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What is the Median and How Do I Find It?

In these set of lessons, we're learning about the measures of central tendency. We'll learn about mode, mean, range, standard deviation (both the overall concept and the method of calculation), statistical significance, and median.

The median of a data set is the value that separates the upper half of the data from the lower half. You can think of the median as the middle value in a data set.

I always think of a highway median - the cement thing that separates one side of the road from the other. In math, a median is the same thing. It travels down the middle of a set of values, separating one half from the other.

So, how do you find the median of a dataset?

The first thing that you need to do is to put the list of numbers in the dataset in order from smallest to largest.

Datasets with an Odd Number of Values:

Now that we have put the values in order, we can find the median! Here's how you find the median when there are an odd number of values:

Cross off a value from the right side of the list, then a value from the left side of the list. Keep doing this until there is one value left. That is the median!

Our values in order are:

$$7, 8, 8, 9, 10, 10, 11, 11, 12, 13, 14$$

Now we alternate crossing off values on each side of the list until we only have one value left, which is the median.

$$\cancel{7}, \cancel{8}, \cancel{8}, \cancel{9}, \cancel{10}, \boxed{10}, \cancel{11}, \cancel{11}, \cancel{12}, \cancel{13}, \cancel{14}$$

So, $10$ is the median!

Let's try another one:

To find the median, we first put the values in order.

$$1.69, 3.45, 7.50, 11.71, 12.43, 20.20, 64.59$$

Next, we alternate crossing off values from both ends of the list until there is one value standing - the median.

$$\cancel{1.69}, \cancel{3.45}, \cancel{7.50}, \boxed{11.71}, \cancel{12.43}, \cancel{20.20}, \cancel{64.59}$$

Another Method to Find the Median

Sometimes, you'll be working with datasets that are much larger than the ones we've used so far.

Here's a trick to find out where in your ordered dataset the median is located when you have an odd number of values.

Step One: Take the number of values and add $1$.

Step Two: Divide that sum by $2$.

Step Three: The result is the location of your median in the ordered dataset.

Let's try that with our pharmacy dataset from earlier. This isn't a huge dataset, but it will help demonstrate this method.

Here is our ordered dataset: $$1.69, 3.45, 7.50, 11.71, 12.43, 20.20, 64.59$$

Let's the follow the $3$ steps listed above:

Step One: Take the number of values and add $1$.

There are $7$ values in our dataset. $7+1=8.$

Step Two: Divide that sum by $2$.

$8/2 = 4.$

Step Three: The result is the location of your median in the ordered dataset.

So the median is the $4$th value in our ordered data set. Indeed, it is $11.71!$

Data sets with an even number of values:

Not a whole lot changes when you've got an even number of values in your dataset.

The first thing you need to do, as ALWAYS, is put your data points in order from smallest to largest.

When you have an even number of data points, there's no single middle value. Instead, there are two values that separate the dataset in half.

To calculate the median, find the average of the two middle values.

(Video) Find the Median

by MySecretMathTutor

This video by My Secret Math Tutor defines the median.

$\large{\text{Summary}}$

The Median is the middle of a data set. If you cut a set right down the middle, the median is where you will make your cut. If there and odd number of points, then it is easy and it is the middle number. If there are an even of points, you take the average of the middle two.

Median

The median of a data set is the value that separates the lower half of the data from the upper half of the data. In other words, it is the "middle value."

To find the median of a set with an odd number of data points, sort the data, then find the value in the exact middle. To find the median of a set with an even number of data points, sort the data, then find the mean of the middle two values.

Click here for more information on when to use the median!

To find the median of the data set $31, 65, 241, 1, 99,$ we first sort the values. $$1, 31, 65, 99, 241$$ Then we find the value in the exact middle. The common way to do this is to alternate crossing off a data point from each end of the data set until there is one value left. This is the median! $$\cancel{1}, \cancel{31}, \boxed{65}, \cancel{99}, \cancel{241}$$

To find the median of the data set $31, 65, 241, 1$ we first sort the values. $$1, 31, 65, 241$$ Then we find the mean of the middle two values. This is the median! $$\cancel{1}, \underline{31, 65,} \cancel{241}$$ $$\frac{31+65}{2} = \frac{96}{2} = \boxed{48}$$

(Video) Finding the Median

by Anusha Rahman

This video by Anusha Rahman goes over how to find the median of an odd set of numbers and an even set of numbers.

Summary

The median is the middle value in a set of numbers. It is the middle number of numbers.

Example 1: Odd set of values You're given the following set of numbers: $$5, 4, 3, 5, 6, 7, 8, 9, 16, 11$$ Step 1: Put the numbers in numerical order, from least to greatest. $$3, 4, 5, 6, 7, 8, 9, 11, 16$$ Step 2: Cross off one number from each side. You're left with $7$ as the median.

Example 2: Even set of values You're given the following set of numbers: $$6, 7, 10, 14, 13, 4$$ Step 1: Put the numbers in numerical order, from least to greatest. $$4, 6, 7, 10, 13, 14$$ Step 2: Cross off one number from each side. With even numbers, you're left with two values, $7$ and $10$. So what do you do? Step 3: Find the average of the two numbers! The average of $7$ and $10$ is $8.5$. Therefore, the median is $8.5$.