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# Always Go To the Toilet...how averages work in real life

## Adam's Law

You're about to head out of the house and you think "I won't bother..." but you get half way down the road and realise that you should have applied Adam's Law! What do we mean by Adam's Law? Not now that he's 14 but in the past when he was little this became the rule we introduced before going anywhere in order to avoid that sudden plea..."I need the toilet" Adam's Law is applied to ensure that we don't need the toilet when only half way down the road.

Now we talk about applying Adam's Law before we leave the house, the bar, the restaurant...well anywhere really. We don't want to be caught out.

The likelihood of those embarrassing moments happening now are far removed from when Adam was perhaps 3 years old, but those moments when we realised that we were too far away meant we gave the trip to the toilet before leaving the house it's own name.

## Is this really a Law?

Adam's Law is not a real law but of course the Laws of Averages created the law. Most of the time once we are far enough away from a toilet so that we could do nothing about it, Adam would need to use the toilet! Most of the time, see that word MOST then I think that refers to an average!

## In the shoe shop

When I was about to teach averages I would ask the pupils about their shoe sizes and then calculate the average shoe size. I now think I was teaching an average that could not be found.

My lesson would look a little like this:

We would add up all their shoe sizes, we would count the number of children in the class and we would divide total shoe size by number of children. For many years I thought this was fine and then one day I thought this is ridiculous, I'm not actually using the correct average here. We could find the mean average length of a pupils foot, but a shoe size...? A shoe size is a description of the shoe that would fit your foot, a label. My size 7 foot is probably a completely different shape and size to your size 7 foot. But, we would both wear a size 7 shoe.

If we want Maths to be real then we have to consider the maths we deliver. I don't think shoe shops use a mean average shoe size because shoe sizes describe the shoe and do not describe something numerical. Instead, I think they may consider two different averages.

I could consider the mode or modal shoe size. This would be the one that I think most shoe shops must use for stock control. I would record the shoe sizes of all the pupils but I would then look for the most common shoe size. This would tell me that my shoe shop needs to stock more shoes that are size 6 for example.

I could consider the median shoe size, this is the shoe size in the middle. This could get difficult to manage in a class but imagine getting all the pupils to stand in order of their shoe size - size 1 at one end, size 12 at the other and everything else in between. One by one from alternate ends I ask pupils to sit down until there is only one or two students left standing. These students are in the middle, they have the median shoe size.

Median shoe size is again an average that seems reasonable to consider. We could begin to get really technical with our shoe shop, we might begin to look at cumulative frequency and the interquartile range for the shoes.

## What are the averages?

Over the last few years, pupils have told me there are 4 averages. Mode, median, mean and range, they name them all but can't necessarily tell me what they mean.

Mode - the most common (see the video)

Median - the middle when placed in order of size

Mean - sharing things out equally (see the video)

Range - difference between biggest and smallest values

## Mode by Mummymaths

## Mean by Mummymaths

## Deceiving averages

Mode, median and mean I will agree with as being averages. But, the range? The range tells us about the spread of data. It might tell us that the lowest paid member of staff earned £14,000 and the highest paid earned £72,000. The range of salaries would then be £58,000 but this isn't an average. It just tells us the difference between the biggest and smallest salaries.

The average would be determined by the number of people working there. If there were only two people working at this company then the mean average would be £36000 and so would the median. There would be no mode.

So let's introduce a few more people to the imaginary company (maybe it could be my shoe shop!) I think I might employ 10 people...

1 person earns £72000 (that's a lot of shoes)

6 people earn £14000

2 people earn £25000

1 person earns £40000

So now we can easily see the mode, that's £14000 because 6 people earn £14000. The median is £14000. If we put the 10 workers in order the middle person would earn £14000. But, the mean average - well that changes it all.

The salaries total £246000, so if that money was shared out equally then each person would earn £24600. That would be a nice increase for the lowest paid but a bit of a drop for the higher earner.

In an advert for a new employer the shoe shop might choose to use the mean as an average. I'm sure you'd rather see an average salary of almost £25000 than £14000.

## Average facts

On average men's noses are 10% longer than women's.

Which average? And is this is a reasonable average to consider. Does it matter, does this mean that if my nose is 6cm long my husbands is almost 7cm?

Over the last week I have walked 2.1 miles a day, on average. Of course, some days I walk more and others I walk less. Over the month I've walked 3 miles on average. These are mean averages - calculated on my phone.

Then we think about that mode average. 8 out of 10 cats, that's referring to most. Making a particular cat food the mode.

Average contents, average use, average miles per galloon and my favourite average speed cameras.

## Average speed cameras

Now here's an interesting one. When and what are they measuring. Can you travel above the speed limit for a while and then reduce your speed.

I assume they use a mean average, that is over a set distance how long has it taken you to travel that distance. To be safe in a 40mph zone you should perhaps travel at 39mph. Or you could try and be mathematical and calculate how much you would need to reduce your speed by if you went too fast for a while.

I remember my journey to Bedford when I was starting my PGCE teaching qualification. I was in the car with my uncle, my aunty was in her car. There were 8 of us travelling because my cousins wanted to visit the Doctor Martins shop in Bedford. As would be expected the two cars became separated, for part of our journey we tried to decide how to adjust the average speed to ensure we could catch up with my aunty or my aunty could catch up with us.

Eventually, as was usual, we just happened upon each other. But, that speed calculation was tough when travelling, so don't rely on it for the cameras.

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