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Does 50th Percentile mean Average?

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The 50th percentile of any given data set is the middlemost value of the given values. Exactly half of the values are greater than the 50th percentile and the remaining half are smaller than the 50th percentile. That is, the 50th percentile is nothing other than what is commonly known as the “median” of the data set.

There are three most commonly used kinds of averages – these are the mean, median, and mode. Since the 50th percentile is equal to the median we can say that it is indeed a kind of an “average” in the sense that it represents the central tendency of the data.

On the other hand, sometimes the word “average” is used to exclusively refer to the mean. The mean can be found by taking the sum of all the data values and then dividing it by the number of observations. If by the word average, one refers to the mean of the data then the 50th percentile is not equal to the average. This is because the mean and the 50th percentile (median) may not be equal. Thus we must be clear about what definition of “average” is being used to decide whether the 50th percentile is equal to the average.

Example:

Consider the following set of data values: 1, 3, 5, 8, 9.

The mean of the given data values is, \text{Mean }= \frac{1+ 3+ 5+ 8+ 9}{5}= \frac{26}{5} = 5.2

On the other hand, the 50th percentile (median) can be found as, \text{Median }= \left(\frac{n+1}{2}\right)^{th}\text{ term }= \left(\frac{5+1}{2}\right)^{th}\text{ term} = 3^{rd}\text{ term } = 5

The value 5 is indeed the 50th percentile since exactly 2 values lie above it and 2 values lie below it. It divides the data into two equal parts. As we can see the value of the mean/average is not equal to the value of the 50th percentile.

When is 50th Percentile equal to the Mean?

If the given data values are symmetrically distributed then the 50th percentile (median) is equal to the mean of the given data values. In the above example, we see that the distribution is not symmetric which is why the mean and the median do not match.

If we are given a sequence of values distributed symmetrically about the central values then the mean, median and the mode are all equal.

Unimodal Symmetric Histogram

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