The limits of mean, median, mode, and range

Study guide: Understanding the limits of basic statistical methods

Now that we are familiar with basic statistical methods like mean, median, mode, and range, we are going to learn about their limits. In other words, while these methods may potentially tell us a lot about something, they may also fall short in being able to explain the complete picture of a situation. There may be other underlying causes, effects, and possible alternative explanations at play that these methods, alone, can’t get to the heart of. Let’s briefly review what these methods are, and then we’ll get into discussing when they are useful and when they might not tell us the whole story.

1. Mean (average)

What It Is:

The mean is what you get when you add up all the numbers in a set and then divide by how many numbers there are.

When It’s Useful:

Example: Imagine you want to find the average score on a math test. If you add all the test scores together and divide by the number of students, you get the mean score. This helps you know the overall performance of the class.

Limitations:

The mean can be affected by really high or really low numbers (called outliers).

Example: Suppose most students scored around 80, but one student scored 20. The mean might drop significantly, giving the impression that the class did worse than it really did. In situations like incomes, a few very high salaries can make the mean much higher than what most people earn.

2. Median (middle value)

What It Is:

The median is the middle number in a list of numbers that have been arranged in order.

When It’s Useful:

Example: If you arrange the ages of children in a classroom from youngest to oldest, the median age tells you the middle age. This is good when you have numbers that might be very high or very low, because the median won’t be as affected by them as the mean is.

Limitations:

The median only shows one value and does not give any information about the other numbers.

Example: If you know the median income of a group of people, you still don’t know if there are lots of people who earn much more or much less than that median income.

3. Mode (most frequent value)

What It Is:

The mode is the number that appears most often in a set of numbers.

When It’s Useful:

Example: If a teacher wants to know which test score was most common, the mode will tell you which score happened the most. This can help show what most students did on the test.

Limitations:

There might be no mode at all if no number repeats. Sometimes, a data set can have more than one mode, and that can be confusing.

Example: In a survey about favorite ice cream flavors, if two flavors are equally popular, then there are two modes. This might not give a clear answer about which flavor is the overall favorite.

4. Range (difference between the highest and lowest)

What It Is:

The range is the difference between the largest and the smallest numbers in a set.

When It’s Useful:

Example: If you look at the temperatures during a week, the range tells you how much the temperature changed from the coldest to the hottest day.

Limitations:

The range only considers two numbers (the highest and lowest) and ignores everything in between.

Example: Two classes might have the same range of test scores, but one class might have most students scoring around the middle, while the other class has scores spread out. The range alone wouldn’t show these differences.

Real-world situations: Where they work and where they fall short

Test Scores in a Class:

• Useful: The mean gives a quick idea of how well the class did on average.
• Falls Short: A few very low or very high scores can distort the mean. The median might be better if the scores are very spread out.

House Prices in a Neighborhood:

• Useful: The mean or median can tell you about the general cost of houses.
• Falls Short: A few extremely expensive houses can make the mean much higher than what most people pay. The median might hide how varied the prices really are.

Favorite Foods Survey:

• Useful: The mode shows which food is most popular among the respondents.
• Falls Short: If people have many different favorite foods and no food is chosen often, the mode might not tell you much about overall preferences.

Sports Statistics:

• Useful: A player’s average score (mean) can show their overall performance.
• Falls Short: The mean might hide important details like a few games where the player scored very low, even though they usually scored high. Looking at the range or the list of scores can give more insight.

Why knowing the limits is important

• Incomplete Picture: Each statistic gives us just one view of the data. They can help us summarize information quickly, but they don’t always show everything.
• Outliers: Extreme values (very high or very low numbers) can change the mean and range, but might not affect the median as much.
• Different Stories: Two sets of numbers can have the same mean or range but tell very different stories about the data.

By understanding the limits of mean, median, mode, and range, you can learn to look at data in more than one way. Sometimes, you might need to use several of these tools together to get a complete picture of what the numbers are really telling you.

Remember, statistics are like different tools in a toolbox. No single tool can do all the work, so it’s important to know which one to use and when to use another one for a better understanding.