What are some examples of a median

Speed ​​control


Image: JENOPTIK Robot GmbH


With a speed control in one 60 zone the following values ​​were determined for seven vehicles:

$$ 60 frac {km} {h} \; \ 53 frac {km} {h} \; \ 168 frac {km} {h} \; \ 59 frac {km} {h} \; \ 52 frac {km} {h} \; \ 55 frac {km} {h} \; \ 57 frac {km} {h} \ $$


Tony wants to know how much the drivers drove on average. She calculates:

$$ frac {60 + 53 + 168 + 59 + 52 + 55 + 57} {7} = frac {504} {7} = 72 $$

Her comment: “They drive on average $$ 72 frac {km} {h} $$. That is $$ 12 (km) / h $$ too much! They're all RASERS !!!! "

Attention! Look again at the readings.
There is only one speeder (driver 3). The others drive fine. Such values ​​are called Runaway. They influence that Average clear.

Outliers influence the mean. if there are outliers in the data, the mean is not very meaningful.

A new mean: the median

And now? There are other metrics besides the mean. These are better suited for data with outliers. One example is that Median.

Arrange the measured values ​​according to size:
52   53   55     59   60   168

The Indian center lying value of ordered sizes describes the control measurement better: A clear majority of the vehicles adheres to the speed limit with $$ 57 frac {km} {h} $$.
This value is called Median. The symbol is $$ tildex $$. Say: x snake.

You determine the median as follows:

Step 1: Arrange all data according to size.
2nd step: Find the median.
a) The value that is exactly in the middle is the median.
b) If there are 2 values ​​in the middle, you form the mean of these 2 values.

Example 1:

Order of numbers: 1 3 3 6 8 9
The median is exactly in the middle: 5

Example 2:

Ordered number series: 2 3 8 9
There are 2 numbers in the middle. Calculate the median: $$ (3 + 5) / 2 = 8/2 = 4 $$

The Median (or Central value) is a good metric when the data has an outlier. Order the data according to size. Is the number of dates odd, the median is the value in the middle. Is the number just, the median is the mean of the two middle values.

Median for a frequency list

This point list is given:

values3 4 5 8
absolute frequency 2 5 3 3



Step 1: Arrange all data according to size.

3   3   4   4   4   4   4   5   5   5   8   8   8

It would also be possible to reverse the order:

8   8   8   5   5   5   4   4   4   4   4   3   3

2nd step: Find the median.
a) The value that is exactly in the middle is the median.
b) If there are 2 values ​​in the middle, you form the mean of these 2 values.

The number of dates is odd.
3   3   4   4   4   4     5   5   5   8   8   8
The Median is $$ tildex = 4 $$.

A median divides a number of values ​​in half. The values ​​of one half are less than or equal to the median. The others are greater than or equal to.

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Comparison of median and mean

Example 1:


The data includes a Runaway. The value is 16. This value has a strong influence on the mean value $$ barx = 6 $$. The median $$ tildex = 3 $$ characterizes the amount of data better.

Example 2:

There are two already ordered series of numbers.

Series of numbers without Outliers: 74 78 84 85 87 90

Average: 83
Median: 84.5
Mean and median are quite close together.

Example 3:

Series of numbers With Outliers: 0 1 3 5 5 7 9 11 15 99
Average: 15.5
Median: 6th
Mean and median are further apart.

If the outlier 99 is not taken into account, the values ​​are closer together:
Average: 6.2
Median: 5th