Applied Descriptive Statistical Analysis
Selecting Central Measures
Beyond the Average Average
You already know how to calculate the mean, median, and mode. They each give you a sense of a dataset's "center." But picking the right one is more of a strategic choice than a math problem. The shape of your data's distribution is the most important factor in this decision. Averages can be misleading if you don't account for the data's overall pattern.
Think about a dataset of employee salaries at a small company. Most employees might earn between $45,000 and $65,000. But the CEO's salary is $400,000. If you calculate the mean salary, that one high value will pull the average up significantly, making it seem like the typical employee earns more than they actually do. This is called skewness—when a few extreme values, or outliers, pull the data in one direction. In this case, the data is "right-skewed."
The more skewed the distribution, the greater the difference between the median and mean, and the greater emphasis should be placed on using the median as opposed to the mean.
Because the mean incorporates every single value in its calculation, it's very sensitive to —data points that are unusually high or low compared to the rest of the set. The median, on the other hand, is simply the middle value. It isn't affected by how extreme the outliers are, only by their position in the ordered list. This makes the median a more "robust" measure for skewed data. It gives a better picture of the true central point when a few outliers might otherwise distort the story.
Specialized Averages
Sometimes, not all data points are created equal. Imagine you're calculating your final grade. Your homework average is 95%, your midterm score is 75%, and your final exam score is 85%. A simple mean would be (95+75+85)/3 = 85. But what if the final exam is worth 50% of the grade, the midterm 30%, and homework only 20%? You need a weighted mean.
Using the weighted mean, your grade would be (0.20 * 95) + (0.30 * 75) + (0.50 * 85) = 19 + 22.5 + 42.5 = 84. The weighted mean provides a more accurate picture when some values contribute more to the overall average than others.
Another special case is when you need to average values that are multiplied together, like growth rates. If an investment grows by 10% one year and 50% the next, you can't just average the percentages. You need the to find the constant growth rate that would yield the same result over the period.
Averages for Categories
What about data that isn't numerical? For categorical data, your choice of central tendency is limited by the type of data you have.
| Data Type | Description | Best Measure | Why? |
|---|---|---|---|
| Nominal | Categories with no order (e.g., eye color, country of birth) | Mode | You can't calculate a mean or find a median of non-numeric, unordered categories. The mode simply tells you which is most common. |
| Ordinal | Categories with a clear order (e.g., survey ratings from 'disagree' to 'agree') | Median | The data can be ordered, so you can find the middle value. A mean is still inappropriate because the intervals between categories aren't necessarily equal. |
Choosing a measure of central tendency is an act of interpretation. It requires you to understand the nature of your data and what you're trying to communicate. A skewed distribution doesn't mean your data is flawed; it just means the simple mean might not be the most honest way to tell the story.
A real estate website reports the average house price in a neighborhood. The neighborhood has many similarly priced homes but also includes one exceptionally expensive mansion. Which measure of central tendency would be most misleading if you wanted to understand the price of a typical home?
When is it most appropriate to use a weighted mean?