Don't be too mean on yourself if you don't know what the "mean" means in mathematics.
Many people are just as baffled as you are when it comes to trying to describe what this concept represents. However, the concept of mean is rather easy to understand. Essentially, the mean refers to the average of something.
Since we have chosen to focus on the mean and on averages, you may have come to realise that the mean in mathematics is a concept that you should know well as it is very important.
Let's explore the concept the mean in more detail.
First, let's see when exactly it becomes vital to focus on the mean in mathematics. As a high school mathematics student, you may recall that in the data handling section of maths, you are expected to calculate the mean, median, and mode, as well as the range of a respective set of data.
Now let's describe and learn how to calculate the mean.
If you are struggling with the data handling section of mathematics or if concepts like mean, median, and mode seem too overwhelming, you may want to reach out to one of our incredible Superprof mathematics tutors. A great maths tutor will be able to break down the section for you so that it becomes easier to grasp.
What is the Meaning of Mean in Maths?
We have come to understand that the term mean refers to an average.
In mathematics, when you have a set of data, you can rather easily calculate the mean. More technically, when we describe mean in that way, the correct term to use is arithmetic mean.
Nevertheless, to calculate the mean or rather the arithmetic mean, you will need to add the sum of all the numbers in a data set. Thereafter you need to divide this total by how many numbers there are in a data set.
Thus, it is fair for us to define the arithmetic mean as the value obtained by summing the values of collection of numbers and then dividing them by the count of the numbers in the collection.
We calculate and learn about the mean because the arithmetic mean value is the yardstick for a set of information. Teachers calculate the arithmetic mean every term when they take their learners test scores, add them all up and divide this sum by the number of learners in the class or grade. In this way, every term teachers calculate the average mark of all the learners in their classes and in their grades.
We also use mean to calculate which countries have the highest average rainfall per year. The mean calculation is used to work out when on average a particular country gets hotter. By knowing the arithmetic mean, we are able to work out certain trends in a respective data set.
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Explain the Mean Differently
The concept of the arithmetic mean may be slightly easier to understand now that we see it as the yardstick for a set of numerical information.
However, we can use the synonym for arithmetic mean, average, to understand the concept better.
When we set out to calculate the average value, we are working on summarising a set of data. This central value in a set of data is necessary to provide the bigger picture of all the data values. It becomes apparent that every single value in a respective data set has an impact on the average. If a certain number in a data set changes, the average will also change as well. In some cases where we round the average off, the change in the average may not be as apparent. However, when calculating the mean, every value becomes extremely important.
While arithmetic mean has become somewhat easier to understand, we also get geometric mean and harmonic mean. The geometric mean has a more complex meaning. It refers to the nth root of products of numbers. Geometric mean is more complex to calculate than the arithmetic mean.
There is also something called harmonic mean. The harmonic mean is used to calculate the average of rates and ratios. This different calculation method is done since the harmonic mean equalises the weight of each data point.
Let us explore how we can calculate each type of mean.
How to Calculate Mean?
To recap, we calculate the arithmetic mean by using the sum of a collection of numbers and dividing this sum by the number of numbers in the calculation.
To calculate the geometric mean, you have to multiply all the numbers in a data set together. Thereafter, you have to find the nth root of the multiplied numbers. The nth root value depends on how much data you have. If you have four data values, you will take the 4th root.
Things start to get slightly more complicated when you need to calculate the harmonic mean. You need to start by calculating the reciprocal of each value. Once this is done, you find the average (or mean) of the reciprocals that were calculated. Lastly, you take the reciprocal of the average that was calculated.
Thankfully, at a high school level, it is just arithmetic mean that you would have to calculate more often. The geometric mean and harmonic mean are left to statisticians and others.
It is always preferable to use the mean calculation when your data is distributed in a symmetrical way with not many extreme outliers. If your dataset includes many outliers, you are better off using the median formula for the calculation. However, we will leave the discussion of median for another day.
Again, if you find the calculation of mean to be too complex to grasp, reach out to one of our awesome mathematics or statistics Superprof tutors.
Exploring the Mean Formula
It doesn't feel like maths unless we explore some formula. We start to analyse a set of observations more closely when we start working with formulas.
The arithmetic mean formula is simply the sum of all the observations divided by the total number of observations.
We can pen down the formula for arithmetic mean as:
∑ (a1 +a2+ a3+ ..../ n)
In this way, we are adding the sum of all our values in a data set and dividing it by n. n stands for the number of values in a data set.
The arithmetic mean formula is relatively easy to use, but it may be impossible to use if we are calculating a very large sets of data.
While it is great to set off to find the mean of almost everything, you must note that there are certain flaws when it comes to applying the mean formula to every situation.
For instance, if we calculated the average salary of people living in Johannesburg, our mean calculation is affected by extreme outliers that may skew our data to the extreme end. We have a mix of people who earn below the minimum wage and also those who earn millions. When using the mean formula, every value counts and thus the average or mean salary amount may be too high depending on which side our data is skewed towards.
You will have to play it by ear when it comes to deciding which are the best cases to whip out the mean formula.
When statisticians aim to work out the population of a certain country, they are more inclined to work out the mean for a sample population as opposed to working out the mean for the whole population. Statisticians justify this calculation on the basis of the law of large numbers in statistics. This law states that as the sample size grows, the sample average tends to be close to the population value.
In summary, don't be too quick in thinking that the mean or average that you have calculated always summarises the full picture exactly. The type of data that you have and what exactly you are wanting to gain from your data set determines whether using the mean formula will provide the most accurate picture or not. It is, however, interesting to note that so much can be said about the mean in mathematics. The next time that you are expected to calculate the mean of a set of data in your math's class, do give consideration as to how much there is to consider when you have to calculate this mean or average value.
If you still feel unsettled by the idea of applying the mean formula, we suggest that you reach out to one of our excellent mathematics tutors today!
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