“Go down deep enough into anything and you will find mathematics.” - Dean Schlicter
In maths classes in school, students will learn about statistics, geometry, and how to calculate the average and the median. Often misunderstood, the median is an important indicator when analysing probabilities, geometry, and statistics.
Maths can be complicated and for many students, private academic support tutoring is one of the best ways for them to succeed. In this article, we’ll be looking at how to find the median.
What is a Median in Mathematics?
To learn how to calculate the median, you’ll need to know exactly what it is.
The median is the value that separates a series of values into equal parts and the median is regularly used in statistics as it can be more useful in certain cases than the mean average.
The median is the central value of a statistical series and to get this value, you’ll need to put all your data into order. From there, you can easily find the value that sits perfectly in the middle of your series.
Let’s use a swimming coach as an example.
Say that the coach has nine swimmers doing two lengths of the pool and they’ve provided the following times in seconds:
30.6, 29.1, 32.9, 35.1, 30.0, 36.4, 31.7, 35.5, 33.9.
Once ranked, the values look like: 29.1, 30.0, 30.6, 31.7, 32.9, 33.9, 35.1, 35.5, 36.4.
The median time would be 32.9.
This means you can split swimmers into groups on either side of this value.
What happens if another student joins the class?
Imagine they swam two lengths in 28.7 seconds. This will leave us with an even number of swimmers (10 data points). The “middle” of this set would sit between two values: 31.7 and 32.9.
To calculate your median with an even number of values, you’ll have to calculate the mean between the two numbers (31.7+32.9)/2 = 32.3.
- If the sample size is odd, you pick the middle value.
- If the sample size is even, you take the mean of the two values on either side of the real central value.
Pretty simple, right?
Finding the Median in a Discrete Statistical Series
In statistics, a variable is said to be discrete when it’s a real value that can be counted. For example, the results of students in a maths test on descriptive statistics.
Let’s say there were a total of 20 marks up for grabs and the students scored: 5, 12, 11, 10, 6, 17, 11, 12, 10, 13, 9, 11, 12, 8, 7, 10, 11, 10, 12, 11, 9, 10, 8, 11.
Order the Statistical Values in a Table
Finding the median requires ordering the values in ascending order in one row and the number of students with said value on the second row.
This will leave you with the following:
- Row 1: 5, 6, 7, 8, 9, 10, 11, 12, 13, 17.
- Row 2: 1, 1, 1, 2, 2, 5, 6, 4, 1, 1.
- Take each value in Row 2 and add the previous value to it. 1, 2, 3, 5, 7, 12, 18, 22, 23, 24.
5 students got 10, one student got 17, and 4 students scored 12, etc.
By adding the previous values, you end up with what is known as the cumulative frequency. This is useful for analysing certain series of data.
For example, 12 students scored less than 10 or 50% on the test.
In some cases, you can use cumulative frequency to calculate results as a percentage, which is particularly useful when using the global population.
The median salary in the UK is £31,460, which means that half of the population earns less than this amount.
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Determining the Median
If the number of data points is odd, the median is middle value or (N+1)/2.
In our case, we have 24 values.
The medium is the number in the middle or the average exactly between the two “middle” values: N/2 and N+1/2.
The average in our case is between the 12th and 13th values: 10.5.
Calculating the Median in Continuous Variables
In a lot of cases, we’ll have a set of continuous variables.
These are data points that can always be measured to an endless degree of accuracy.
Take temperatures, for example. This is a continuous variable as your value between 30°C and 31°C could be 30.1°C, 30.5°C, 30.99999°C or 30.00001°C, etc.
The first thing you need to do is create a cumulative frequency curve to establish the median and quartiles.
Let’s say you want to see how many people earn between £500 and £2,100 a month.
Imagine the following data series:
- 40 people earn between £500 and £800 a month.
- 31 people are on between £800 and £1,100.
- 25 earn between £1,100 and £1,200.
- 52 are on between £1,200 and £1,500.
- 37 people earn between £1,500 and £1,800.
- 18 get £1,800 and £2,000 a month.
- 27 earn between £2,000 and £2,100.
N is 230. We can plot the cumulative frequency against the values. You can put the cumulative frequency on the y-axis and the upper values of the ranges on the x-axis.
In this case, you can read the data without needing mathematical formulae. Just draw a line outwards from halfway point of the y-axis (either 115 or 50% if you’ve converted it to a percentage).
Similarly, the quartiles can be found at 25% and 75% or 57.5 and 172.5 respectively. In our series, the 25% and 75% values are £970 and £1,700 respectively. We can deduce that 25% of the sample earn between £1,700 and £2,100.
Here are some things we can learn from our data: 25% of the sample earns £970 or less a month and 75% of the sample earns less than £1,700.
As we said, we can see that 25% of the sample earns between £1,700 and £2,100 a month.
If you have nothing better to do, you could always break your data down into centiles.
Calculating the Median in Geometry
In geometry, the median is also used quite often, especially in triangles.
Some students may struggle with the concept of the median of a triangle.
Essentially, the median is a line that joins a vertex to the midpoint of the opposite side. In a triangle labelled ABC, the median leaves the A vertex and joins up on the line between B and C.
As the line ends in the middle of the other side, the distance between the vertices B and C and the line we’ve drawn will be equal in length. This also means that if you do this for each vertex, you’ll end up with pairs of equal triangles.
Apollonius’ theorem states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side".
The six triangles you create by taking the median of a triangle are also congruent pairs of triangles. The point where they intersect in the middle is also the centre of gravity for the triangle.
The median can be used to see if a triangle is isosceles, too. If two medians are of the same length, the triangle is isosceles.
In a right triangle, the median from the right angle can be used to find the middle of the hypotenuse, the famous side from Pythagoras’ theorem. If the length of the median is half the length of the following side, you’ve got yourself a right triangle.
Don’t Confuse the Mean with the Median
Don’t make the mistake of confusing the mean with the median. While both can be used to calculate averages for a series of data, the mean is more greatly affected by extreme values. The mean is calculated by adding up all the values in the series and dividing it by the number of values in the series.
With the median, the larger and smaller numbers at the edges of your sample will have a smaller influence on the value. As we showed you earlier, the median can also be used for some useful statistical analysis.
If we go back to the example of earnings, the mean won’t show you the whole picture. The mean might hide statistics from very high and very low earners. A small number of very high earners such as millionaires and billionaires can easily skew the data, for example, which is why most of the data for the average earnings use the median. This also allows you to see the quartiles in the data and better understand parts of the data.
As you can see, descriptive statistics in mathematics can be useful for analysing data in everyday life.
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