Summary statistics

Mean

The mean is the same thing as the average when referring to a list of numbers. If one term is easier to remember than the other, then just replace it with that word each time you see it, so that it is easier to remember. The average of a list of numbers indicates the number you get after adding up all the numbers and diving by the number of numbers used:

Consider the number list $2,3,5,6,8$.

The average of this number list would be the number you get after adding up all the numbers ($2+3+5+6+8=24$) and dividing it by the total amount of numbers or values used (there are five numbers in the list, so $\frac{24}{5}$ ($4.8$) would be the mean, or the average, of this number list.

Median

The median represents the middle number of an ordered number list. You can remember this by the root of the word (Median, like Medium) or by the idea of the “median” in a road. The median is the strip of concrete that separates one side of the road from the other, which is, to state the obvious, the middle of the road.

Consider the number list $5,9,13,15,14$.

In this first example, it is quite easy to point to the middle number ($13$), but we have to explore some less common examples as well:

What about the number list $5,1,3,9,12$?

In this example, we cannot simply point to the number in the middle because the list is not in order. We must FIRST rearrange the list so that it’s in numerical order:

$1,3,5,9,12$

Now, we can point to the center number, which is $5$.

Consider the number list $1,3,5,7,9,11$.

In this number list, we cannot point to a center number because there are an even amount of values. There are two numbers in the middle of the list that compete to be the median ($5$ and $7$). So, we find the average, or the middle, between these two numbers ($6$).

Mode

The mode is often considered the easiest statistics category to find because there is no complicated formula involved in finding it. The mode is simply the most frequent number in a number list.

Consider this number list: $1,2,2,4,8$

The most frequent number is $2$, so $2$ is the mode.

Now check out this new number list: $1,2,2,4,8,4$

You will see that there are two $2$s and two $4$s. This means that both $2$ and $4$ are the mode of this number list.

So, this one is easy: just look for the number that shows up the most!

Range

The range of a number list is very similar to the normal definition of the word “range.” It is how far the number list goes. If our smallest number was $2$ and our largest number was $4$, then we know that the range from $2$ to $4$ would be the difference between them: $2$. You can think of this as a difference, as a distance, or anything that helps you remember that it is NOT simply the largest number—it is the space between the smallest and largest numbers.

Consider this number list: $1,2,5,9,4$

We know that to find the range we need to find the space between the largest number ($9$) and the smallest number ($1$). The easiest way to do this is $9-1=8$. However, if you are caught in the moment you could always count up from $1$ to $9$ and count a space of $8$ between the two numbers. In any case, however you may solve it, $8$ is the range of this number list.

Standard deviation

Standard deviation is a way to measure how spread out a number list is. While the mean tells us where the numbers are centered, the standard deviation tells us how far the numbers typically are from that center. You can think of it as a measure of how “clustered” or how “spread out” the data is.

A small standard deviation means the numbers are close together and close to the mean. A large standard deviation means the numbers are more spread out and farther from the mean.

Consider the number list $4,5,6$.

The mean of this list is $5$, and all the numbers are very close to that mean. Because the numbers stay tightly grouped, this list has a small standard deviation.

Now consider the number list $1,5,9$.

The mean is still $5$, but the numbers are much farther away from the mean. Since the values are more spread out, this list has a larger standard deviation.

On the ACT, you are often asked to compare standard deviations rather than calculate them. If two number lists have the same mean, the list with values that are farther apart will have the larger standard deviation.

Multiple topic questions

These types of questions involve the use of more than one of these terms we have talked about, and typically include the use of algebra in the solution. Most often, a question will want you to use the formula for the mean of a number list to find the answer in an example like the following:

Five scores were taken. Three of those five scores are $12$, $14$, and $19$. The median of the five scores is $16$ and the mean is $17$. What is the value of the fifth test score?

To solve this, we must first figure out what to do with the numbers we are given. There are five numbers and we have been given three of them. First, let’s determine what it means that the median is $16$: the middle number must be $16$, meaning that the final, fifth number must be $16$ or greater. Now we have to use the formula for the mean to find the last number. We know this because we are given the mean without knowing all the numbers. So, the best way to do this is to rewrite the formula with the numbers we are given:

$$\text{Mean}=\frac{\text{The sum of all\hspace{0.33em}the scores}}{\text{The total number of scores}}$$

$$17=\frac{12+14+16+19+x}{5}$$

Now we can solve for $x$:

$$\begin{array}{rl}17\ast 5& =12+14+16+19+x\\ 85& =12+14+16+19+x\\ 85-12-14-16-19& =x\\ x& =24\end{array}$$