How To Find An Average

In mathematics, what people usually call the "average" is properly known as the "mean" or the "mean number." There are actually two other types of averages – the "mode" and "median" – that you'll learn about when you study statistics in high school and beyond. But for most mathematical applications, the term "average" tells you to seek out the mean, which can be calculated with basic addition and division. Additionally, it is useful to know that a sum is the total value of all the terms in a list of numbers added together.

\(\text{Average Value} = \frac{\text{\ \ \ \ Sum of the Terms \ \ \ \ }}{\text{Number of Terms}}\)

What does it mean to calculate the average or mean? Technically, you're dividing the sum of the values you're working with by the count (or quantity) of numbers in that set. But in real-world terms, it's more like distributing the value of the entire set evenly among each of its numbers, and then stepping back to see what value the numbers all ended up at across that number of values.

This type of average is useful for making sense of large data sets or estimating where an entire group stands. For example, you might be asked to calculate the average percentage grade in your class, the average GPA among the other high school students, the average temperature for a certain city or location, or the average price of donuts at the supermarket. Nearly anywhere quantifiable data (data that can be counted) can be found, calculating the average is very important.

Does the idea of how to find averages make sense? The formula is a little clunky to write out in words, but working through a few examples will bring the concept home.

Example 1:

Find the average grade in your math class. There are 10 students, and their cumulative percentage final grades are: 77, 62, 89, 95, 88, 74, 82, 93, 79 and 82.

Start by adding up all of the students' scores:

\(77 + 62 + 89 + 95 + 88 + 74 + 82 + 93 + 79 + 82 = 821\)

Next, divide that total by the number of scores you added. (You could count them, or you could just take note that the original problem tells you there are 10.)

\(\frac{821}{10} = 82.1\)

The result, 82.1, is the average score in your math class.

What is the average of 2, 4, 6, 9, 21, 13, 5 and 12?

You aren't being told what real-world context these numbers might exist in, but that's okay. You can still perform the mathematical operations to find their average. Start by adding them all the given numbers together to find the sum:

\(2 + 4 + 6 + 9 + 21 + 13 + 5 + 12 = 72\)

Next, count up how many numbers you added together. There are eight, so your next step is to divide the total (72) by how many quantity of numbers given (8):

\(\frac{72}{8} = 9\)

So the average of that data set is 9.

Of the students in your class, seven take the bus to and from school. (The others are driven by their parents.) All told, those seven students spend a total of 93 minutes walking to and from the bus each day. What is the average walk time for the students in your class?

Normally your first step would be to add all the students' walk times together, but that's already been done for you; the problem tells you that the total of their walk times is 93 minutes.

The problem also tells you how many pieces of data you're dealing with (seven – one for each student). So if you read the problem carefully, all you have left to do to find the average is divide the sum or total of the data (93 minutes) by the number of data points (7):

\(\frac{93 \text{ minutes}}{7} = 13.2857 \text{ minutes}\)

Most people don't care about whether you've walked 13.2857 minutes or 13.2858 minutes, so in a case like this you'll almost always round your answer to make it more useful. This rounding process is very important when reporting statistics in science and math, the number of digits included is often known as the number of significant figures in scientific fields.

If rounding is allowed, your teacher will tell you what decimal place to round to. In this case, let's round to the tenths place, which is one spot to the right of the decimal. Because the number in the next place (the hundredths place) is greater than 5, you'll round the number in the tenths place ‌up‌ when you truncate the decimal.

So, your answer, rounded to the tenths place, is 13.3 minutes.

The Technical Formula

The generalized formula for a classical average known in mathematics as an arithmetic mean is:

\(A = {\frac{1}{n}} \sum_{i=1}^{n} {a_i} \ = \ \frac{a_1 + a_2 + \cdots + a_{n-1} + a_{n}}{n}\)

To understand this formula, we can look at the sum sign (Σ), the set ‌a‌, and the value ‌n‌. In this context ‌a‌ is a set of numbers and ‌n‌ is the size of the set (or the number of numbers in the set).

Statistics is all about understanding complicated sets of data with fewer and more representative values, like averages. Means and medians are both measures of central tendency, meaning they provide a single value that can give a general picture about a data set.

Data sets often need additional values to best represent their spread. Two data sets could have the same mean, but they can be spread out over vastly different values. Statistics uses things like standard deviation, sample size, and selecting for outliers to account for this variance.

There is an incredible amount of data everywhere, and understanding how to find an average is a crucial tool to understanding our world.