# Median Definition and Calculation

This page explains median, its interpretation and how to find median of a data set. You can easily calculate median, percentiles, standard deviation, skewness, kurtosis, and other measures in Excel using the Descriptive Statistics Calculator.

## Median Definition

Like arithmetic average, geometric average, or mode, median is one of the measures of central tendency of a population, a sample, or a probability distribution.

Median is the value which divides a data set in two halves – one with values lower than the median and the other with higher values.

## Median Calculation

The word "calculation" might sound as an overstatement with median, as there is in fact nothing you need to calculate. You only need to find the value that divides the data set exactly in two halves (half of the observations is below and the other half is above the median).

The algorithm to find median of a data set has two steps:

1. Sort the values.
2. Find the middle value.

The second step is slightly different depending on the number of values being odd or even.

### Median of Odd Number of Values

Let's say a fund recorded the following performance in the last five years:

• Year 1: 5%
• Year 2: -3%
• Year 3: 19%
• Year 4: 8%
• Year 5: 1%

What is the median of the annual performance during this five-year period?

First, we need to sort the values from lowest to highest – we have: -3%, 1%, 5%, 8%, 19%.

When the number of values (years in our case) is odd, it is easy to find the middle observation. The median is 5%.

### Median of Even Number of Values

It is less simple when there is an even number of observations, because in this case there is no value which could be considered exactly "middle".

Let's add another year to the example above: In year 6 the fund's performance was +17%. What is the median now?

First, we sort the data: -3%, 1%, 5%, 8%, 17%, 19%.

There are six values now. Because the number of values is even, two of them are in the middle: 5% and 8%.

In other words, the border between the lower and upper half of the data set runs between the values 5% and 8%.

In this case median equals the mean (arithmetic average) of the two middle values: 6.5%.

## Calculating Median in Excel

In Excel, you can calculate median using the built-in MEDIAN function.

In the Descriptive Statistics Calculator, median is calculated in cell D10.

## Multiple Observations as the Median

Sometimes there are multiple identical values equal to the median.

For example, the median of the data set 1, 2, 3, 3, 7 equals 3.

In this case there are two of the observations equal to the median. Technically, the definition of median dividing a data set in two halves is a bit blurred here.

## Median and Quantiles

If you are familiar with the concept of quantiles (statistical measures which divide a data set into a certain number of equally sized parts), you can see that median is in fact a specific case of quantiles – the 2-quantile.

Median is also the same value as the second quartile (4-quantile). There are three quartiles, which divide a data set into four quarters. Median is the second, or middle, of them.

Similarly, median is the same as the fifth decile (10-quantile) or the 50th percentile (100-quantile).

Median performs the same function as mean or mode. It gives us an idea about the overall level, central value, or central tendency of a data set. You might have noticed reports about median household income or median house prices in the media.

In comparison with arithmetic average, median has some strengths and weaknesses. It addresses some of the well-known disadvantages of the arithmetic average.

In particular, median works better with skewed data sets and it is less sensitive to extreme values (which might result from errors in measurement).

Its greatest disadvantage relative to arithmetic average is ease and practicality of calculation – median is much less possible to be further processed and used in further calculations compared to the arithmetic average.

You can easily calculate median, percentiles, standard deviation, skewness, kurtosis, and other measures in Excel using the Descriptive Statistics Calculator.

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