When we look at data, we often want to understand it better using different methods. Three common ways to describe data are the mean, median, and mode.

However, there are times when some very high or low numbers in the data, called outliers, can mess with our understanding. Outliers can really change the mean, but they don’t impact the median and mode as much.

Let’s break it down!

The Mean

The mean is what we usually think of as the average. To find it, you add all the numbers together and then divide by how many numbers there are.

For example, let’s look at this data set:

$$\{2, 3, 4, 5, 100\}$$

First, we add up the numbers:

$$2 + 3 + 4 + 5 + 100 = 114$$

Next, we divide by how many numbers there are, which is 5:

$$\text{Mean} = \frac{114}{5} = 22.8$$

Here, the large number 100 makes the mean very high.

If we only looked at the smaller numbers:

$$\{2, 3, 4, 5\}$$

The mean would be:

$$\text{Mean} = \frac{2 + 3 + 4 + 5}{4} = \frac{14}{4} = 3.5$$

See how different the means are? This shows how an outlier can really change the average. A mean of 22.8 doesn't really represent the group because 2, 3, 4, and 5 are much closer in value.

The Median

Now, let’s talk about the median. The median is the middle number when we arrange the data from smallest to largest.

For our example:

$$\{2, 3, 4, 5, 100\}$$

When we put it in order, the middle number is 4. So, the median is:

• Ordered list: 2, 3, 4, 5, 100
• Median = 4

Even if we remove that outlier and just look at:

$$\{2, 3, 4, 5\}$$

The median, which is the average of the two middle numbers, would be:

$$\text{Median} = \frac{3 + 4}{2} = 3.5$$

The median does a better job of showing the middle of the data because it isn't swayed by that big outlier.

The Mode

Lastly, we have the mode. The mode is the number that appears the most often in the data set.

For instance, look at this data:

$$\{1, 2, 2, 3, 100\}$$

The mode here is 2, since it shows up twice, while 100 is just there once.

If we look at another set:

$$\{1, 2, 2, 3\}$$

The mode is still 2. This shows us that the mode is real stable and doesn’t change just because of an outlier. It focuses on what is most common.

Summary

Here’s how outliers affect the different ways to look at data:

1. Mean:

   • Big outliers can really change it.
   • Gives us an average, but can be misleading in certain situations.

2. Median:

   • Not much affected by outliers.
   • It really shows the center of the data better.

3. Mode:

   • Stays the same unless the outlier changes how many times a number shows up.
   • Helps us see what happens the most often.

In real life, knowing how to look at these different measures helps us understand data better. If we spot outliers, it’s a good idea to report all three measures to get a clearer picture of what’s going on. Whether in science, economics, or social studies, understanding data is essential. These concepts help us make better decisions based on what the numbers really tell us!