Summary statistics

Mean

The mean is the same as the average for a list of numbers. It tells you what you get when you add all the values and divide by how many values there are.

Consider the number list $2,3,5,6,8$.

Add the numbers:

• $2+3+5+6+8=24$

Count how many numbers are in the list:

• There are $5$ numbers.

Divide the sum by the number of values:

• $\frac{24}{5}=4.8$

So, the mean (average) of this list is $4.8$.

Median

The median is the middle number in an ordered number list.

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

In this example, it’s easy to identify the middle number as $13$, but you also need to handle cases where the list isn’t already in order.

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

Here, you can’t pick the middle value until you put the list in numerical order. First rearrange the list:

$1,3,5,9,12$

Now the middle number is $5$, so the median is $5$.

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

This list has an even number of values, so there isn’t one single middle number. The two middle numbers are $5$ and $7$. In this case, the median is the mean of those two values:

• $\frac{5+7}{2}=6$

So, the median is $6$.

Mode

The mode is the most frequent number in a number list.

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

The number that appears most often is $2$, so the mode is $2$.

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

Here, $2$ appears twice and $4$ appears twice. That means the list has two modes:

• $2$ and $4$

So, to find the mode, look for the value(s) that show up the most.

Range

The range tells you how far the number list goes. It is the difference between the largest and smallest values.

It’s not the largest number by itself - it’s the space between the smallest and largest numbers.

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

Identify the smallest and largest values:

• Smallest: $1$
• Largest: $9$

Subtract:

• $9-1=8$

So, the range of this list is $8$.

Standard deviation

Standard deviation measures how spread out a number list is.

• The mean tells you where the data is centered.
• The standard deviation tells you how far the values typically are from that center.

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 close to $5$. Because the values are tightly grouped, this list has a small standard deviation.

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

The mean is still $5$, but the values are much farther from $5$. 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 questions use more than one of the terms above and often require algebra. A common setup is that you’re given the mean but not all the values, so you use the mean formula to solve for the missing number.

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?

Start by using the median information. If the median is $16$, then the middle score (the third score when ordered) must be $16$. That means one of the five scores is $16$.

Now use the mean formula. Since you are given the mean but one score is missing, set up an equation.

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

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

Now solve for $x$:

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

Key points

• Mean (average). Sum divided by amount. Typically the most used in word problems out of all the terms
• Median. The middle number of an ordered number list. Make sure the list is in numerical order before picking the middle number
• Mode. Easiest one to find: most frequent number
• Range. The difference between the highest and lowest numbers in a number list
• Standard deviation. A measure of how far spread apart a list of numbers is from the center (mean or average)
• Multiple topic questions. Use the formula for the average in an algebra problem.