Sample Median [Sample Statistics]

The Problem

The median is the middle number of the sample when the values are ranked from lowest to highest. It is used to estimate the center of a population when the population is highly skewed. To illustrate calculating the median, assume that you collected the following 7 data values:

57, 97, 61, 73, 73, 47, 35

Calculate the median value of this sample, $\tilde{x}$ .

Your Answer

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.

Your guess:

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Assistance

Show Formula

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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.

$\tilde{x} = \left\{ x \mid P[X \le x] \ge 0.500 \cap P[X \ge x] \ge 0.500 \right\}$

In this formula, there are several symbols to know:

$\tilde{x}$

the sample median

$\left\{ \cdots \right\}$

the set of

$\mid$

such that

$\cap$

the intersection of the two conditions, read as “and”

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.

Show Solution

How to calculate the median for this data
Step 1:	Sort the data from smallest to largest:	35, 47, 57, 61, 73, 73, 97
Step 2:	Determine if n is even:	n = 7 is not even
Step 3a:	Determine (n+1)/2:	4
Step 3b:	Determine data value number 4:	61
Step 4:	The median is this middle value:	61

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The Sample Median

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Another Set of Data

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