What is statistics? – Median

Definition of Statistical terms
Probability (P): Probability is the chances that an event will happen or not happen e.g. what is the Probability (P) of a dice that you will throw only once to result a “6”? -Answer: P=1/6 or what is the probability to have open or close your eyes now? -Answer: 50% or 1/2.

What is Median value – History
The first source that referred to “Median value” was in an old Navigation paper written in 1599 by Edward Wright. He tried to find a spot by throwing arrows and: “who to find out the place where the mark stood, shall seek out the middle place amongst all the arrows”.

The usefulness of Median value or the Mean value is that can tell you what is the “central trend” value of a dataset.

What is Median value – Theoretical definition
The Median value is another measurement of the Central Tendency. The Median value can be defined as the number that its position is in the middle of an arithmetic data set arranged in an ascending order. Note that the Median value, by definition, divides a dataset in half: 50% of its values will be before the Median and the rest 50% of values will be after the Median when it is arranged in an ascending order.

Median value – Odd number of values
When a dataset includes an odd number of values e.g. 15 or 23 or 57 or 239, or 1001 values, then one Middle number exists.

Median value – Even number of values
However, when a dataset includes an even number of values e.g. 4 or 8 or 26 or 1002 values, then, the Median value is the average of these two middle values.

Statistical definition of Median value
The statistical definition of Median value includes the probability term. The 50% of the numbers must be left or right of this value: P(X\leq m)\geq \frac{1}{2} \hspace{5} and\hspace{5} P(X\geq m) \geq \frac{1}{2}

Note that m is the median value, while x_{i} is a symbol used to show each element / number of an arithmetic set e.g. x_{1},x_{2}... \hspace{5}...x_{n}.

The formula for the Median says that: In order a value to be named as Median value, it must satisfy that the probability of m as a number to be \frac{1}{2} or otherwise 50% lower or higher than the existed numbers/values in an arithmetic dataset X.

Statistical Example I
You are a freelancer and you would like to calculate the Median value of your incomes for 5 months, which are the following ones: 200, 300,200,400,400, In order to find the Median value:

i)you must arrange the numbers in an ascending order: 200,200,300,400,400.

ii) Because the size of your dataset is Odd, that is, it includes five values: 5, one Median value exists.

iii) The middle value is the Median value which is the 3rd value and it is equals to 300.

median_ex_I

Statistical Example II
You have a shop and you have written down the number of customers that visited your shop in the last 10 days e.g.: 1, 2,  2, 1, 2, 0, 100, 100 , 100, 100.

i) In order to find what is the Median value, the numbers must be arranged in an ascending order: 0, 1, 1, 2, 2, 2, 100, 100, 100, 100.

ii) Your dataset includes an Even number of values, that is, it includes: 10, then:

iii) Two middle values exist. By counting the numbers from Right to Left, we have as a result the value of 2. By counting the numbers from Left to Right, we have as a result the value of 2.

iv) The Median value is the average number of these two values: 2+2=4=>\frac{4}{2}=2, which is 2

median_example_II

Median and the Mean
In example I, the mean value of this dataset is \frac{1500}{5}=300 which is the same value as the value of the Median that we have found: (m=300).

median

In statistical example II, the Median was 2. The mean value of this dataset is: \bar{x}=\frac{0+1+1+2+2+2+100+100+100+100}{10}=\frac{408}{10}=40.8

median and mean

In the 1st example, the Median and the Mean had exactly the same value. However, in the 2nd example, the values of the Arithmetic Mean and Median are not the same ones. By definition, the Mean depends on the size of the Dataset while -By definition- Median is Independent from the size of the Dataset. Therefore, both values is useful to be calculated.

An historical phenomenology of mean and median