Mean, Median, Mode, and Range

When working with data, it is helpful to use single numbers to summarize a whole set of information. The four most common ways to describe a data set are the mean, median, mode, and range.

Mean (The Average)

The mean is the average of the numbers. To find it, add up all the numbers in the data set, then divide by how many numbers there are.

Example: Find the mean of $4, 7, 9, 3, 12$.

1. Find the sum: $4 + 7 + 9 + 3 + 12 = 35$
2. Divide by the count ($5$ numbers): $35 \div 5 = 7$

The mean is $7$.

Median (The Middle)

The median is the middle value when the data is ordered from least to greatest.

Example 1 (Odd amount of numbers): Find the median of $3, 5, 7, 8, 10$. Since there are $5$ numbers, the middle one is the 3rd number. The median is $7$.

Example 2 (Even amount of numbers): Find the median of $3, 5, 7, 8, 10, 12$. Since there are $6$ numbers, there are two middle numbers: $7$ and $8$. To find the median, calculate the mean of these two numbers: $\frac{7 + 8}{2} = 7.5$ The median is $7.5$.

Mode (The Most Frequent)

The mode is the number that appears most frequently in a data set.

• A data set can have one mode, more than one mode, or no mode at all (if every number appears the same amount of times).

Example: Find the mode of $2, 4, 4, 6, 8, 4, 9$. The number $4$ appears three times, which is more than any other number. The mode is $4$.

Range (The Spread)

The range tells you how spread out the data is. It is the difference between the largest value and the smallest value.

Example: Find the range of $3, 5, 7, 8, 10, 12$.

1. Largest value: $12$
2. Smallest value: $3$
3. Subtract: $12 - 3 = 9$

The range is $9$.

Working Backward with Mean

Sometimes you know the mean, but are missing a value. You can use the total sum to find it.

Problem: A data set of $5$ values sums to $35$. A 6th value is added, making the new mean $9$. What is the 6th value?

1. The sum of the first $5$ values is $35$.
2. If the new mean of $6$ values is $9$, the new total sum must be: $9 \times 6 = 54$
3. Subtract the old sum from the new sum to find the 6th value: $54 - 35 = 19$

The 6th value is $19$.