Mean, Median, and Mode

The mean, median, and mode are three ways to describe the “center” of a dataset. They are called measures of central tendency because each one identifies a central or typical value. Knowing which measure to use — and when — is one of the most practical statistics skills you can have.

Mean (Average)

The mean is the sum of all values divided by the number of values. It is what most people think of when they hear “average.”

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

Example 1: Test Scores

A student’s five quiz scores are: 78, 85, 90, 88, 94.

Step 1: Add all the values.

$78 + 85 + 90 + 88 + 94 = 435$

Step 2: Divide by the number of values (5).

$\text{Mean} = \frac{435}{5} = 87$

Answer: The mean score is 87.

Example 2: Daily Customers

A small shop tracks daily customers over six days: 42, 38, 55, 40, 37, 48.

$\text{Mean} = \frac{42 + 38 + 55 + 40 + 37 + 48}{6} = \frac{260}{6} \approx 43.3$

Answer: The shop averages about 43.3 customers per day.

Median (Middle Value)

The median is the middle value when all values are sorted from least to greatest. If there is an even number of values, the median is the mean of the two middle values.

Finding the Median

1. Sort the values from smallest to largest.
2. If the count is odd, the median is the single middle value.
3. If the count is even, the median is the average of the two middle values.

Example 3: Odd Number of Values

Data: 12, 7, 3, 15, 9

Step 1: Sort the values: 3, 7, 9, 12, 15

Step 2: The middle value (3rd of 5) is 9.

Answer: The median is 9.

Example 4: Even Number of Values

Data: 22, 18, 35, 27, 14, 31

Step 1: Sort the values: 14, 18, 22, 27, 31, 35

Step 2: The two middle values (3rd and 4th of 6) are 22 and 27.

Step 3: Average them.

$\text{Median} = \frac{22 + 27}{2} = \frac{49}{2} = 24.5$

Answer: The median is 24.5.

Mode (Most Frequent Value)

The mode is the value that appears most often. A dataset can have one mode, multiple modes, or no mode at all.

• One mode: 4, 7, 7, 9, 12 — the mode is 7
• Two modes (bimodal): 3, 3, 5, 8, 8, 10 — the modes are 3 and 8
• No mode: 2, 4, 6, 8, 10 — every value appears once, so there is no mode

The mode is the only measure of central tendency that works for categorical data (non-numeric data like colors, brands, or responses).

When to Use Each Measure

Key insight: When data has outliers, the median is usually more representative than the mean.

Consider five employees with salaries: $35,000,$38,000, $40,000,$42,000, $250,000.

$\text{Mean} = \frac{35{,}000 + 38{,}000 + 40{,}000 + 42{,}000 + 250{,}000}{5} = \frac{405{,}000}{5} = \$81{,}000$

$\text{Median} = \$40{,}000$

The mean ($81,000) is inflated by the one high salary. The median ($40,000) better represents what a typical employee earns. This is why income data is almost always reported using the median.

Real-World Application: Nursing — Interpreting Blood Pressure Readings

A nurse records a patient’s systolic blood pressure at the same time each day for seven days:

The Thursday reading (138) is noticeably higher — possibly due to stress or a missed medication.

Mean:

$\text{Mean} = \frac{118 + 122 + 119 + 138 + 120 + 121 + 117}{7} = \frac{855}{7} \approx 122.1$

Median:

Sorted: 117, 118, 119, 120, 121, 122, 138

The median (4th of 7) is 120.

Mode: No value repeats, so there is no mode.

The mean (122.1) is pulled upward by the one high reading. The median (120) gives a better picture of the patient’s typical blood pressure. A nurse might report both values and flag the 138 reading as an outlier for the doctor to review.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Mean = sum divided by count. Best for symmetric data without outliers.
• Median = middle value when data is sorted. Best when outliers or skewness are present.
• Mode = most frequently occurring value. The only measure that works for categorical data.
• When in doubt, report both the mean and the median — the gap between them tells you how skewed the data is.
• A large difference between the mean and median signals that outliers are present.

Return to Statistics for more topics in this section.