## Stem and Leaf – is there a point?

Stem and leaf diagrams, or “Those leafy stem things”, as one of my former pupils used to call them, have long been an annoyance of mine. I’d never heard of them until I was brushing up on the GCSE syllabus ahead of my PGCE and when I did come across them I couldn’t see anything that they brought to the party that couldn’t better be shown using alternative methods.

You can imagine my feelings then as the KS3,4 and now 5 curricula jettisoned them, meaning the end was in sight for the need to teach them. I let my feelings on this be known in my recent post around the new A level curriculum and this led to further discussion around them on twitter. Then Jo Morgan (@mathsjem) wrote this fantastic piece which supports their place in a classroom and gives some great activities to use in teaching them.

It got me thinking, are my feelings unfounded? Should I be writing off stem and leaf diagrams? I’ve long been an advocate of maths for maths sake, see this defence of circle theorems for one example, so why is this feeling bot the sane for stem and leaf?

Perhaps it’s that it falls under the banner of “stats”, a very applied area of maths. This suggests that there should be an application associated with it. The use mentioned in Jo’s blog for bus and train timetables is the best example I’ve seen, but I think a normal timetable will be easier to read for the majority if people, as the majority of folk aren’t familiar with stem and leaf. Hannah (@missradders) suggested that they were used a lot in baseball, but I can’t see any reason that they would be better than a bar chart or a boxplot.

Colin Wright (@ColinTheMathmo) suggested during the twitter discussion that they could be used to build understanding around data, even though they are no use for any real data sets which would be far too big. Jo also uses this idea in her defence, saying they could provide a good introduction to the ideas of skew, quartiles and outliers. I can see this argument, but I still think there are better, more visual and less convoluted ways to introduce these to pupils, such as the aforementioned bar charts and box plots along with scattergraphs and a host of other data presentation methods (but not pie charts, they’re just as bad, if not worse! But that’s a topic for another day!)

*I really enjoyed Jo’s post, if you haven’t read it I would advise you do. It made me think and look hard at my views. In the end though, I still see no need in stem and leaf diagrams and will be glad to see the back of them. If you have opinions either way I would love to hear them, especially if you have further real life uses!*

## A probability puzzle

*“Colin and Dave are playing a game. Colin has a probability of 0.2 of hitting the target with any given shot; Dave has a probability of 0.3. Whoever hits the target first, wins. Colin goes first; what is his probability of winning?”*

Yesterday I listened to the latest edition of “Wrong, but useful” (@wrongbutuseful), and the above is Is the puzzle set by the cohost Dave “The king of stats” Gale (@reflectivemaths).

It was pretty late on in the evening, but I decided to have a quick attempt at the puzzle nonetheless.

Here you can see my back of the envelope workings, complete with the word “frustum” on the top of the envelope as I had added an extra r into it in my recent post. You will also see that approaching midnight on a Saturday after a day at the NTEN-RESEARCHED-YORK conference is not the most idea time for solving maths puzzles! My thinking was fairly valid, I think, but my tired brain has made an absolute ton of mistakes! *(If you haven’t spotted them, go and have a look before reading on!)*

When I looked at it this morning the first thing that jumped out at me was “0.14×0.028 does NOT equal 0.0364. Then I thought, “ffs, 10/43 is very definitely not 0.43!” This was closely followed by: “and why have you used 0.14 as r, the 0.2 is the probability he hits, if he hits the game is over, Cav you ploker!”

So I had another go:

I think this is right now. (Although do feel free to correct me if you spot another error!) The probability Colin wins on the first go is 0.2, on his second go is the product of him missing (0.8) Dave missing (0.7) then him hitting (0.2) so 0.8×0.7×0.2. As the sequence goes on you are multiplying by 0.7 and 0.8 each time, so it gives rise to a geometric sequence with first term (a) as 0.2 and common ratio (r) as 0.56.

The total probably of Colin winning is the sum of the probabilities of him winning each time. This is because the probability he wins his the probability he wins on his first go OR his second go, OR his third etc. The game goes on until someone wins, so is potentially infinite, thus we need to sum the series infinitely.

As the series has a common ratio of 0.56, which is less than one, we can sum the series to infinity using s = a/(1-r) which gives 0.2/(1-0.56) = 0.2/0.44 = 5/11. Thus the probability Colin wins is 5/11 or 0.45recurring.

## What does it mean?

Today my year 11s were busy revising ahead of tomorrow’s mock exam and one of them started singing the averages song. You know the one:

*“Mean is average, mean is average, mode is most, mode is most, median’s the middle, median’s the middle, range high low, range high low.”*

This got me thinking about the words we use. I’ve always disliked this song as a mnemonic as it encourages people to think of the mean as the “average” when actually the mode and the median are also averages too. The median in particular is a very useful one and we need pupils to understand the distinction. I have been very impressed in recent staff meetings to hear the principal, an English teacher by trade, use the term “national median” rather than “national average”!

As I was thinking about this, though, I had the sudden realisation that I should also be feeling the same way about the term “mean”! Granted, at GCSE level we only talk about one mean, the arithmetic mean, but that doesn’t mean the geometric mean doesn’t exist. (Nor the root mean square nor harmonic mean for that matter! *Other means are available*)

This is a hypocrisy in the way we treat certain words. I’m not the only maths teacher who dislikes the way mean and average have become synonymous. But no one has ever mentioned that the word arithmetic is missing from the term every time we use it.

I worry that we may be setting students who go on to further study statistics up for confusion in the future by simply referring to the arithmetic mean as the mean.

Have you ever used the term arithmetic mean, or even geometric mean, with your students? Have you shared my worry? Or do you think I’m being overly pedantic and it doesn’t matter? I’d love to hear your opinion.

## Statistical Deception

When teaching and talking about statistics I always emphasise the need to be careful what you believe and to always ask yourself “what agenda does the person presenting this data have?”

I’ve written before about how stats can legitimately be manipulated to serve different points of views, especially when there are false variables at work. But recently I’ve noticed at darker art in statistical manipulation, one that is, at its heart, lying.

We are less than six weeks away from local elections now, and it is becoming silly season for party political leaflets coming through our letterboxes. Now we all know that the political parties will present data in a way that makes them look better, they are trying to win your vote afterall, but we would expect them not to lie. For the data to be accurate and presented correctly. Unfortunately, however, this is not always the case:

Exhibit AThis popped up a number of times in my twitter feed from a variety of sources. I believe it is from a Lib Dem leaflet in Manchester. As you can see, they have presented a bar chart with proportions labelled as percentages. The first screaming error is that the red bar and the orange bar are massively different heights, yet are both emblazoned by the label 39%. The second glaring error is that the percentages add up to more than 100%. The first implies that either the Lib Dems are deliberately trying to mislead voters into thinking they are in a stronger position in the ward than they are, or that they don’t realise that 39% is equal to 39%. I’m not sure which is worse?!

Here’s an excel interpretation of what the graphs

shouldlook like:Exhibit BThis graph came through my door in Leeds North West parliamentary constituency. The first thing that caught my eye was that although the gap between the number of votes between Lib Dems and Labour; and between Labour and Conservative is almost the same, the difference in the gaps between the bars was almost 5 times as big, which would imply almost five times as many less votes! An obvious fallacy. Either it’s a deliberate attempt to mislead, or they can’t draw a bar chart. If it’s the latter, do we want them in charge of our local authority budgets?! (or the entire economy for that matter!!)

Something else that struck me as deciving, although this time mathematically correct at least, was the choice of data. This was a leaflet issued in the run up to a local election, and the data set used was from the last local election. Why then, is the data that for the parliamentary constituency rather than the council ward? The ward makes up around a quarter of the constituency, and the vote share in the ward is radically different to that of the constituency. The sitting councilor is conservative and sits on a huge majority, and the Lib Dem candidate last time out cane third. To issue a leaflet in the run up to a local election which implies the conservatives can’t win in a ward where they have a large majority and back it up with local election data for a parliamentary constituency is deliberately deceptive and misleading.

Here’s an excel interpretation of what this one

shouldlook like:Exhibit CThis one comes from

“across the pond”and is another which was viral. This one seemed to appear constantly for a few days everywhere I looked. If you are still wondering what’s wrong with it, take a little look at those numbers down the left hand side…. See it? The y axis goes upwards to zero! Drew Barker (@twentythree) made this version which gives a much better picture as to what’s going on.I can’t wait to see what my classes make of these!

nb I haven’t “selected” these graphs as an attack on the Lib Dems, it’s just they are the only party who have sent me a leaflet with incorrect maths. I’ll gladly expose any of the parties if they themselves do. I do collect these, so if you spot anything similar, do send me it!## Share this via:

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