Measures of Center: Mean and Median

18 min read

Quick Reference

Try This First

A small tutoring center records how many minutes each of seven sessions lasted in one afternoon.

\[ 38, \quad 42, \quad 45, \quad 40, \quad 41, \quad 39, \quad 43 \]

Before reading any definition, work these on paper.

1. Add the seven numbers and divide by 7. Write down the result.
2. Now put the seven numbers in order from smallest to largest. Circle the number that sits exactly in the middle of the ordered list.
3. Predict, before checking: will the two numbers you found be close together, or far apart? Then keep reading.

You have just computed both measures of center by hand. The mean is the balance point of the values; the median is the middle of the ordered values. Those two ideas become precise below, along with the one situation, an outlier, where choosing between them matters.

An Intuitive Introduction

A measure of center answers a single question about a quantitative variable: if the whole distribution had to be reported as one typical number, what would that number be? There are two standard answers, and they describe “typical” in two different ways.

The mean is the arithmetic average. Think of the data values as weights placed along a ruler. The mean is the point where the ruler balances. Because every value pushes on the balance point, moving any one value moves the mean. A single very large value pulls the balance point toward itself.

The median is the physical middle of the ordered list. Half of the observations sit at or below it, and half sit at or above it. The median only cares about which value is in the middle position, not about how large the values on the ends are. Pushing the largest value out to infinity does not move the median at all, because the middle position has not changed.

Both numbers report a center. They agree when the distribution is roughly symmetric. They disagree when the distribution is skewed or contains an outlier, and that disagreement is itself information: the gap between the mean and the median is a signal about the shape of the data.

Prerequisite Hub

Builds On

Unlocks

Cross-Course Prerequisites

None. The summation notation and the arithmetic of ordering numbers are in the review nodes above.

The Official Definitions

Mean ($\bar{x}$). To find the mean of a set of observations, add their values and divide by the number of observations. If the $n$ observations are $x_1, x_2, \ldots, x_n$, their mean is \[ > \bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}, > \] which in summation notation is \[ > \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i. > \]

(Moore, McCabe, Craig, IPS 9th edition, Section 1.3, Describing Distributions with Numbers; corroborated by OpenStax Introductory Statistics 2e, mean as a measure of central tendency commonly called the average, https://openstax.org/books/introductory-statistics-2e/pages/2-key-terms.)

Median ($M$). The median $M$ is the midpoint of a distribution, the number such that half the observations are smaller and the other half are larger. To find it, arrange the observations in order of size from smallest to largest.

• If the number of observations $n$ is odd, the median is the center observation in the ordered list.
• If the number of observations $n$ is even, the median is the mean of the two center observations.

(Moore, McCabe, Craig, IPS 9th edition, Section 1.3, Describing Distributions with Numbers; corroborated by OpenStax Introductory Statistics 2e, median separates ordered data into halves, https://openstax.org/books/introductory-statistics-2e/pages/2-key-terms.)

Section 1.3 of IPS states no theorem here. The content is two definitions and the practice of computing each one correctly, plus the comparison between them. Three constraints inside the definitions are easy to miss and worth holding onto.

• The mean uses every value, so it responds to every value, including an outlier.
• The median requires the data to be in order first. Computing a median on an unsorted list is the single most common arithmetic error here.
• When $n$ is even there is no single middle value, so the median is the average of the two middle values, and that average need not be one of the observed data points.

The Locating Rule for the Median

A clean way to find the position of the median in an ordered list of $n$ values is the locating index $\frac{n+1}{2}$, counting from the smallest value.

For $n = 7$, $\frac{n+1}{2} = 4$, so the median is the 4th value. For $n = 8$, $\frac{n+1}{2} = 4.5$, so the median is the average of the 4th and 5th values. The locating index gives the position; it is never the median value itself.

Reading the Two Measures Side by Side

Both numbers report a center. Read the data list, compute the mean and the median, and explain when they agree and when they split.

Worked Examples

Example 1: Mean and Median of an Odd-Length List

A campus bike-share station records how many bikes were available at seven hourly checks during one morning.

\[ 12, \quad 9, \quad 15, \quad 11, \quad 14, \quad 10, \quad 13 \]

Predict first. Before any arithmetic, the values run from 9 to 15 with nothing extreme, so the mean and the median should land near the middle of that range, somewhere around 12.

Step 1: Compute the mean. Add all seven values and divide by 7.

\[ \bar{x} = \frac{12 + 9 + 15 + 11 + 14 + 10 + 13}{7} = \frac{84}{7} = 12 \text{ bikes}. \]

Step 2: Order the data and locate the median. In order:

\[ 9, \quad 10, \quad 11, \quad 12, \quad 13, \quad 14, \quad 15. \]

Here $n = 7$, so the locating index is $\frac{7+1}{2} = 4$. The median is the 4th value: $M = 12$ bikes.

Read it back. The mean is 12 and the median is 12. They match exactly, which fits the prediction: this list is symmetric around 12, so its balance point and its middle position are the same number.

Example 2: Median of an Even-Length List

A coffee stand records the number of customers in each of eight five-minute intervals.

\[ 7, \quad 4, \quad 9, \quad 6, \quad 8, \quad 5, \quad 11, \quad 6 \]

Predict first. There are eight values, an even count, so there is no single middle value. The median will be the average of the two middle values, and it may not equal any value in the list.

Step 1: Order the data.

\[ 4, \quad 5, \quad 6, \quad 6, \quad 7, \quad 8, \quad 9, \quad 11. \]

Step 2: Locate the two middle values. Here $n = 8$, so the locating index is $\frac{8+1}{2} = 4.5$. The median is the average of the 4th and 5th values. The 4th value is 6 and the 5th value is 7.

\[ M = \frac{6 + 7}{2} = 6.5 \text{ customers}. \]

Step 3: Compute the mean for comparison.

\[ \bar{x} = \frac{4 + 5 + 6 + 6 + 7 + 8 + 9 + 11}{8} = \frac{56}{8} = 7 \text{ customers}. \]

Read it back. The median is 6.5 and the mean is 7. The median, 6.5, is not one of the recorded counts, which is expected for an even-length list. The mean is slightly higher than the median because the value 11 pulls the balance point up.

Example 3: One Outlier Splits the Mean and the Median

A real-estate listing reports the sale prices, in thousands of dollars, of five homes that sold on one street.

\[ 210, \quad 225, \quad 240, \quad 230, \quad 1{,}900 \]

The value 1,900 is a mansion among modest homes: a clear outlier.

Predict first. The median ignores the size of the extreme value and reports the middle of the ordered list, so it should stay near the cluster around 225. The mean adds the 1,900 into the total, so it should be pulled far above the cluster. Predict which number is the better description of a typical home on this street.

Step 1: Order the data.

\[ 210, \quad 225, \quad 230, \quad 240, \quad 1{,}900. \]

Step 2: Find the median. Here $n = 5$, so the locating index is $\frac{5+1}{2} = 3$. The median is the 3rd value: $M = 230$ thousand dollars.

Step 3: Find the mean.

\[ \bar{x} = \frac{210 + 225 + 230 + 240 + 1{,}900}{5} = \frac{2{,}805}{5} = 561 \text{ thousand dollars}. \]

Read it back. The median is 230 thousand dollars, right inside the cluster of ordinary homes. The mean is 561 thousand dollars, more than double every ordinary home on the street. No house actually sold for anything near 561 thousand dollars, so the mean is a misleading “typical” value here. The single outlier dragged the balance point far from the bulk of the data, while the median did not move. For a skewed list like this, the median is the honest measure of center.

The mean uses every value, which makes it sensitive to an outlier. The median uses only the middle position, which makes it resistant to an outlier. The gap between them ($561 - 230 = 331$ thousand dollars) is a flag that the distribution is strongly skewed to the right.

Common Misconceptions

Practice Problems

Mastery Checklist

Novice (Level 1-2):

• I can compute the mean of a list by adding the values and dividing by the count.
• I can order a list and find the median of an odd-length list using the middle position.
• I can state, in words, that the mean is a balance point and the median is the middle of the ordered values.

Competent (Level 3-4):

• I can find the median of an even-length list by averaging the two middle values.
• I can use the locating index $\frac{n+1}{2}$ to find the position of the median.
• I can explain why the mean is sensitive to an outlier while the median is resistant to it.

Proficient (Level 5):

• I can read the relationship between the mean and the median as a signal of the shape (symmetric, left-skewed, or right-skewed).
• I can construct a data set to meet a target relationship between the mean and the median.
• I can justify, in terms of how each measure is computed, why low or high values move the mean but not the median.

Connections

Looking back:

• Sigma (Summation) Notation supplies the $\sum$ symbol used to write the mean compactly.
• Basic Algebra Review supplies the ordering and arithmetic behind both measures.
• Describing a Distribution: Shape, Center, Spread establishes center as one of the three features of a distribution.

Looking ahead:

• Measures of Spread: Quartiles and the IQR pairs with the median, splitting each half of the ordered data again.
• Measures of Spread: Variance and Standard Deviation pairs with the mean, measuring spread around the balance point.
• Density Curves carry the mean (balance point) and median (equal-area point) from a data list to a smooth model.

Real-world connections:

• Reported “average” incomes are usually the median, because a small number of very high incomes would inflate the mean and misrepresent a typical household.
• Home-price reports use the median sale price for the same reason: one mansion sale would distort the mean for a neighborhood of modest homes.
• A teacher comparing the mean and the median of an exam can detect a tail of very low scores without looking at every paper, because a mean well below the median points to a left-skewed distribution.

Resources

Last updated: 2026-06-16