In mathematics, the term ‘median’ is often floated around along with ‘mean’ and ‘mode’. They all represent different types of ‘averages’, but what does ‘median’ exactly mean in this context? Let’s explore this concept in more detail.

In statistics, the median is the value separating the higher half from the lower half of a data sample. If you have a list of numbers in numerical order, the median would be the value in the middle of this list. It literally stands for the middle point of data. It is one of the many measures of central tendency, which are statistics that give us a centralized view of data.

To understand it better, consider this list of numbers: 1, 3, 3, 6, 7, 8, 9. The median number here is 6 because there are an equal amount of numbers above and below it. However, when you have an even set of numbers, such as: 1, 3, 3, 6, 7, 8. The median would be the average of the two middle numbers, which in this case would be (3+6)/2 = 4.5.

The median is an incredibly useful measure of data because it represents the exact middle of a dataset. Unlike the mean (or average), the median is not skewed by extremely high or low values. This property makes the median a more accurate representation of a dataset, especially when dealing with skewed distributions.

For example, if you were to look at a group of people’s salaries, the median salary would be the amount at which half the people earn more and half earn less. If a billionaire were to join this group, the mean (average) salary would go up significantly, but the median would essentially remain the same. This anomaly could give the impression that everyone in the group earns a very high salary, whereas the median would give a more accurate representation of the typical salary in the group.

Now to calculate the median of any given set of numbers, one must first arrange the numbers in order of size (from smallest to largest), then find the number that is exactly in the middle. If there are an even number of observations, then the median would be the mean (average) of the two middle numbers.

In conclusion, understanding the concept of median provides a more â€˜centralâ€™ or ‘typical’ value representation for any given set of numbers in a statistical study or any math-related field. Moreover, the determination of the median is very broad in applications such as probability theory, mathematical statistics, and descriptive statistics, and extends to subjective perception and decision making. Therefore, the understanding and application of the median is an essential part of mathematics and statistics.