What you will learn
- Calculate the mean, median, and mode of a set of numbers
- Explain what each one measures about the center of the data
- Understand why the median can be more reliable than the mean when data is skewed
- Apply the idea of a center to financial returns
The most basic question you can ask about a set of numbers is: where is the center? What is a typical value? Statistics gives three answers, the mean, the median, and the mode. They usually agree, but when they disagree the difference tells you something important about the shape of the data. We start with simple everyday numbers, then apply the idea to returns.
Mean, median, and mode (Khan Academy)
The gentlest possible introduction to the three measures of center. Watch this first, then work the examples below yourself.
The mean: the balancing point
The mean, also called the average, is what most people picture when they think of a typical value. You compute it by adding up all the values and dividing by how many there are. Picture the numbers as weights on a seesaw. The mean is the point where the seesaw balances. Because every value pulls on that balance point, the mean uses all the data, which is its strength. But that is also its weakness, as you will see, because one extreme value can drag it far away from what is typical.
Key terms
- Mean (average)
- The sum of all values divided by how many there are. The balancing point of the data.
- Median
- The middle value when the data is sorted. Half the values are above it, half below.
- Mode
- The value that appears most often.
- Skew
- When data is lopsided, stretched out more on one side than the other.
Calculating the mean return
A stock had yearly returns of 4 percent, negative 2 percent, 6 percent, 0 percent, and 2 percent over five years. What is the mean (average) return?
- Add up the returns. 4 plus negative 2 plus 6 plus 0 plus 2 equals 10.
- Divide by how many there are. There are 5 returns, so 10 divided by 5.
- Read the result. That is 2 percent.
Why it matters: The mean adds everything and divides, so every value counts. It is the natural summary of a typical return, and it becomes the expected return in the next lesson.
Calculate the mean
A stock had monthly returns of 3 percent, negative 1 percent, 2 percent, and 4 percent. What is the mean return, as a percent?
The median: the middle value
The median is simply the middle value once you sort the data from smallest to largest. Half the values fall below it and half above. If there is an even number of values, the median is the average of the two middle ones. The median does not care how far away the extremes are, only how many values sit on each side. That makes it far less sensitive to a single wild value than the mean.
The mode: the most common value
The mode is the value that appears most often. A dataset can have one mode, several modes, or no repeated value at all. The mode is most useful for categorical data, where mean and median do not apply, such as the most common sector in a group of companies. For continuous data like returns, the mode is used less often, but it still tells you where values cluster.
| Measure | What it is | Best when |
|---|---|---|
| Mean | Sum divided by count | Data is roughly symmetric, no wild outliers |
| Median | The middle sorted value | Data is skewed or has outliers |
| Mode | The most common value | Data is categorical, or you want the peak |
Why the median can beat the mean
Here is the key insight. When data is skewed or has an extreme outlier, the mean gets pulled toward the extreme while the median stays put. Consider five people with salaries of 40, 45, 50, 55, and 60 thousand dollars. Both the mean and the median are about 50 thousand. Now replace the top earner with a billionaire making 1 million thousand dollars. The median barely moves, still around 50 thousand, because it only cares about the middle position. But the mean shoots up into the hundreds of thousands, which is not typical of anyone in the group. This is why news reports about incomes and home prices usually use the median. It resists distortion by extremes.
The mean is pulled by every value, including the outliers. The median stands its ground in the middle. When they disagree, the data is skewed.
Measures of central tendency (CrashCourse Statistics)
Explains when each measure is appropriate and how outliers and skew affect the choice. Watch for why the median is often the honest summary.
Which measure is honest here?
A small fund reports the returns of its five holdings this year: 3%, 4%, 5%, 6%, and 90% (one holding got very lucky). The manager advertises the average return. Is the mean a fair summary of a typical holding?
The single 90 percent return is an outlier that pulls the mean far above a typical holding. The median, 5 percent, is not distorted by the extreme, so it is the more honest summary here.Center is only half the story
The mean is the workhorse for financial returns, because it becomes the expected return you will meet in the next lesson, and it feeds directly into portfolio theory. But knowing the center is only half of understanding a distribution. Two stocks can have the exact same mean return while one is calm and steady and the other swings wildly. To capture that difference you need a measure of spread, which is where variance and standard deviation come in, over the next few lessons.
In your own words
Explain in a sentence or two the difference between the mean and the median, and why a report on typical home prices would usually use the median.
Write an answer before comparing it with the model response.
Model answer
The mean adds every value and divides, so it is pulled toward extreme values, while the median is just the middle value once the data is sorted and ignores how far the extremes are. Home prices are skewed by a small number of very expensive homes, which would drag the mean upward and make it look higher than what a typical home costs. The median stays near the middle of the market, so it gives a more honest picture of a typical price.
You can now find the center of a set of numbers. Next we turn the mean into one of the most important ideas in finance: the expected value, the probability-weighted average you should anticipate from an uncertain outcome.