Mr F’s Maths Notes Statistics 3. Averages

3. Averages and Measures of Spread What are Averages and Measures of Spread, and why do we need them? Averages and Measures of Spread are two of the most important and useful concepts in maths They allow us to look at a huge load of data and make some sense out of it, summarise it, and compare it to other huge sets of data. Averages and Measures of Spread are used every single day, whether it be crowd attendance at Old Trafford, viewing figures for Lost, or the salary of your average poor maths teacher. The usefulness of the different types of Averages and Measures of Spread will be looked at in The Big Cricket Example, so for now, let’s learn how to work them out! 1. The Mean Whenever most people talk about an average, this is the one they… mean! How to work out the Mean: 1. Add up all your data values 2. Divide this total by the number of data values What about a little rhyme?: I’m not really sure What the MEAN is about Just add them all up And share them all out

2. The Median This is the one people tend to mess up!... Don’t let it happen to you! How to work out the Median: 1. Place all your data values in ascending order (biggest to smallest) 2. The piece of data in the middle is your median NOTE: If you have an EVEN number of data values, there will be TWO pieces of data in the middle. No problem, just add them together and divide by two to find the number halfway between them… and this is your median! What about a little rhyme?: I don’t know the rhyme I don’t know the riddle I am the MEDIAN And I’m in the middle! 3. The Mode This is the final type of average, and the easiest one to work out… so long as you remember how! How to work out the Mode: 1. Find the most common piece of data (number or letter) and this is your mode! NOTE: You can have no modes or more than one mode, and you must write them all down! What about a little rhyme?: I don’t want to brag I don’t want to boast I am the MODE I am the most

The Big Cricket Example 4. The Range The Range is a Measure of Spread, and tells you… well, how spread out the data is! How to work out the Range: 1. Subtract the smallest data value away from the biggest data value! What about a little rhyme?: From largest to smallest See how they change Take them away And I am the RANGE NOTE: It’s one thing knowing how to work out the three averages and the measure of spread, but it’s just as important to know how to interpret them! Hopefully this example will help! The Big Cricket Example Andrew Flintoff and Michael Vaughn are having an argument in the pub trying to decide who has had the better season with the cricket bat. Here are their scores: Use your knowledge of Averages and Measures of Spread to decide which cricketer has had the better season 20 35 22 55 60 10 17 32 64 86 14 32 50 24 30 0 0 0 2 0 15 5 370 250 0 3 5 0 1

NOTE: The first point to notice is that by just looking at the scores as they are makes it hard to come to a decision about who has had the better season… that’s why we need Statistics! Secondly, just adding up the total amount of runs and deciding that way would not be fair. Why?... well, because they have not played the same number of games! So, there is only one thing for it… let’s work out some statistics! 1. The Mean 1. Add up all your data values 2. Divide this total by the number of data values Andrew Flintoff total runs scored: 551 total games played: 15 mean: 36.7 runs (1dp) Michael Vaughn total runs scored: 651 total games played: 14 mean: 46.5 runs What does this tell us? – well, it looks like, on average, Michael Vaughn has had the better season Good thing about the mean – notice how every single score was used to calculate the mean – this means it gives a good summary of the whole season. Bad thing about the mean – look at Michael Vaughn’s scores. He only had two decent ones, and yet his mean is far higher than Andrew Flintoff’s! This is because the mean is significantly affected by outliers – pieces of data which stand out for being really low or really high like the two scores of 370 and 250. You could argue that these have distorted the result!

2. The Median 1. Place all your data values in ascending order (biggest to smallest) 2. The piece of data in the middle is your median 10 14 17 20 22 24 30 32 32 35 50 55 60 64 86 Median = 32 runs Note: there are the same number of data (7 pieces) either side of the box! 0 0 0 0 0 0 1 2 3 5 5 15 250 370 Median = 1.5 runs Note: there are still the same number of data (6 pieces) either side of the box! What does this tell us? – well, this time it looks like Andrew Flintoff has had the better season Good thing about the median – because we are only focussing on pieces of data in the middle, outliers don’t have as big an effect, so they cannot distort the results! Bad thing about the median – the problem here is that you are only looking at – at most – a couple of pieces of data from each player. You could argue that the result is not representative as a lot of pieces of data (scores) are just ignored!

3. The Mode 1. Find the most common piece of data (number or letter) and this is your mode! Andrew Flintoff mode: 32 runs Michael Vaughn mode: 0 runs What does this tell us? – well, using the mode it again looks like Andrew Flintoff is on top! Good thing about the mode – very speedy to work out! Bad thing about the mode – can give distorted, or even no results. Imagine if Andrew Flintoff only scored one innings of 32 runs… he would have no mode to compare! Or, imagine if he instead scored a couple of innings of 200… the mode would then say this was his average!

So who is the better cricketer?... 4. The Range 1. Subtract the smallest data value away from the biggest data value! Andrew Flintoff largest value: 86 smallest value: 10 range: 76 runs Michael Vaughn largest value: 370 smallest value: 0 range: 370 runs What does this tell us? – well, one answer that I often here is this: “Michael Vaughn has the biggest range, so he is the best!”… but that’s not quite right. The bigger the range, the more spread out your scores are… so the less consistent (brilliant maths word that always impresses examiners/teacher) your performance is. So, I would argue, because Andrew Flintoff has a smaller range, his performance is more consistent, and therefore he has had the better season! Good thing about the range – gives a very quick measure of how spread out the data is Bad thing about the range – unfortunately, this statistic is vulnerable to outliers as well! Michael Vaughn had a couple of big scores, and look at the effect it had on his range!... This is why mathematicians prefer to measure the spread of data using the Inter-quartile Range or Standard Deviation… but don’t worry about them yet! So who is the better cricketer?... Well, in the end, it’s up to you! The most important thing is that you have shown you can calculate each of the statistics and – not a lot of people can do this - interpret what they mean! If you want my opinion, as a proud Lancastrian, Andrew Flintoff is much better!

Sum of Mid-Point x Freq Mean = Total Frequency Estimating the Mean from Grouped Data Attendance Frequency 0 < A ≤ 5,000 5 5,000 < A ≤ 10,000 12 10,000 < A ≤ 15,000 24 15,000 < A ≤ 20,000 8 Example: Calculate an estimate of the mean number of fans attending the mighty Preston North End football matches from the following table: Okay, now do you see a problem here?... Look at the first group… we know there were 5 matches where between 0 and 5,000 people turned up, but we don’t know exactly how many people were at those matches!... One match could have had 1,309… another 4,510… we just don’t know! So… the best we can do is to make an estimate! And what is our best estimate for that first group?... Well, the MID-POINT… 2,500! And that is how we calculate an estimate for the mean from grouped data: 1. Work out the mid-point 2. Work out the mid-point x Freq for each group 3. Use this formula: Sum of Mid-Point x Freq Mean = Total Frequency Attendance Mid-Point Frequency Mid-Point x Freq 0 < A ≤ 5,000 2,500 5 12,500 5,000 < A ≤ 10,000 7,500 12 90,000 10,000 < A ≤ 15,000 24 300,000 15,000 < A ≤ 20,000 17,500 8 140,000 TOTALS 49 542,500 (nearest whole)

Good luck with your revision!