In statistics, there are three types of averages: the mean, the median and the mode. Measures of spread such as the range and the interquartile range can be used to reach statistical conclusions.
The mean is the most commonly used average. The median is the middle number in a set of data when the data has been written in ascending or descending size order. The mode is the number, or item, which occurs most often in a set of data. In statistics, a range shows how spread out a set of data is. The bigger the range, the more spread out the data. If the range is small, the data is closer together or more consistent.
Okay, we're looking at analysing data. So we'll look at the five main methods that we have for this and then we'll do a couple of example questions at the end. So let's say we've done a hot dog eating contest and we recorded the number of hot dogs eaten per person in 1 minute. So what? Ten people have taken part in this contest and so, like, two people are eating four hot dogs, two people are being five hot dogs, one person is eating six hot dogs, et cetera.
And we've kind of listed all of our hot dogs eaten in order. Well, let's first look at the mean. So let's first look at the kind of the mean number of hot dogs eaten per person. Well, the mean we can remember is called the mean because it's the meanest one to calculate. It takes the longest amount of time.
In this case, what we do is we have to add up all of our numbers and then we divide by the number of numbers or the number of data points. In this case, we've got ten data points, right? Ten people taking part in the contest. And so we first add up on numbers. Well, that's going to be four plus four is eight, plus ten is 18, plus six is 24, plus another 21, 45, plus another 16, that's 61.
So our total number of hot dogs eaten between our ten people is going to be 61 hot dogs that they've all eaten together. Well, if you want to work out the number of hot dogs eaten per person on average, we just take that total number of hot dogs and we divide it by the number of people, which is ten in this case. And so we know the average person has eaten 6.1 hot dogs, right? Ten of those people would give us 61 hot dogs. So our mean is 6.1 hot dogs.
Let's look at the median. So the median is the middle number. Well, you'll see here that we don't really have a middle number, right? We've got five numbers here, five numbers here. We don't have this middle number.
We're missing this middle number. Sometimes textbooks are quite nice and they'll give you an odd number of numbers. I haven't been quite so nice here, but what we do in this case is we kind of make up the data, right? We say, well, what number would be here if there was a number here? And we know it's going to be banging in the middle of six and seven.
And so we say the median is going to be 6.5. We just split the difference between six and seven. And there is a formula that we can use as well. So the formula is we take the number of data points, we add one to it and we divide it by two. In this case, that's going to be ten people.
Add one is eleven divided by two or eleven over two, it's just 5.5. And so again, the formula tells us that we look at the five point fifth person and so we go along 1234, 5.5 people. And again, the formula tells us to look in this area to find our median. Let's look at the mode next. So the mode kind of rhymes with most.
It's the most occurring number. If we look at our data, we've got three sevens, we haven't got any more of any other number. And so our most occurring number is going to be seven. There are three main ways of calculating the average, right? We've got mean, median and mode.
We can split this down the middle and we'll now look at the range and the lower and up quartile. And these are kind of no longer really related directly to the average of our data. So the range is very simple. It's just the maximum minus the minimum. In this case, the maximum is going to be eight and the minimum is going to be four.
So eight minus four gives us four. So there's a range of four hot dogs there faster seeing hot dog eater Hasine formal hot dogs and the slowest hot dog eater. Let's look at the lower and up quartile. So the lower and up quartile are very similar to the median. The median is the very middle.
It's 50% of the way along. The lower quartile is 25% of the way along. So the lower quartile is just a number that's 25% of the way along. But if we look at this, it's halfway between our first half, right? The number that's halfway between our first half is that five.
There isn't that. We've got two either side of it, five playing in the middle. And so our lower quartile is going to be five. And likewise, in our top half of our data, we've got this seven bang in the middle of this top half. And so our upper quartile is going to be seven.
Our inter quartile range, sometimes we can be asked that as well. Interquartile range, it's just going to be our upper quartile minus our lower quartile. In this case, that's going to be seven minus five, which gives us two. The interquartile range is just a way of looking at how groups together the data is right. In this case, it's fairly grouped together.
But if we were to say, look at the heights of students in a class, well, we may have one class where all the students are very closely grouped together on a graph. Most of our students are around 1.6 metres or something. Most are kind of appearing in this gap and there are a few shorter students and a few taller students. Those have a very narrow intercourse range. If we had another class where we had lots of much more discrepancy in our heights, we might have a graph that looks more like this way, more spread out.
And so our course our range is going to be much wider. There are still a few short students and still a few tall students, but a much wider range of our students are in the intercourse range. So it's a very convenient way of looking at how spread out our data is. Let's look at some example questions. So what is the mode of this data set?
Well, the mode, again, is the most occurring number. In this case, we've got a three here and a three here, and so I know my mode is going to be three. The range of the data set, again, is the maximum minus the minimum. In this case, it's going to be ten minus 110 minus one will give us nine. So nine is our range.
Finally, Bob says, from the table, I can tell that the median of the data is in the seven to nine class interval. Why is he incorrect? So he's incorrect. Well, it's a couple of different ways that we can understand this. Right, so the table has already done half work for us, really.
It's kind of put it in order already. Or I've got there this could be, for example, lengths of a piece of strength. To keep it super simple, we've got four pieces of string that are maybe one to three centimetres long. We've got six pieces of strength that are maybe four to six centimetres long, and so on. And we want to find the very middle piece of strength.
Where does that lie, roughly? What's the length of that piece of strength that's in the very middle? If we put all of our lengths out from smallest to longest well, if we look at how many pieces of string we've got in total, just continuing this piece of string algae, we've got four plus six is ten, plus one is eleven plus and four is 15, plus two is 17. So we've got a total of 17 data points or 17 pieces of string in my example. And so we know that we could shove it into our formula if you want to see.
Right, our median point is going to be N plus one over C. In this case, it's 17 plus one over two. Well, 17 plus one is 18 and half is 18 gives me nine. And so I know the 9th number will represent my median number. Well, the 9th number occurs in this class interval.
Right. We've got four numbers in the first one and six numbers in the second car interval. And so if the second to last number in this car interval would represent my median, another way that we can understand this is if we just kind of estimated the length of each string. Let's continue this string example. So let's say, rather than one to three, we're going to estimate and say all the pieces of string in this class interval are two centimetres long.
And so we'd have four two centimetre long pieces of string in that class interval, we then have six pieces of string that are roughly five centimetres long. So we're going to have six five centimetres long pieces of string. And then we're going to have one roughly eight centimetres long pieces of string, 411 centimetres long pieces of string, and two roughly 14 centimetres long pieces of string. And so, again, we've got 17 pieces of string overall, and we want to find the very middle one. So we know that we can kind of count along eight from the left, which is going to be that we can count along eight from the right, which is going to be that and the middle number that's left.
That middle number there is our median, right. That five is in our four to six class interval. So that's another way of thinking about this. Again, it's just a very central number. If we put all of our numbers in the long run, I've just estimated all these string lengths just to prove a point, but that's kind of what's going on behind the scenes when we use this formula.
Hope that makes sense. That's the end of our lesson and that's the end of the lesson detail, so hopefully that's.
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