Using parentheses in math, what the median is

Median clearly explained

Median or mean - that is the question! When do you take what? And what is the median anyway? How is it calculated? For those in a hurry, there's the short version first (sort of a spoiler before the article even starts ...):

The median or central value is the value that lies exactly in the middle of a series of values ​​ordered according to size.

Imagine writing all the values ​​in a data set, sorted by size, lengthways on a piece of paper, from edge to edge. Then you fold the paper exactly in the middle: then the value on the crease is the median.

The median divides a sized dataset into two halves: at least 50% of the data is less than (or equal to) and at least 50% is greater than (or equal to) the median.

It is the value that lies exactly in the middle of the data set - hence the name "central value". It is a measure of the central tendency.

In the case of an odd data record, the central value is contained directly in the data record; in the case of an odd data record it is averaged from the two values ​​in the middle, i.e. created artificially, so to speak.

Other alternative names are the 2nd quartile (Q2) and the 50% quantile.

In which area of ​​statistics does it belong?

The central value belongs in the world of descriptive or descriptive statistics. Descriptive statistics are about reducing the information contained in a data set to a few parameters and also presenting it graphically. In contrast to inferential statistics, no conclusions are drawn about the corresponding population, but only described what can be found in a specific sample.

The median is one of the measures of location that describe the center of gravity of a data set.

To be honest, there isn't so much that can be interpreted here. Only this: up to this point 50% of all values ​​have accumulated and 50% are above. That's it.

When there are outliers or extreme values ​​in the data set and the mean should not be used - or when ordinal scaled data is available. The median is used to determine or quantify the middle of a data set.

Caution: For ordinally scaled data such as B. ranks and even data set, the median cannot be clearly defined: For example, if there was 1st, 2nd, 3rd and 4th place in a beauty contest, the arithmetic median would be between 2nd and 3rd place. That would be the 2.5th place, which however does not exist!

The calculation of the central value is quite simple, but, as usual, it is a bit complicated in the statistics. As you will see in a moment, there are two variants: one for an odd and one for an even data set.

As an example, we take a survey on the fear of missing out on something (FOMO = Fear of Missing Out), surveyed among young people between 14 and 18 years of age, where FOMO is between 0 (no fear, deeply relaxed) and 100 (dies, if not all Social media feeds are checked every minute). These are the dates:

Odd record: 23, 56, 87, 30, 28, 45, 66, 18, 49, 48, 55n = 11

Even data set:23, 56, 87, 30, 28, 45, 66, 18, 49, 48    n = 10

Odd record

Even without formulas, the first step is always to sort the data according to size!

Then you just look which value is exactly in the middle: here it is the 48th bingo.

even record

Here you just have to average the two values ​​in the middle - et voilà!

Calculation of the median with formulas

And now the whole thing with formulas for the two variants. The subscript brackets to the right of the x show position numbers, i.e. where a certain value is located in a data series sorted by size.

You cannot call up the median exclusively, but receive it with the exploratory data analysis.

To do this, go to "Analyze", "Descriptive Statistics", "Exploratory Data Analysis". In the window that opens, select the variables you are interested in, move them to the "Dependent variables" field and press "OK". You can find an example of this here.

Last but not least, a short overview:

  • The median is the value that divides a data set into 50% smaller and 50% larger values
  • Belongs to the world of descriptive statistics
  • Applicable from the ordinal scale upwards
  • With ordinal and straight n not clearly defined
  • If there are outliers, it is more suitable for interpretation than the mean value (the latter reacts very sensitively to outliers and extreme values)
  • Is included in SPSS in Exploratory Data Analysis.
  • The median / central value is the value or line that lies in the middle of the box of the box plot


Turning insight into action ...

How about - is the motivation enough for an immediate application of what you have read? Then you could, for example, enter the values ​​given above into SPSS and let the exploratory data analysis do the calculations.

Or you can think of a fictitious study along with numbers and start calculating wildly.

Option 3: Look for exercises - either on the Internet or in your textbooks or exercise books.

Or you don't do anything ... fine too.

But first, as always: Reward!

Statistics are known to be not that funny. Therefore, you should really enjoy yourself after learning (or during it) ...



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