Learning goals
- Calculate the arithmetic mean, median, and mode for a small data set
- Explain how outliers pull the mean but leave the median more stable
- Choose which average to report when summarizing scores, prices, or survey answers
Three ways to describe a “typical” value
Mean, median, and mode each answer “what is typical?” in a different way. The mean is the balance point of the numbers. The median is the middle value after sorting. The mode is the value that appears most often.
In everyday language people say “average” and usually mean the arithmetic mean. In reports and exams you should name which measure you used, because they can disagree — especially when one extreme value sits far from the rest.
Use the mean for fairly symmetric numeric data. Prefer the median when a few outliers would distort the story (home prices, salaries). Use the mode for categories or when the most common outcome matters more than the numeric center.
Arithmetic mean
mean = (x₁ + x₂ + … + xₙ) ÷ n
Add every value, then divide by how many values there are. Every observation gets equal weight. One very large or very small number can move the mean a lot.
Median (odd and even counts)
median = middle value after sorting (or average of two middle values)
Sort the data from smallest to largest. If n is odd, the median is the value in position (n + 1) ÷ 2. If n is even, average the two central values.
Example 1 — Mean of a short list
Given
Test scores: 72, 88, 90, 85, 75. What is the mean?
Steps
- Add the scores: 72 + 88 + 90 + 85 + 75 = 410.
- Count the scores: n = 5.
- Mean = 410 ÷ 5 = 82.
Answer
The mean score is 82.
Tip: Quick check: all scores sit near 80–90, so a mean of 82 is believable.
Example 2 — Median with an outlier
Given
Home sale prices (thousands): 220, 240, 250, 260, 900. Compare mean and median.
Steps
- Sorted list is already ordered. n = 5 (odd), so median = middle value = 250.
- Mean = (220 + 240 + 250 + 260 + 900) ÷ 5 = 1,870 ÷ 5 = 374.
- The $900.00k sale pulls the mean up to 374 while the median stays at 250.
Answer
Median = 250; mean = 374. The median better describes a “typical” sale here.
Tip: When one luxury listing skews the market story, report the median (or both).
Example 3 — Finding the mode
Given
Shoe sizes sold today: 8, 9, 9, 10, 9, 11, 8. What is the mode?
Steps
- Tally frequencies: 8 appears twice, 9 appears three times, 10 once, 11 once.
- The highest frequency is 3 for size 9.
Answer
The mode is 9.
Tip: A data set can have more than one mode (bimodal) or no unique mode if all frequencies tie.
Calling every summary “the average”
Common mistake
Report “average price $374.00k” from the list above without saying it is the mean — readers assume a typical home.
Better approach
Say “mean $374.00k, median $250.00k” (or lead with the median) so the summary matches the story the data actually tell.
Check your understanding
1.Find the mean of 4, 6, 10, 12.
Answer: (4 + 6 + 10 + 12) ÷ 4 = 8.
2.Find the median of 3, 9, 1, 7, 5.
Answer: Sorted: 1, 3, 5, 7, 9 → median = 5.
3.Find the median of 2, 4, 6, 8.
Answer: Two middle values 4 and 6 → median = 5.
4.Data: 1, 2, 2, 3, 7. Mode and mean?
Answer: Mode = 2; mean = (1+2+2+3+7) ÷ 5 = 3.
Key takeaways
- Mean = sum of values ÷ count; it uses every data point equally.
- Median is the middle value after sorting and resists extreme outliers.
- Mode is the most frequent value — best for categories or “most common” questions.
- When mean and median disagree sharply, investigate outliers before choosing which to publish.