# Are the mean and median the same

## The average is not in the middle

Mean values are important indicators in statistics and applied mathematics in general. I consciously write “mean values” in the plural. There are a number of different mean values, all of which are justified. In order not to be misled by statistics such as those in the news, you need a basic understanding of the most important mean values.

At the end of this page I show a typical case from practical life in which the mean values differ from one another, which often leads to misunderstandings. But first I will explain three simple but important mean values: the mode, the median and the arithmetic mean, which is often also called the average. This should be done using an example - I rolled the dice nine times with the following result:

⚀ ⚀ ⚄ ⚀ ⚃ ⚅ ⚁ ⚀ ⚅

### mode

The mode describes the most common occurrence. If I summarize my dice results according to their characteristics, I quickly see that the number 1 occurs most frequently:

⚀ ⚀ ⚀ ⚀

⚄

⚃

⚅ ⚅

⚁

So ⚀ is the mode. We can always form the mode for finite quantities, even if the characteristics such as different types of candy have nothing to do with numbers and have no natural order. If we examine a box with candies of different types, the mode is the most represented type. If several types of candy are tied, there are several modes.

### Median

The median is obtained by sorting all values in ascending order and then picking out the central value:

⚀ ⚀ ⚀ ⚀ __⚁__ ⚃ ⚄ ⚅ ⚅

In our example, ⚁ is the median. The median can only be formed if there is a natural order among the values, according to which one can sort. This does not have to be expressed in numbers like the eyes of the sides of a dice, but can also be a rank in the police, for example.

With an even number of values, there is no *the one* central value, but two equally central. In the event that they differ, there are different definitions of the median. Sometimes the smaller value is chosen (lower median), sometimes the larger (upper median). Some people also choose the arithmetic mean of the lower and upper median, but this limits the applicability.

### Arithmetic mean

The arithmetic mean is often called the average and is probably the best known mean. It is obtained by adding up all the values and dividing by the number of values. In our example, the average of the numbers is 3:

In order to be able to form the arithmetic mean, you have to deal with numbers - you cannot form an average from two detective superintendents and one senior detective. By the way, the Greek word αριθμός (arithmos) also means number.

In our cube example, the arithmetic mean is the largest of the three mean values considered. In practice this is not true for rolling the dice, but it is typical for many distributions. In principle, however, any order of mode, median and arithmetic mean is possible. The mean values can also coincide.

### Average values in practice

News broadcasts provide us with regular information on economic key figures. But other sources also provide us with statistical data. My electricity provider, for example, supplements the annual accounts by comparing my consumption data with the usual ones.

Many of the statistical data are arithmetic means, for example the average temperature of the month, the annual CO₂ emissions per person, life expectancy, asylum seekers per day, population density, per capita debt, the average time spent in front of the television ...

### Distribution of income

In 2003 that was **average** Net monthly income of a household in Germany 2833 euros. Five years later, the average was 2914 euros. So a slight growth; the income had apparently increased. But what did an average household look like? In fact, it was found among the moderately higher earners. The arithmetic mean was not in the middle of society!

He tells us how income was in the middle of society **Median** the net monthly income of households. In 2003 this was 2394 euros, almost 440 euros below the average. It is exciting to look at the development: the median decreased to 2369 euros by 2008. The median and average therefore drifted in different directions; the gap widened to over 540 euros.^{1}

The figure illustrates the income distribution behind the different mean values: A large proportion of households concentrate on a few low income classes with the **mode** at 1500–2000 euros and an almost equally strong class at 1000–1500 euros. This explains the median of 2369 euros, i.e. that the income of half of the households was below or at this amount.

Higher incomes are spread across many income brackets, including those that are well above the middle of society. The graphic is cut off at 6,000 euros, but actually the incomes extended in decreasing numbers to even higher amounts. Even in small numbers, these high incomes made a significant contribution to the sum of all incomes and thus raised the average.

### assessment

Economic indicators, which are arithmetically averaged over the population, say more about the totality of the economy than about a citizen in its midst. If you want to know where you stand in comparison to the middle of society, you should disregard the average and inquire about the median.

But there are also economically sensible applications for the arithmetic mean. For example, per capita values can be used to compare the performance of the economies of several countries with different populations.

- Data source: Federal Statistical Office, Fachserie 15 Heft 6, EVS 2008, p. 31

In its calculation, the Federal Statistical Office only included households whose monthly net income did not exceed 18,000 euros, since too few households were included in the income and consumer sample for the extrapolation. It can therefore be assumed that the arithmetic mean and median were even further apart in reality. - Data source: Federal Statistical Office, Fachserie 15 Heft 6, EVS 2008, p. 41

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