Mean, median and mode are the three most common ways of representing a set of numbers. There are many other ways of representing a set of numbers in statistics but the only other one that we are interested in is the range.
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What is a set of numbers or data set?
A set of numbers (also called a data set) is a list of numbers in a data set that represents some common characteristics or relate to a particular topic or subject.
If we collect the height of every pupil in a class and write it down then we will call that as a data set of pupil’s height in a class.
If we conduct a test and collect the scores of everyone who took the test then that would be a data set of scores.
The postcodes of all houses in London with more than four family members would form another data set of families bigger than four.
We collect the data sets so we can analyse it and study it. For example, if we collect the data set of heights of pupils in a class then we can study it to find out what should be the size of the chairs and benches to sit on.
For fun, let’s imagine creating a dataset of how many cookies each pupil in a class can eat. We know that some children can eat more than others but that is fine, we are simply interested in taking of note.
This is how our fun data set may look like,
| Pupil Name | Cookies |
| Martha | 5 |
| Jack | 0 |
| Daniel | 5 |
| Reshma | 3 |
| Pupil Name | Cookies |
| Marcin | 1 |
| Silvia | 2 |
| Rob | 5 |
| Aval | 3 |
What is observation in data set?
Some times you may hear people talking about observations in a data set so let’s understand what it means. In our fun data set example, the number of cookies that each pupil ate was the observation.
In other words, you can say that we observed that Martha ate five cookies and Reshma ate three cookies. The numbers that you collect for the data sets are called observations. These numbers are then later analysed using the mean, median, mode and range methods.
What is the mean?
Mean is the average of all the numbers (or observations) in a data set. It is a typical value in a given data set.
Let’s assume that we are preparing for the summer fair at our school and the headteacher suggested that we should have cookies for the whole school. Which sounds like a great idea but how do we know how many cookies to buy from the superstore. This is a big challenge.
Our Maths teacher saved the day. She had the fun data set of her class and she calculated the mean so that we do not have to remember the whole data set. Instead, we use the magic mean to remember just one number and use that to calculate how many cookies to buy from the store.
To calculate the mean we will use this formula,
Mean = Sum of all numbers ÷ total number of observations
Sum of all numbers = 5 + 0 + 5 + 3 + 1 + 2 + 5 + 3
Total number of observations = 8
Mean = 24 / 8
Mean = 3
What this means is that on an average a pupil eats 3 cookies. In to other words, we can also say that the mean of 8 observations in a class is 3 cookies.
Now, we can use this knowledge to calculate the total number of cookies we need for the summer fair.
So let’s assume that there are 100 pupils in the schools then we will need to buy,
3 x 100 = 300 cookies
This is how mean can be a useful tool.
What is the median?
Median is the middle value in a data set. So that sounds like easy to do but here is the trickery. In order to calculate the median, you must first arrange the numbers from smallest to largest or sort them in the ascending order.
Sorting (or sort) means arranging the numbers.
Ascending means increasing or from smallest to largest. Think of ascending as floating upwards and the opposite descending means floating downwards.
So let’s again take the example of our fun data set and arrange the eight observations in the ascending order.
0, 1, 2, 3, 3, 5, 5, 5
This is an interesting case because there isn’t a middle number. For a number to be exactly in the middle the number of observations on the left and right should match. If we pick the first 3 then there are three observations on the left and four on the right. If we pick the second 3 then there are four observations on the left and three on the right. That means none of those are in the middle.
In cases like these we will take both numbers and then find the average of them,
Median = (3 + 3) ÷ 2
Median = 3
But what if the data set looked like this,
0, 1, 2, 3, 4, 4, 5, 5, 5
In this example, 4 is in the middle as there are four observations on the left and four on the right so the median for this data set is 4.
Did a notice a pattern?
If the number of observations are odd then you simply find the middle number but if the number of observations are even then you will need to add the two observations in the middle and find the average.
What is the mode?
The mode is the number that occurs most of the times in the data set. To remember this definition rhyme mode with most.
In French “a la mode” means in fashion. So the number that occurs most of the times becomes “a la mode“.
In our fun data set 5 occurs three times whereas the other numbers appear less than three times. So we can say that 5 is the mode or in other words, 5 is the number that occurs most of the time.
5, 0, 5, 3, 1, 2, 5, 3
But let’s make things a little interesting. What if the data set looked like this? What is the mode?
1, 2, 6, 7, 4, 5, 3, 9, 8
There is no mode because no values are repeating. The mode is about most occurrences and if the number only occurs once then there is no mode.
OK, what if the data set looked like this.
4, 1, 1, 1, 1, 3, 3, 3, 3, 7
Here we have two modes 1 and 3 because both 1 and 3 are repeating the same number of times.
What is the range?
The range of a list of numbers is the difference between the largest number and the smallest number in the data set.
In our fun data set, the largest number is 5 and the smallest number is 0. So the range of our fun data set is 5 – 0 = 5.
OK, let’s wrap it up
Let’s quickly recap the definitions of mean, median, mode and range.
Mean is the average of all numbers is a data set.
Median is the middle value in a data set.
Mode is the value with most occurrences.
Range is the difference between the largest and the smallest value.
Maths Courses
Our learning app and courses go deeper into the statistics topics and presents you with several problems on mean, median, mode and range.
More Reading
To learn more about mean, median, mode and range you can read the following articles.
