What is the definition of the average?

On this website you can find a wide variety of averages. But what is the average? What do we mean exactly when we talk about the average? The average is a calculated central value in a dataset and the definition is:  The average is a single value (for example the mean, mode, or median) that summarizes/represents the general significance of a set of unequal values.

What is the average in math?

The most common value when talking about the average is the mean. You can find the mean by taking the sum of all numbers in a particular data set, divided by the amount of numbers there are in the data set. In statistics and mathematics this mean is called the arithmetic mean. It is called the arithmetic mean and not just the mean because in maths and statistics there are many other types of means, like the quadratic mean, the geometric mean, the harmonic mean, the truncated mean and many more. The arithmetic mean is the mean mostly used in analysing results of experiments or surveys. The median and mode can also be used to calculate the average of a particular dataset.  

How to calculate the average?

You can calculate the average, as in arithmetic mean, by using the following formula:

  = ∑ / n 


 x̅ = arithmetic mean/average

X = sum of all numbers in the dataset / sample

n = sample size

So if you have a dataset with the following numbers: 1,3,4,7,8,9,9,10,12, the mean is:

(1+3+4+7+8+9+9+10+12)/9 = (63/9) = 7. So the mean of this sample is 7.

There are tools that can help you calculate the average. Excel is a commonly used tool for this purpose. If your dataset values are in the A1:A10 range in your Excel sheet, you can calculate the average with the following average function: =AVERAGE(A1:A10).
SQL is a database programming language also used a lot in analysing datasets. The average function in SQL looks like this: SELECT AVG(column_name) FROM table_name. Both functions explained here will provide you with the arithmetic mean.
Other popular software you can use in analysing datasets and finding the average are: Python, R and Matlab.

When to use the mean?

The mean is a very good measure of the average if the dataset doesn’t contain extreme outliers and the numbers in the dataset are evenly spread. If this is not the case, then the mean might not give a good representation of the dataset analysed. The mean is also not useful in analysing ordinal or nominal scales. When the mean is not the right measure of the average you can think of using the median or modus.

What is the median, modus and range?

When talking about the average and mean then also the median, modus and range are often discussed. The median is the middle number in a sorted list of numbers. In several cases people prefer using the median instead of the mean since the median is less sensitive to outliers. The modus is the number with the highest frequency in a data set. The modus can be very useful when analysing ordinal and even nominal scales. The range is the difference between the lowest and highest values in the dataset.
Of the dataset in the example above the median is 8, the modus is 9 and the range is 11 (12-1).

The average mostly discussed in articles on this website is the arithmetic mean.

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