Averages and spread (IQR/SD as required) - Mathematics IGCSE Study Notes

Imagine you've just eaten a bag of sweets, and you want to tell your friend about them. You probably wouldn't list every single sweet's flavour, would you? Instead, you might say something like, 'Most of them were strawberry,' or 'On average, they were pretty good.' This is exactly what averages do – they give us a single, easy-to-understand number that sums up a whole bunch of information.

There are three main types of averages, like three different ways to describe your sweets:

• Mean: This is what most people think of as the 'average'. You add up all the numbers and then divide by how many numbers there are. Think of it like sharing all your sweets equally among everyone. If you have 10 sweets and 5 friends, everyone gets 2! It's good for when all your numbers are pretty close together.
• Median: This is the 'middle' number when you line all your numbers up from smallest to largest. Imagine you line up all your sweets by size, from smallest to biggest. The median is the one right in the middle! It's great when you have some really big or really small numbers that might mess up the mean.
• Mode: This is the number that appears most often. If most of your sweets were strawberry, then 'strawberry' would be the mode. It's useful for things like favourite colours or shoe sizes, where numbers might repeat a lot.

But what if you tell your friend the average number of sweets you ate per day last week was 5, but one day you ate 20 and another day you ate 0? The average alone doesn't tell the full story! That's where spread comes in. Spread tells you how much the numbers in your group vary. Are they all clustered close to the average, or are they really spread out, like some are super high and some are super low? It's like knowing if all your sweets were roughly the same size, or if you had some tiny ones and some giant ones!

Let's say you and your friend, Alex, both played 5 games of a new video game. You want to see who is generally better. Here are your scores:

• Your Scores: 10, 12, 11, 9, 13
• Alex's Scores: 2, 20, 10, 18, 5

Let's find the mean (average) for both of you:

1. Your Mean Score: (10 + 12 + 11 + 9 + 13) / 5 = 55 / 5 = 11 points
2. Alex's Mean Score: (2 + 20 + 10 + 18 + 5) / 5 = 55 / 5 = 11 points

Wait a minute! Both of you have the same mean score of 11. Does that mean you're equally good? If you only looked at the mean, you might think so. But look at the individual scores again. Your scores are all very close to 11 (9, 10, 11, 12, 13). Alex's scores, however, are all over the place (2, 5, 10, 18, 20)! Alex had some really bad games and some really good games.

This is where spread helps us. It tells us that even though your average scores are the same, your performance is much more consistent (less spread out) than Alex's. Alex's scores are much more spread out, meaning they are less consistent. So, while your average might be the same, you're probably the more reliable player!