticians will average these middle two values to obtain a unique median.

For instance, if the sorted data are 1, 3, 5, 6, then the mathematical median is any number between 3 and 5. The statistical median is the average of 3 and 5, which is 4.

When dealing with data, the median is determined visually. The first step is to sort the values from smallest to largest. Then, if the sample size n is odd, the median is the (n+1)/2 th value. If the sample mean n is even, then the median is the average of the n/2 th and n/2+1 th values.

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Show the R Code

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Copy and paste the following code into your R script window, then run it from there.

In the R output, the median is the number output by the script.

Show the Excel Code

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Copy and paste the following code into your Excel spreadsheet window, making sure your cursor is in A1 when you paste.

Note that the function in Excel is MEDIAN.