In mathematics, the mean is a kind of average, but it's so much more than that and in mathematics and statistics, in particular, it can be incredibly useful.
In this article, we'll be looking at what averages are, everything you need to know about the mean, when you should use it, and when you may want to use another type of average.
What Are Averages?
In both mathematics and statistics, averages are used to represent the most typical value from a set of several values.
We say “try” because the utility of averages is dependent on what kind of value we're trying to achieve and also the kind of data that we're working with.
Averages are used in mathematics to provide summaries to sets of data as it can sometimes be more useful to look at the bigger picture from data values than trying to look at every value independently.
You can also use averages to compare multiple sets of data rather than trying to compare individual values independently. For example, if you wanted to look at which country is hotter, Australia or New Zealand, you could use average temperatures from both countries to compare. You could also take various averages from different months and times of year.
This could help you to make a decision on the best time of the year to visit New Zealand. After all, you mightn't be able to predict exactly which days will have the best weather, but you could look at average temperatures, typical rainfall, and other averages to give yourself a better idea of when would be the best time to go.
You can also use these averages to work out trends. For example, if you had lots of these averages, you could see whether these places are getting warmer.
There are 3 (or 4) main types of averages: the mean, the median, and the mode. There's also the range, but since the range gives a completely different type of average to the others, we won't be looking too closely at it.
Instead, we'll be focusing on just the mean in this article, but we have a whole article series on averages and the different kinds you can use.
How Do You Calculate the Mean?
The mean is one of the three main kinds of averages that we'll be looking at. It's probably the most commonly used average and in most instances, when somebody is talking about the average, they're likely referring to the mean.
To calculate the mean, you need to find the sum of all the values in your data set and then divide the result by the number of values.
For example, if we took the highest temperatures recorded in Sydney this week (at the time of writing), we could work out the average highest temperature for the week: 28, 24, 21, 26, 21, 21, and 22.
- The first thing we'd have to do is find the sum of all these numbers, which is 163.
- From there, we'd need to divide this number by how many values we have. Since it's the values for each day of the week, we know that there are 7 of them.
- 163 divided by 7 is 23.2857.
Since the original data was given as whole numbers, we can round this down to 23 degrees, but you don't have to if you'd like a greater degree of accuracy.
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What Are the Advantages of Using the Mean?
There are advantages and disadvantages to using the mean to calculate an average.
One of the biggest advantages is that every single value in the data will have an effect on the average. If any of these values were to change, the overall average would change. This mightn't be fully reflected if we round the average, but the real average would change.
The mean is also really simple to calculate and while large sets of data won't be something that you can calculate in your head, the formula itself is particularly simple.
Since the mean considers every value, outliers will affect it, but the effect of each outlier is reduced as you increase the number of values. In our previous example, the lowest values were only a couple of degrees lower than the average but the highest was 4 degrees higher. If we took even more days into consideration, the effect of outliers would be greatly reduced.
What Are the Disadvantages of Using the Mean?
While the mean is the most common kind of average used, that doesn't mean that it's without its flaws.
The fact that the mean considers every value in a dataset means that it's affected by extreme outliers. For example, if you took the mean for the average salary of every Australian, the number would be significantly higher because of millionaires and billionaires who earn orders of magnitude more than the typical Australian.
The mean salary in Australia wouldn't likely be representative of a number that most Australians feel like they earn or that a typical person would earn and in instances like this, other averages are needed.
In the case of every average, you don't want the average to be affected by outliers or unusual values in a set of data. In our earlier example, if one day had been 40 degrees, our average would have been significantly different and not particularly reflective of the temperatures experienced throughout the week.
What About the Other Averages?
Since every kind of average has its pros and cons, in certain circumstances, it might be better to use an average other than the mean.
The average you use depends on your data and what information you want to get from said data, but the median is the most common type of average that's used instead of the mean.
The median is calculated by putting all the values in numerical order and then choosing the middle value. For our earlier example of Sydney's highest temperatures, the median is 22.
This is because we took the values 28, 24, 21, 26, 21, 21, and 22 and put them in order: 21, 21, 21, 22, 24, 26, 28.
Since there are 7 values, the fourth value is the middle one. For datasets with an even number of values, you take the two values on either side of the true centre and then take the mean of the two.
Here our median is only 1 degree colder than the mean, but we can imagine that the hottest day (which was 28 degrees) skewed the mean upwards.
In the example of Australian salaries we mentioned earlier, the median would have potentially been a better choice.
The median is often used for this kind of data like salaries, house values, and other datasets that have outliers because of high-net-worth individuals. The median is particularly valuable for any kind of dataset that has extreme outliers and not just examples involving financial data.
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When Should I Use the Mean or Median?
You need to think about what you're using your data for and the information you want to ascertain from taking an average.
When to Use the Mean
You should use the mean when the distribution of your data is symmetrical. This means that most of the values lie towards a middle value and there are more values towards the central values than large numbers of high or low values.
You can use the mean when there aren't many extreme outliers. This is why the financial examples prefer the median as there are several very extreme outliers.
The mean also provides an exact value for the average that can lie between values in the dataset.
When to Use the Median
The median is often the alternative average used as its advantages can deal with the disadvantages that are found when using the mean.
For example, when the distribution of data is skewed with a lot of extreme values, the median becomes particularly useful as it essentially ignores the very highest and lowest values by only focusing on the middle of the dataset.
The median is also less sensitive to extreme values so if you think you have some values in your dataset that will skew the mean, you can choose the median.
Which is More Accurate?
Neither the mean nor the median is more accurate as it completely depends on the data you have and the kind of information you're trying to get from it.
With every average, it's important you know when are the best instances for using a particular average and the pros and cons of each.
If you'd like to learn more about mathematics and averages, don't forget that there are plenty of qualified and experienced mathematics tutors on the Superprof website, where you can find plenty of incredible in-person and online maths tutoring!
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