Ever wondered why your teacher asks for the "average" of test scores? It’s not just a random number – it tells a story about the whole set.
In simple words, the mean, median and mode are three ways to describe a group of numbers. The mean adds everything up and divides, the median finds the middle, and the mode spots the number that shows up most often.
Mean – the average of a list
The mean (also called the arithmetic average) is what most people think of when they hear "average". You add up all the values and then split the total equally among the items.
How to find the mean (step‑by‑step)
- Write down every number in the data set.
- Add them together to get the total sum.
- Count how many numbers there are.
- Divide the sum by the count.
Example
Suppose a class scored these marks in a quiz: 78, 85, 92, 67, 74.
| Step | Calculation |
|---|---|
| Sum of scores | 78+85+92+67+74 = 396 |
| Number of scores | 5 |
| Mean | 396 ÷ 5 = 79.2 |
So the average (mean) mark is 79.2.
Median – the middle value
The median is the number that sits right in the middle when the data are arranged from smallest to largest. It’s useful when you have extreme values that could skew the mean.
Steps to get the median
- Sort the data in ascending order.
- If the count of numbers is odd, pick the middle one.
- If the count is even, average the two middle numbers.
Example
Take the same marks: 78, 85, 92, 67, 74.
First, sort them: 67, 74, 78, 85, 92.
There are 5 numbers (odd), so the third one, 78, is the median.
Mode – the most frequent number
The mode is the value that appears most often in the data set. A set can have one mode, more than one, or none at all.
Finding the mode
- List each distinct number.
- Count how many times each appears (this is called frequency).
- The number(s) with the highest frequency is/are the mode(s).
Example
Imagine the scores are: 70, 80, 70, 90, 80, 80.
Frequency table:
| Score | Frequency |
|---|---|
| 70 | 2 |
| 80 | 3 |
| 90 | 1 |
Since 80 shows up three times, the mode is 80.
Quick Comparison
| Measure | Best when | How to calculate | Typical exam question |
|---|---|---|---|
| Mean | Data are evenly spread | Sum ÷ Count | Find the average of given numbers |
| Median | Data have outliers | Middle value of ordered list | Identify the middle score |
| Mode | Looking for most common value | Count frequencies | State the most frequent number(s) |
Key Tips for Exams
- Always check if the data are already ordered – if not, sort them first for median.
- Write down the total sum before dividing – it avoids arithmetic slip‑ups.
- Make a quick frequency table for mode; it’s faster than eyeballing.
- Remember: if two numbers share the highest frequency, the data are bimodal (two modes).
- When the question says "ungrouped data", it means the numbers are listed individually, not in intervals.
📝 Likely Exam Questions
- Find the mean of 45, 52, 63, 71, 84.
Sum = 315, Count = 5, Mean = 315 ÷ 5 = 63. - Determine the median of 12, 7, 9, 20, 15, 8.
Ordered: 7, 8, 9, 12, 15, 20. Even count (6), median = (9+12)/2 = 10.5. - What is the mode of the data set: 5, 9, 5, 12, 9, 9, 7?
Frequency: 5→2, 7→1, 9→3, 12→1. Mode = 9. - A class of 8 students scored: 68, 74, 68, 80, 85, 74, 90, 74. Find mean, median and mode.
Mean: Sum = 613, Mean = 613 ÷ 8 = 76.625.
Median: Ordered = 68,68,74,74,74,80,85,90 → middle two = 74 and 74 → Median = 74.
Mode: 74 appears 3 times, so Mode = 74. - Explain why median can be a better measure than mean when the data include 2, 3, 4, 5, 100.
Mean = 22.8, which is pulled up by the outlier 100. Median = 4, which reflects the central tendency of the majority of values.