# What is a median 1

### 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:

values | 3 | 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

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