When we want to summarize a group of numbers, two important ways to do this are the mean and the median. Each of these measures can tell us something different about the data. However, there are times when the median gives us a clearer picture than the mean. Let’s look at when the median is better to use.

1. Outliers

One big reason to use the median is because it is not affected by outliers. An outlier is a number that is very different from the others in the group. For example, look at these incomes (in thousands of dollars):

• $30, 35, 40, 45, 50, 1000$

If we want to find the mean, we first add up the numbers:

$$30 + 35 + 40 + 45 + 50 + 1000 = 1200$$

Then, we divide by how many numbers there are (6):

$$\text{Mean} = \frac{1200}{6} = 200$$

This mean of 200 makes it seem like everyone is earning a lot, but that’s because of the outlier of $1000.

If we find the median, we look at the middle numbers when we put the list in order. The middle numbers here are 40 and 45:

$$\text{Median} = \frac{40 + 45}{2} = 42.5$$

So, the median is 42.5. This number gives a much better idea of what a typical income is in this group.

2. Skewed Distributions

Sometimes, data is not evenly spread out. This can happen when the mean gets pulled in one direction by extreme values. For example, with these exam scores:

• $50, 52, 54, 56, 58, 70, 95, 98, 100, 100$

Calculating the mean here looks like this:

$$\text{Mean} = \frac{50 + 52 + 54 + 56 + 58 + 70 + 95 + 98 + 100 + 100}{10} = \frac{683}{10} = 68.3$$

Now, to find the median, we look at the 5th and 6th scores:

$$\text{Median} = \frac{58 + 70}{2} = 64$$

In this case, the mean is 68.3, which is higher. The median of 64 gives us a better understanding of what a typical score looks like.

3. Ordinal Data

The median is also great for ordinal data. This means the data can be ranked, but we can't say how much better one rank is than another.

For instance, if people rated their satisfaction from 1 to 5 like this:

• $1, 1, 2, 3, 4, 5, 5, 5, 5, 5$

If we try to find the mean, it wouldn’t give us a good picture since the gaps are not equal.

Instead, the median shows:

$$\text{Median} = 5$$

This means that at least half of the people rated their satisfaction as 5, which helps us understand the overall satisfaction better.

Conclusion

In summary, while the mean gives us a broad view, using the median can be clearer in cases with outliers, uneven distributions, or ordinal data. By using the median, we can better understand what the data really shows. For those of you interested in data science, it’s important to use the right statistics for the right situations. This will help you make better decisions and get clearer insights.