TL;DR:
The mean is the balance point, the median is the middle value, and the mode is the most common value.
One extreme value drags the mean a long way but barely moves the median.
When data is skewed or has outliers, report the median, and say which one you used.
A street has nine houses. Eight sell for between $300,000 and $380,000. One is a mansion at $2,500,000. An estate agent says the "average" price on the street is over $580,000. Is that fair? It depends on which average you mean, and this is a good case for telling the three apart.
Three ways to find a typical value
Mean. Add all values and divide by how many there are. It is the balance point of the data.
Median. Sort the values and take the middle one. With an even count, take the average of the two middle ones.
Mode. The value that appears most often. A set can have no mode, one mode, or several.
Worked example: house prices on a street with one mansion
Prices in thousands of dollars, sorted: 300, 320, 330, 340, 350, 360, 370, 380, 2500.
Mean: the sum is
300 + 320 + 330 + 340 + 350 + 360 + 370 + 380 + 2500 = 5250. Divide by 9:5250 / 9 ≈ 583.Median: with 9 values, the middle is the 5th, which is 350.
Mode: every value appears once, so there is no mode.
The mean says $583,000, but only the mansion costs more than that. Eight of the nine houses cost less than the mean. The median, $350,000, is much closer to what a typical house costs.
Remove the outlier
Take the mansion out and the eight remaining prices have sum 5250 − 2500 = 2750. The mean drops to 2750/8 ≈ 344, and the median becomes (340 + 350)/2 = 345. The mean fell by 583 − 344 = 239 thousand, while the median moved by only 350 − 345 = 5 thousand. One extreme value changed the mean enormously and the median hardly at all.
Take the values 2, 4 and 9. The mean is 15/3 = 5. The distances from the mean are −3, −1 and +4, which add to 0. That is what balance means: the shortfalls below the mean exactly match the excesses above it. A far-off value like 2500 needs a large pile of small values on the other side to balance, which is why it drags the mean so far.
Skew: the mean follows the tail
A data set with a long tail to the right is called right-skewed, and its mean is pulled above its median. Prices and incomes often look like this. A long tail to the left pulls the mean below the median. If the two are close, the data is roughly symmetric. So comparing the mean and the median is a quick test for skew.
Try one yourself: test scores
Scores are 70, 75, 80, 85, 90 and 40. The sum is 70 + 75 + 80 + 85 + 90 + 40 = 440, so the mean is 440/6 ≈ 73.3. Sorted, the scores are 40, 70, 75, 80, 85, 90, so the median is (75 + 80)/2 = 77.5. The mean is below the median because of the one low score, a left-skewed pattern. Neither summary alone tells the full story, so give both and mention the 40.
Be careful when combining groups. If 2 students average 10 and 8 students average 20, the overall mean is not 15. It is (2 × 10 + 8 × 20)/10 = 180/10 = 18, because the larger group counts for more.
Where the mode is the right answer
The mode shines for categories. A shoe shop wants to know which size to stock most. Sizes sold: 7, 8, 8, 8, 9, 10. The mode is 8. The mean is (7 + 8 + 8 + 8 + 9 + 10)/6 = 50/6 ≈ 8.33, a size that does not exist. The median is (8 + 8)/2 = 8. Here the mode and median agree, and the mean is the odd one out because it gives a number nobody can buy.
Reporting honestly
When you give one number, say which one and why. For the street, a fair sentence is: "The median price is $350,000; one sale at $2,500,000 pulls the mean up to about $583,000." This tells the reader both the typical house and the reason the two figures differ. If the mean and median are close, either is fine. If they are far apart, the gap is itself a finding worth stating, because it tells the reader the data has a tail or an outlier.
How to check your answer
Count. Is your median the middle of a sorted list?
Range. The mean and median must lie between the smallest and largest values.
Plausibility. If the mean is far above most values, look for an outlier.
Say which. Name the measure you used.
Common mistakes
Finding the median without sorting first.
Averaging the median of two groups to get the overall median.
Treating the mean as always the best typical value.
Reporting a mode when every value is different.
Forgetting that the mean can be a value that nobody in the data has.
Where you will use it next
The mean and median give the centre, but not the spread. For that, see standard deviation by hand. Ideas about reading data with context are also used in conditional probability with a screening test.
Frequently asked questions
Which one is the average?
All three are averages in the loose sense. In maths class, "average" usually means the mean. In the news, always check which is meant.
Can the mean be below every value except one?
Yes. In the street example, eight of nine values are below the mean.
Get started
