What Is the Formula for Finding the Median of Ungrouped Data?

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Assuming the data has N elements in order of magnitude, the median is found by taking the ((N+1)/2)th element if there are an odd number of elements. If there are an even number of elements, then the median is an average of the (N/2)th and (N/2 + 1)th element.

To determine the median of ungrouped data with N elements, one should first order the data from lowest to highest. The median is determined by finding either the middle element in the data or the average of the middle two elements, depending on whether there are an odd or even number of elements in the data.

For example, suppose a set of data has the numbers {4, 8, 7, 10, 100}. First, one would reorder the elements from lowest to highest so that it appears as {4, 7, 8, 10, 100}. There are an odd number of elements in the set; i.e. N=5. Since (5+1)/2 = 3, in this case, the third element in the set is the median, which is 8.

Suppose the ordered data set has a sixth element of 1000, so it now looks like {4, 7, 8, 10, 100, 1000}. In this case, N=6; therefore N/2=3 and N/2 + 1 = 4, so the median is the average of the third and fourth data element. The median is (8 + 10)/2 = 9.