Describing Data with Numbers—Center

What is “typical”? It is a surprisingly tricky question. Suppose someone asks, “What is a typical salary at this company?” You could answer with the average, but what if the CEO makes 50 times more than everyone else? That one enormous salary would pull the average way up, making it seem like everyone earns more than they actually do. Suddenly, “average” does not feel so typical anymore.

This is exactly why statisticians have developed multiple ways to describe the center of a data set. Each measure captures something different about what is “typical,” and knowing which one to use—and why—is a skill that will serve you well beyond any math class. Whether you are comparing job offers, understanding news reports about income inequality, or figuring out your grade average, these tools help you see through the numbers to the story they are really telling.

Core Concepts

Why We Need a Single Summary Number

Imagine trying to describe your city’s weather to someone who has never been there. You could list every daily temperature for the past year—365 numbers—but that would be overwhelming and not very helpful. Instead, you might say something like, “It is usually around 70 degrees.” That single number gives a quick, useful picture of what to expect.

This is exactly what measures of center do for any data set. They take a collection of values and summarize them with one representative number. But here is the catch: there are different ways to find that representative number, and each method tells a slightly different story. The three most common measures are the mean, median, and mode.

Mean: The Balance Point

The mean (often called the average) is what most people think of first. You calculate it by adding up all the values and dividing by how many values you have.

$$\bar{x} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + \cdots + x_n}{n}$$

The notation $\bar{x}$ (read as “x-bar”) represents the mean of a sample. The symbol $\sum$ means “add up all of,” and $n$ is the number of values.

Think of the mean as a balance point. If you placed all your data values as weights along a number line, the mean is where you would put the fulcrum to balance them perfectly. Every value contributes to the mean, pulling it in its direction—larger values pull it up, smaller values pull it down.

The strength of the mean: It uses every piece of data, so nothing gets ignored.

The weakness of the mean: It uses every piece of data—including extreme values that might distort the picture.

Median: The Middle Value

The median is the middle value when you arrange all your data in order from smallest to largest. It splits your data in half: 50% of values fall below the median, and 50% fall above.

Finding the median depends on whether you have an odd or even number of values:

• Odd number of values: The median is simply the middle value.
• Even number of values: The median is the average of the two middle values.

The median is remarkably stable. No matter how extreme your largest or smallest values are, the median stays right in the middle, unaffected. A billionaire moving into your neighborhood would dramatically change the mean income, but the median would barely budge.

The strength of the median: It is not thrown off by extreme values (we say it is resistant).

The weakness of the median: It ignores how far values are from the center—it only cares about position.

Mode: The Most Common Value

The mode is the value that appears most frequently. Unlike the mean and median, which can only be calculated for numerical data, the mode works for any type of data—including categories like “red,” “blue,” or “favorite pizza topping.”

A data set can have:

• One mode (unimodal): One value clearly appears most often
• Two modes (bimodal): Two values tie for most frequent
• Multiple modes (multimodal): Three or more values tie
• No mode: Every value appears the same number of times

The mode is particularly useful when you want to know what is most common or popular. “What is the most requested shoe size?” is a mode question.

The strength of the mode: It works for categorical data and tells you what is most common.

The weakness of the mode: For numerical data, it might not exist or might not be near the center at all.

Comparing Mean and Median: What Skewness Tells You

When the mean and median are close together, your data is probably fairly symmetric—values are evenly distributed around the center. But when they differ, something interesting is happening.

If the mean is greater than the median: Your data is skewed right (or positively skewed). This means there are some unusually large values pulling the mean up. Picture a long tail stretching to the right. Income data often looks like this—a few very high earners pull the mean above the median.

If the mean is less than the median: Your data is skewed left (or negatively skewed). There are some unusually small values dragging the mean down. Imagine a long tail stretching to the left.

If the mean equals (or nearly equals) the median: Your data is approximately symmetric.

Knowing how skewness affects these measures helps you choose the right one. For skewed data, the median often gives a more honest picture of “typical.”

The Effect of Outliers

An outlier is a value that is unusually far from the rest of the data. Outliers can dramatically affect the mean but leave the median largely unchanged.

Consider test scores: 78, 82, 85, 88, 90. The mean is 84.6, and the median is 85. Now suppose one student scored only 10 points (perhaps they were sick that day): 10, 78, 82, 85, 88, 90. The median is now 83.5 (average of 82 and 85)—only a small change. But the mean drops to 72.2—a dramatic shift that makes it seem like the typical student scored much lower than they actually did.

This is why income statistics, house prices, and similar data are almost always reported as medians rather than means. A few extremely wealthy individuals or luxury properties would otherwise distort the picture.

Resistant vs. Non-Resistant Measures

A resistant (or robust) measure is one that is not heavily affected by extreme values.

• The median is resistant. Change the largest value to a million, and the median stays the same.
• The mean is non-resistant. That same change to a million would wildly alter the mean.
• The mode is resistant. Extreme values do not affect which value is most common.

When you suspect your data might contain errors, outliers, or extreme values, resistant measures give you a more reliable picture.

Choosing the Right Measure of Center

There is no single “best” measure—the right choice depends on your data and what you want to communicate.

Use the mean when:

• Your data is roughly symmetric
• There are no extreme outliers
• You need a measure that accounts for every value
• You will be doing further statistical calculations

Use the median when:

• Your data is skewed
• There are outliers or extreme values
• You want a resistant measure
• You are describing incomes, house prices, or similar data

Use the mode when:

• Your data is categorical (non-numerical)
• You want to know what is most common or popular
• You are looking at discrete data with repeated values

Often, it is wise to report multiple measures. Saying “the median income is $52,000 while the mean is $78,000” tells your audience that the distribution is skewed—some people earn substantially more than most.

Weighted Mean

Sometimes not all data values are equally important. A weighted mean accounts for this by giving some values more influence than others.

$$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}$$

The most familiar example is your GPA. A 4-credit class should count more toward your GPA than a 1-credit class. If you earned an A (4.0) in a 4-credit class and a B (3.0) in a 2-credit class:

$$\text{GPA} = \frac{(4 \times 4.0) + (2 \times 3.0)}{4 + 2} = \frac{16 + 6}{6} = \frac{22}{6} \approx 3.67$$

The 4-credit A has twice as much influence as the 2-credit B.

Weighted means appear everywhere: calculating final course grades (where exams might count more than homework), finding average stock prices (weighted by number of shares), or computing index values in finance.

Notation and Terminology

Term	Meaning	Example
Mean ($\bar{x}$)	Sum of values divided by count	$\bar{x} = \frac{\sum x_i}{n}$
Median	Middle value when data is ordered	50th percentile
Mode	Most frequently occurring value	The “peak” of data
Outlier	Unusually extreme value	150 in {10, 12, 11, 13, 150}
Resistant	Not affected much by outliers	Median is resistant
Non-resistant	Sensitive to outliers	Mean is non-resistant
Weighted mean	Mean accounting for different weights	GPA calculation
Skewed right	Tail extends toward larger values	Mean > Median
Skewed left	Tail extends toward smaller values	Mean < Median
Symmetric	Balanced distribution	Mean $\approx$ Median
$\sum$ (sigma)	“Sum of”	$\sum x_i$ means add all $x$ values
$n$	Number of values in the data set	Sample size

Examples

Key Properties and Rules

Formulas for Measures of Center

Mean: $$\bar{x} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + \cdots + x_n}{n}$$

Weighted Mean: $$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$$

Median:

• For odd $n$: The $\frac{n+1}{2}$th value when data is ordered
• For even $n$: Average of the $\frac{n}{2}$th and $\frac{n}{2}+1$th values

Mode:

• The value(s) with the highest frequency
• May not exist (no repeats) or may be multiple values

Properties of the Mean

1. Uses all data values: Every observation contributes to the mean.
2. Affected by outliers: Extreme values can dramatically shift the mean.
3. Balance point: The sum of deviations from the mean equals zero: $\sum(x_i - \bar{x}) = 0$
4. Additive: Adding a constant $c$ to every value shifts the mean by $c$.
5. Multiplicative: Multiplying every value by $c$ multiplies the mean by $c$.

Properties of the Median

1. Uses only position: Only the middle value(s) matter, not their magnitudes.
2. Resistant to outliers: Extreme values do not affect the median.
3. Divides data in half: Exactly 50% of values lie on each side.
4. Minimizes absolute deviations: The median is the value that minimizes $\sum|x_i - m|$.

Properties of the Mode

1. Frequency-based: Identifies the most common value.
2. Works for categorical data: Unlike mean and median, can be used with non-numerical data.
3. May not be unique: Data can be bimodal, multimodal, or have no mode.
4. Resistant to outliers: Adding extreme values does not change which value is most common.

Comparing Measures: Quick Reference

Situation	Best Measure
Symmetric data, no outliers	Mean
Skewed data	Median
Data with outliers	Median
Categorical data	Mode
“What is most popular?”	Mode
Need for further calculations	Mean
Income or housing data	Median

Relationship Between Mean and Median in Skewed Data

• Right-skewed (positive skew): Mean > Median
• Left-skewed (negative skew): Mean < Median
• Symmetric: Mean $\approx$ Median

Real-World Applications

Test Scores and Academic Performance

Teachers often report both the mean and median of exam scores. The mean tells you the overall class performance, while the median tells you what a typical individual student scored. If the median is 75% but the mean is 70%, you know some students scored very low, pulling the mean down. Your GPA is calculated using weighted means, where courses with more credits count more heavily.

Income and Wealth Statistics

Income data is almost always reported as median rather than mean. Why? Because income is notoriously right-skewed. A small number of very wealthy individuals (CEOs, celebrities, tech entrepreneurs) have incomes hundreds or thousands of times larger than typical workers. Including them in a mean would make “average income” seem much higher than what most people actually earn.

When you hear that “the median household income in the US is about $75,000,” it means half of households earn more and half earn less. The mean household income is substantially higher—pulled up by the wealthy—but less useful for understanding what a typical family earns.

House Prices

Real estate listings emphasize median home prices for the same reason. A few luxury mansions or waterfront properties can dramatically inflate the mean, making an area seem more expensive than it is for typical buyers. The median gives house hunters a more realistic expectation.

Sports Statistics

Sports are filled with measures of center:

• Batting average in baseball is a mean (hits divided by at-bats)
• Points per game in basketball is a mean
• ERA (Earned Run Average) in baseball is a weighted mean, accounting for innings pitched

Commentators sometimes compare a player’s mean performance to their median to identify consistency versus variability. A basketball player who scores 20, 20, 20, 20, 20 (mean = median = 20) is more consistent than one who scores 5, 10, 15, 25, 45 (mean = 20, median = 15).

Grade Calculations

Understanding weighted means helps you strategize your studying. If your final exam is worth 40% of your grade but you have only spent 10% of your study time preparing for it, you might want to reconsider your priorities. Knowing which assignments carry more weight helps you allocate your effort wisely.

Consumer Decision Making

When reading product reviews, the average rating (mean) can be misleading if there are many extreme ratings. A product with mostly 4-star and 5-star reviews but some angry 1-star reviews might have a lower mean than a product with consistently mediocre 3-star reviews. Looking at the mode (most common rating) or the distribution of ratings often gives better insight.

Self-Test Problems

Summary

• Measures of center summarize a data set with a single “typical” value. The three main measures are mean, median, and mode.

• Mean ($\bar{x}$) is calculated by adding all values and dividing by the count. It uses every data point but is sensitive to outliers.

• Median is the middle value when data is ordered. For an even number of values, average the two middle values. It is resistant to outliers and preferred for skewed data.

• Mode is the most frequently occurring value. It works for categorical data and can identify what is most common.

• When mean > median, data is skewed right (positive skew). When mean < median, data is skewed left (negative skew). When they are equal, data is approximately symmetric.

• Outliers are extreme values that can dramatically affect the mean but leave the median largely unchanged.

• Resistant measures (median, mode) are not heavily influenced by outliers. Non-resistant measures (mean) are sensitive to extreme values.

• Weighted mean gives different values different levels of importance, useful for GPA calculations, portfolio values, and grade calculations.

• Choose your measure wisely: use mean for symmetric data, median for skewed data or when outliers exist, and mode for categorical data or finding the most popular value.

• In real-world contexts like income and housing prices, the median typically provides a more honest picture of “typical” than the mean.