Averages and spread (IQR/SD as required)

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!

Let's break down how to calculate the main averages and one measure of spread, the Interquartile Range (IQR).

Calculating the Mean (The 'Fair Share' Average)

1. Add them all up: Sum all the numbers in your data set.
2. Count them: Find out how many numbers there are in total.
3. Divide: Divide the sum from Step 1 by the count from Step 2.

Calculating the Median (The 'Middle' Number)

1. Order them: Arrange all your numbers from the smallest to the largest.
2. Find the middle position: If you have an odd number of values, the median is the single number exactly in the middle. If you have an even number of values, the median is the average of the two middle numbers.

Calculating the Mode (The 'Most Popular' Number)

1. Count repeats: Look for the number that appears most often in your data set.
2. Identify: The number that shows up the most is your mode. You can have more than one mode (if multiple numbers tie for most frequent) or no mode (if all numbers appear only once).

Calculating the Interquartile Range (IQR) (How 'Spread Out' the Middle Half Is)

1. Order them: Arrange all your numbers from the smallest to the largest (just like for the median).
2. Find the Median (Q2): Locate the middle number of the entire data set. This is also called the second quartile (Q2).
3. Find the Lower Quartile (Q1): Look at the first half of your ordered data (all numbers before the median). Find the median of this lower half. This is your lower quartile (Q1).
4. Find the Upper Quartile (Q3): Look at the second half of your ordered data (all numbers after the median). Find the median of this upper half. This is your upper quartile (Q3).
5. Calculate IQR: Subtract the Lower Quartile (Q1) from the Upper Quartile (Q3). So, IQR = Q3 - Q1. This tells you how spread out the middle 50% of your data is.