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Common Core: 6.SP.5

# Median

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Author: Sophia Tutorial
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This lesson will explain finding the median of a data set.

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Tutorial

This tutorial will discuss the concept of the median. You're going to learn about:

1. Median
2. Extreme Values
3. Median Class

## 1. Median

Suppose you have the players from the Chicago Bulls basketball team, and you wonder what would be a typical height for someone on this team. Here is the list of the players and their heights:

You might notice that many of the players are 81 inches tall. Can you therefore call that height typical of the Chicago Bulls?

In order to answer that question, you'll need to calculate a median.

Median

The value that is in the "middle" of a data set when the set is arranged from least to greatest.

A median is simply a measure of center for the data set that actually finds the middle value in a sorted list. It's the middle number when the data set is arranged from least to greatest or greatest to least.

In the above list, the players were sorted alphabetically. To find the median, you need to have that list ordered from least to greatest. The first step is therefore to reorder those numbers, which will look like this:

In order to find the middle number, start by crossing off the lowest and highest numbers:

and continue working our way in until we get just one number left:

That last number is the median: 79.

Notice is that half the values in the list are at or below 79, and half the values in the list are at or above 79.

You can also use technology to figure out the median of a data set. Place the list of heights, not ordered, in a spreadsheet. Use the autosum "equals median." Then, select the full range of numbers for which you want to find the median hit Enter.

Using this method, your spreadsheet will give you a median of 79, just like the last time.

Suppose you have a class of 10 students and you have a 10-point quiz. Here are the scores from each of the students:

What is the median?

The first step you would take is to reorder these scores, which should result in a list that looks like this:

As you cross out the highest and lowest numbers, working toward the center, you will notice that there are two middle values:

In a case like this, you have to average those two numbers. So to take the mean of 8 and 9: (8+9) / 2 = 8.5.

So the median is 8.5.

## 2. Extreme Values

How is the median affected by extreme values? Suppose that you have another 10-point quiz for a different class. Here are the scores, in order:

Obviously, one of these values is completely out of range, perhaps because of a typo. Despite this typo, however, the median of this data set is 7 because that's the middle number. If you correct the typo, changing that 90 to a 9, the median will still be 7. So the median is not all that affected by outliers or extreme values.

## 3. Median Class

Another way to figure out a median would be if you have data summarized in a frequency table, which can help you find the median class.

Median Class

The bin that contains the median value. This is the most precise measurement we can obtain when we are looking at data that have already been categorized.

When is it best to use a frequency table?

Here is information about the number of days that the temperature was in a particular range in Chanhassen, Minnesota in 2009:

For example, you can see that there were 8 days that had a temperature of between 0 and 9 degrees Fahrenheit.

Using this table, there are a couple of different ways to find not exactly what the median temperature is, but which bin it's in.

• You can see that the 183rd day of the year falls in the 50-59 category

That means that there are 182 days that were as cold or colder than that particular day, and there are 182 days that were at least as warm as that particular day, which means that these are semi-ordered by temperature. The 182 below and 182 above form the two halves. This the median is somewhere in the 50's.

You can't be 100% sure exactly where in the 50's it is, but we can be sure that it's in the 50's. Notice the number 183, when you look at the cumulative frequency, falls between the 156 and the 202. By the time you've gotten there, you haven’t accounted for half the days in terms of ordered temperatures. But you have accounted for more than half the days by the time you finish the 50's, which means that the median is somewhere in the 50's.

• If you look at the relative cumulative frequency column, you can see the same thing:

By the time you have finished the 40's, you've accounted for less than 43% of the data. By the time you finish the 50's, however, you will have accounted for over 55% of the data.

Where's the 50th percentile? You don't know what the number is, but again, you know it's somewhere in the 50's: 50% of days fall in or below the 50's.

Therefore, call the 50's the median class because you know the median is somewhere in that bin. Because of the way this data is presented, you don't know exactly what the median is, but we can tell that it's in the 50's and not anywhere else.

The median identifies the middle number in a set of ordered data. If there's an even number of data values, we're going to take the mean of those two middle numbers. Even for datasets with extreme values, the median will still be the middle number. And if the data are on a frequency table, you can find the median class, but you can't find the median directly.

Thank you and good luck!

Source: THIS WORK IS ADAPTED FROM SOPHIA AUTHOR JONATHAN OSTERS

Terms to Know
Median

The value that is in the "middle" of a data set when the set is arranged from least to greatest.

Median Class

The bin that contains the median value. This is the most precise measurement we can obtain when we are looking at data that have already been categorized.