Median in statistics: concept, properties and calculation

In order to have an idea of this or that phenomenon, we often use average values. They are used to compare the level of salaries in various sectors of the economy, the temperature and the level of precipitation in the same territory over comparable periods of time, the yield of grown crops in different geographic regions, etc. However, the average is by no means the only generalizing indicator - in a number of cases, such a value as a median is suitable for a more accurate evaluation. In statistics, it is widely used as an auxiliary descriptive characteristic of the distribution of a feature in a single population. Let's figure out how it differs from the average, and what is the reason for using it.

Median in statistics: definition and properties

Imagine the following situation: the firm employs 10 people together with the director. Simple workers receive 1000 UAH each, and their leader, who is also an owner, is 10000 UAH. If we calculate the arithmetic mean, it turns out that on average the salary at this enterprise is 1900 UAH. Will this statement be fair? Or take this example, in the same hospital ward there are nine people with a temperature of 36.6 ° C, and one person at whom it is equal to 41 ° C. The arithmetic mean in this case is: (36.6 * 9 + 41) / 10 = 37.04 ° C. But this does not mean that everyone is sick. All this pushes the idea that one middle is often not enough, and that's why, in addition to it, the median is used. In statistics, this indicator is called the variant, which is located exactly in the middle of the ordered variation series. If you count it for our examples, you will get 1000 UAH accordingly. And 36.6 ° C. In other words, the median in statistics is a value that divides the series in half in such a way that on both sides of it (down or up) the same number of units of the given set is located. Because of this property, this indicator has several more names: the 50th percentile or the quantile 0.5.

How to find a median in statistics

The way of calculating this value depends largely on the type of variation series we have: discrete or interval. In the first case, the median in the statistics is quite simple. All you need to do is find the sum of the frequencies, divide it by 2 and then add to the result ½. It is best to explain the principle of calculation in the following example. Suppose we have grouped fertility data, and we need to find out what the median is equal to.

Having carried out simple calculations, we find that the sought value is: 195/2 + ½ = 98, i.e. 98th option. In order to find out what this means, one should consistently accumulate frequencies, starting with the smallest variants. So, the sum of the first two lines gives us 30. It is clear that there are no variants here. But if you add the frequency of the third option (70) to the result, you get the amount equal to 100. It is exactly the 98th version, and so the median will be a family that has two children. As for the interval series, the following formula is usually used here:

M _e = X _Me + i _Me * (Σf / 2 - S _Me-1 ) / f _Me , in which:

• X _Me - the first value of the median interval;
• Σf - the number of the series (the sum of its frequencies);
• I _Me is the value of the median range;
• F _Me is the frequency of the median range;
• S _Me-1 is the sum of the cumulative frequencies in the ranges preceding the median.

Again, without an example here is difficult to understand. Suppose there is data on the amount of wages.

To use the above formula, we first need to determine the median interval. As such a range, choose one whose cumulative frequency exceeds half or all of the sum of the frequencies. So, dividing 510 by 2, we get that this criterion corresponds to an interval with a wage value of 250,000 rubles. Up to 300,000 rubles. Now you can substitute all the data in the formula:

M _e = X _Me + i _Me * (Σf / 2 - S _Me-1 ) / f _Me = 250 + 50 * (510/2 - 170) / 115 = 286.96 thousand rubles.

We hope our article has proved useful, and now you have a clear idea of what a median is in statistics and how to expect it.