📖Definition
Mean, median, and mode are measures of central tendency - they describe the center of a data set. Mean is the average, median is the middle value when ordered, and mode is the most frequent value.
📐Formula
Mean: add all values and divide by count. Median: arrange in order and find the middle. Mode: find the value that appears most often.
📝Step-by-Step Guide
Calculate the Mean
Add all values together, then divide by the number of values.
Find the Median
Order values from least to greatest. For odd count, median is the middle value. For even count, average the two middle values.
Identify the Mode
Find which value(s) appear most frequently. There can be no mode, one mode, or multiple modes.
Choose the Right Measure
Mean is affected by outliers. Median is better for skewed data. Mode is useful for categorical data.
🧭How to find the mean
The mean is what most people call the average.
Add up all the data points, then divide the sum by the number of numbers in the set.
Take 4, 8, 6, 10 and 7. The sum is 35. There are 5 data points, so the answer is 35 ÷ 5 = 7.
Dividing the sum this way spreads the total evenly across every value. If all five numbers were 7, the total would still be 35.
🧭How to find the median
The median is the middle value. Sort the data points from smallest to largest before you do anything else.
With an odd count, the median is the value in the middle. In 3, 7, 9, 12, 15 the median is 9.
With an even count, take the two middle values and average them. In 4, 6, 9, 11 the middle pair is 6 and 9, so the median is 7.5.
Sorting from smallest to largest is the step students skip most often. Do it every time, even when the numbers look close to ordered already.
🧭How to find the mode
The mode is the value that appears most often.
In 2, 5, 5, 7, 9 the mode is 5, because it is the only value that repeats.
A set can have two modes, or more. In 3, 3, 8, 8, 10 both 3 and 8 are modes.
A set can also have none. If every value appears exactly once, there is no mode to report.
This is the only average that works on words and categories, such as favourite colour or most common shoe size.
🧭Which average should you use?
Use the mean when the data points are evenly spread and nothing looks extreme.
Use the median when one or two values sit far from the rest. A single outlier drags the mean but barely moves the middle value.
Salaries are the classic case. One very large salary lifts the mean above what most staff actually earn, so the median gives the fairer picture.
Use the mode for categories, or whenever you want the most common choice rather than a calculated centre.
🧭All three measures in one worked example
Here are seven test scores: 12, 15, 11, 15, 19, 9, 15.
For the mean, the sum is 96. Dividing the sum by the number of numbers gives 96 ÷ 7 ≈ 13.7.
For the median, order the scores from smallest to largest: 9, 11, 12, 15, 15, 15, 19. The fourth value is the middle one, so the answer is 15.
For the mode, 15 appears three times, more than any other score. So the answer is 15.
The three results differ because the low score of 9 pulls the average down. Comparing the measures is what tells you the data is skewed.
⚠️Common Mistakes to Avoid
- Forgetting to order data before finding median
- With even numbers, not averaging the two middle values for median
- Confusing mean, median, and mode
- Thinking there must always be a mode
- Letting outliers mislead when using mean
✏️Practice Problems
Find the mode of: 3, 5, 3, 7, 3, 8, 5
Answer: 3
Find the median of: 14, 8, 21, 6, 19, 12
Answer: 13 (average of 12 and 14)
A set of 5 numbers has a mean of 12. Four of the numbers are 8, 11, 14 and 15. Find the fifth number.
💡 Hint: A mean of 12 across 5 values means the sum must be 60.
Answer: 12
Find the mean of: 4, 8, 6, 10, 7
Answer: 7
Order these from smallest to largest and find the median: 9, 2, 7, 4, 5
Answer: 5 (ordered: 2, 4, 5, 7, 9)
Find the mode of: 6, 2, 6, 9, 2, 4
Answer: 2 and 6 (the set is bimodal)
A data set is 3, 4, 5, 6, 92. Which is the better measure of centre, mean or median? Why?
Answer: Median (5), because 92 is an outlier that pulls the mean up to 22
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Worked Examples
❓Mean, Median, Mode FAQ
What is the difference between mean, median and mode?
The mean is the total divided by how many values there are. The median is the middle value once the data points are in order. The mode is the value that appears most often.
How do I find the median of an even number of values?
Sort the values from smallest to largest, then take the two in the middle and average them. For 4, 6, 9, 11 that is (6 + 9) ÷ 2 = 7.5.
Can a data set have more than one mode?
Yes. Two values tied for most frequent makes the set bimodal, and more than two makes it multimodal. A set where every value appears once has no mode at all.
Which average is affected by outliers?
The mean. One unusually large or small value shifts it noticeably, because every data point enters the calculation. The median and mode are far more stable.
Is the mean the same as the average?
In everyday use, yes. In statistics "average" is the broader word, and the mean, median and mode are all averages. The mean is just the one people reach for by default.