Determining the median

In math classes at school, students learn about statistics, geometry, and how to calculate the average and median. The median is an essential indicator when analysing probabilities, geometry, and statistics despite being frequently misunderstood.

One of the best ways for many students to succeed in math because it can be difficult is through private academic tutoring. In this article, we'll look at how to find the median.

What does a numerical median indicate?

To work out the median, you must first understand what it is: The average value

The median is an average that can be useful when the range of values includes extreme values.

Statistics frequently use the median, which divides a set of values into equal parts and is occasionally more useful than the mean average. You must arrange all of your data to determine the median, or the central value of a statistical series. Finding the value that perfectly balances your series will be easy from there.

Take a swimming instructor as an illustration.

The nine swimmers were told to go around the pool twice, and they gave the coach the times listed below in seconds:

30.6, 29.1, \s32.9, \s35.1, \s30.0, \s36.4, \s31.7, \s35.5, \s33.9.

Following ranking, the values are: 29.1, 30.0, 30.6, 31.7, 32.9, 33.9, 35.1, 35.5, and 36.4.

We'd use a median time of 32.9.

This allows for the division of swimmers into groups on either side of this value.

What happens if a fresh student is enrolled in the course?

If you will, they covered two pool lengths in 28.7 seconds. When that happens, we'll have an equal number of swimmers (10 data points). This set's " middle " would be between 31.7 and 32.9.

When there are an even number of values, you must find the mean between the two numbers (31.7+32.9)/2 = 32.3 in order to calculate your median.


If the sample size is unusual, you should pick the middle value.

You can take the mean of the two values that are on either side of the actual central value if the sample size is even.

Pretty elementary, no?

Where does the median come from?

Sorting your data will help you locate them more quickly. The originator is Papazacharias.

Take a look at our article on algorithms.

If the students received the following scores out of a possible 20, for example: 5, 12, 10, 13, 9, 12, 11, 10, 6, 17, 11, 12, 8, 7, 10, 11, 10, 12, 11, 9, 10, 8, 11.

Sort the data in the table by statistics.

To find the median, it is necessary to arrange the values in one row in ascending order and count the number of students who have each value on the second row.

After that, you'll have the following configuration:

5 in row one, 12 in row two, 17 in row nine, and 6 in row ten.

To each of the values in Row 2, add the value that came before. 1, 2, 3, 5, 7, 12, 18, 22, 23, 24.

Five students received 10, four students received 12, etc., while one student received a score of 17.

The cumulative frequency is created by summing the previous values. For analysing specific data series, this is beneficial.

As an illustration, 12 students received test results of less than 10, or 50%.

In some cases, cumulative frequency can be used to calculate results as a percentage when using the entire world's population.

The median income in the UK is £31,460, which means that half of the population earns less than this amount.

Determining the median

The median is the middlemost value when there are odd numbers of data points,or (N+1)/2. 

There are 24 values in the case we are in.

The exact average between N/2 and N+1/2, or the middle value, is referred to as the medium.

The average in our case is 10.5, which is between the 12th and 13th value.

How to Calculate the Median for Continuous Variables

Often, we'll have a set of continuous variables.

Where does the median come from?

 you must take the mean of the two middle values if there are an even number of values.

These are variables that can be precisely measured indefinitely.

Think about the temperature right now. This is a continuous variable because the temperature between 30°C and 31°C could be anywhere, such as 30.1°C, 30.5°C, 30.99°C, or etc.

You must first create a cumulative frequency curve in order to calculate the median and quartiles.

Let's say you're interested in learning how many people earn between £500 and £2,100 per month.

Think about the data series below:

40 people earn between £500 and £800 each month.

31 people will receive between £800 and $1,100.

25 earn between $1,100 and $1,200 per year.

52 people subsist on $1,200 to $1,500 per month.

37 people earn between $1,500 and $1,800 annually.

18 are paid between 1,800 and 2,000 per month.

27 make between £2,000 and £2,100 a year.

N is 230. Values and the cumulative frequency can be plotted. The cumulative frequency can be plotted on the y-axis, and the upper values of the ranges can be plotted on the x-axis.

To read the data in this case, mathematical formulas are not required. Simply draw a line starting at the midpoint of the y-axis, which is 115 (or 50% if you've converted it to a percentage).

The quartiles are positioned at 57.5 and 172.5, or 25 and 75 percent, respectively. In our series, the 25% and 75% values are £970 and £1,700, respectively. We can assume that 25% of the sample's employees earn between £1,700 and £2,100 annually.

The following are some conclusions we can make based on our data: 25% of the sample's participants earn £970 or less a month, and 75% earn less than £1,700.

As mentioned earlier, we can see that 25% of the sample earns between £1,700 and £2,100 per month.

You can always divide your data into centiles. too

Calculation of the Geometric Median

In geometry, the median is frequently used, especially in triangles.

How is the geometric median calculated?

The median is also useful in geometry. (Source: Myriams-Fotos)

Learn how to calculate a quotient.

Some students might have trouble understanding the concept of a triangle's median.

A vertex and the middle of the other side are connected by the median, which is essentially a line. The median joins the line that connects B and C after leaving the triangle with the letters ABC's A vertex.

Because the line we've drawn ends in the middle of the opposing side, the distance between the vertices B and C and it will be equal in length. If you carry out this procedure for each vertex, you will end up with pairs of equal triangles.

By taking the median of a triangle, which yields six triangles, congruent pairs of triangles can also be created. Where their lines intersect in the middle is the triangle's centre of gravity.

The median can also be used to determine whether a triangle is isosceles. If the two medians' lengths match, the triangle is isosceles.

By using the median from the right angle, it is possible to locate the middle of the hypotenuse, the well-known side from Pythagoras' theorem, in a right triangle. When the median is half the length of the side it follows, a right triangle is created.

To be distinguished from the Median

Avoid making the common error of confusing the mean and the median. Both can be used to calculate averages for a collection of data, but the mean is more strongly affected by extreme values. To find the mean, divide the sum of all the series' values by the total number of series' values.

When using the median, the larger and smaller numbers at the extremes of your sample will have less of an effect on the value. As we previously showed, the median can also be used for some useful statistical analysis.

To return to the example of earnings, the mean won't give you the whole picture. Data from extremely high and extremely low earners may be hidden by the median. Because a small number of extremely high earners, such as millionaires and billionaires, can easily skew the data, the median is used in the majority of data for average earnings. This can also help you understand some of the data's components and the data's quartiles.