How is the median calculated?
The median is the middle value in a data set and is situated at the central point of distribution. That means that there are as many values above the median as there are below it.
The median is a useful measure of average values that discounts outliers that would skew a reading found by calculating the mean of a data set of values.
Along with the mode, the mean, and the range, the median is one of the measures of average value that is taught at school level. You can also learn about standard deviation, but that is usually saved for students studying A-level Mathematics.
We are going to take you through how to calculate and find the median, what it is used for, and how it differs from other average measures.
How is the median calculated?
To calculate the median you put all of the numbers in a data set in ascending order and number them from 1 up to the highest point in the set. Then, you simply find the middle number of your set and that is the median value.
You can calculate the median by using n as the number of values in your data set. If n is an even number then the median is the value at n / 2. If n is an odd number and you want to end on a whole number, you can either use the middle pair of values as a joint median or you can calculate the median formula as (n + 1) / 2.
What are the mean, the mode, and the range?
The mean, median, mode, and range are all measures of finding the average number from a data set of values. However, when people mention the ‘average’ the measure they are usually referring to is the mean.
To find the mean you add up all the values in a data set to find the total number in that set and then divide by the number of values in the data set. For example, if your data set consisted of the numbers 9, 12, and 15, you would simply do:
- 9 + 12 + 15 = 36
- 36 / 3 = 12
- the mean equals 12.
The mode refers to the most frequently occurring number in a data set. If your data set consisted of the numbers 9, 12, 14, 15, 15, 18, 18, 18, 22, 25, then the mode would be 18 the number occurs three times in the list, which is more than any other value.
The range is simply the difference between the highest value in a data set and the lowest. It looks at the extreme data points to produce a whole new data point. The range is a good measure of variability in a particular data set. So, again, if your data set consisted of the numbers 9, 12, 14, 15, 15, 18, 18, 18, 22, 25, to find the range you would do:
- 25 – 9 = 16.
- The range equals 16.
The median is simply the middle number in a data set. So, using the same numbers from the previous example, if your data set consisted of the numbers 9, 12, 14, 15, 15, 18, 18, 18, 22, 25, then the median would be 15 as there are 10 numbers and 15 is in the fifth position.
Each of the different measures has value in certain scenarios and drawbacks in others.
You can find calculators for each formula online.
What is the median value used for?
As with many mathematical formulas you learn in school, a common question that is asked by pupils is, ‘when will I actually use this?’. Well, the truth is many of us can go our whole lives without needing to know how to calculate the median value of a data set, but that doesn’t mean it isn’t incredibly useful for people in certain lines of work or in particular situations.
Finding the median value is particularly useful for measuring data sets with skewed distributions or with outliers. That means data sets where most of the numbers are similar, but there are a few that are significantly higher or lower than the rest and would skew the average if the mean value were found instead.
For example, income levels are often explained by finding the median value. In 2020, the mean UK salary was £38,600. However, this takes into account the relatively few people who earn six or even seven-figure salaries. Lower earners may also skew the data slightly, but a salary of £15,000 per year is going to skew the data less than a salary of £15 million per year.
In the same year, the median salary in the UK was £31,461. This is a far better measure of what the average person earns in the UK as it takes the salaries of huge earners out of the final figure.
What are the drawbacks of the median?
There are also drawbacks of the median that make it less helpful in certain situations.
For example, the median does not convey any information regarding the minimum and maximum values of a data set. This means analysts are less certain of the range of the values than they would be in a reading that used the range or the mean of the data set.
The median can also lead to false impressions. For example, let’s say the median age a child says their first word is 24 months. This also means that half of all children do not say their first word until they are older than 24 months. But if a parent hears their child is outside of the ‘norm’ they may begin to worry. In this instance, the mode may be a better measure.
For very accurate measurements, the median can also be less useful as it is not possible to find middle numbers in a data set with an even number of values. Instead, you must either present two middle values as the median, though this is not accurate.
For example, if you have 10 data points in your data set, you would have 1 to 5 at one end of the scale and 6 to 10 at the other. So would the median be 5 or 6?
Although this may not seem important, when analysing scientific data, such small margins can make a very big difference.
Online mathematical resources
There are some great resources online that can help you find the median, mean, and mode, and assist you with many other mathematical problems.
Here we have included a list of some of our favourites:
- Mean, median, mode calculator analyses your data set and finds the different types of average for you.
- Percentage Calculator allows you to work out a range of different questions relating to percentages.
- Good Calculators has all manner of calculators free to use.
- GCF-LCM is a great calculator for finding greatest common factors and lowest common multiples.
- Desmos is an advanced online scientific calculator that can perform complex equations and offer guidance to users.
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