The median and the mode

The middle value

The median is the value in the middle when the data is sorted from smallest to largest. Half of the values are below it, and half are above.

Always sort first. The middle of an unsorted list means nothing.

• With an odd number of values, the median is the one in the middle. For $148, 150, 152, 155, 160$ it is $152$.
• With an even number of values, there are two in the middle. The median is halfway between them. For $11, 12, 14, 15$ it is $(12 + 14) \div 2 = 13$.

With $n$ values, the median sits at the position $\dfrac{n + 1}{2}$. For $5$ values that is the third. For $4$ values it is the position $2.5$, halfway between the second and the third.

You should find that the position goes up by a half each time you add a value. For an even count, the red dot sits between two blue dots, not on one.

Try it in Python

The median and extreme values

Take the values $1, 2, 3, 4, 100$. The mean is $22$, bigger than four of the five values. The median is $3$, and it describes the group much better. The median does not care how large the biggest value is, only that it is above the middle. This is why it is called a resistant average.

The mode

The mode is the value that appears most often. It is the only average that works for words: the most popular colour, the most common size, the favourite sport.

Some data has two modes, or none at all if every value appears once. For $1, 1, 2, 2, 3$ the modes are $1$ and $2$.

Practise with new numbers

Common mistakes

• Forgetting to sort before finding the median.
• Taking the middle position as the median. The median is the value at that position, not the position itself.
• Saying the mode is the number of times. The mode is the value, such as size $7$, not the count, $4$.
• Thinking that every data set has a mode. If every value appears once, there is none.

Remember these

• Median: sort first, then take the middle. With two in the middle, take the halfway value.
• Mode: the value that appears most often.
• The mode also works for words and labels.
• Extreme values hardly change the median.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the median of $9, 2, 7$?

2. What is the median of $3, 5, 8, 10$?

3. What is the median of $1, 2, 2, 3, 100$?

4. What are the modes of $1, 1, 2, 2, 3$?