25 July 2014

Long Live Stem and Leaf

I planned to write a post about teaching stem and leaf diagrams. But then I realised that many maths teachers consider them to be pointless, a waste of teaching time and a welcome removal from the curriculum.

Now let’s think about this for a minute. Admittedly stem and leaf diagrams are rarely used in ‘real life’. In fact the only common examples we can find are timetables, such as the Japanese train timetable shown in the picture below (more examples here). But does this ‘uselessness’ really mean that stem and leaf diagrams shouldn’t be taught? If that’s the case then a number of the topics we teach in maths should be taken off the curriculum. 

Japanese train timetable. Source: Wikipedia

Much of the maths we teach at Key Stage 3 helps pupils access more advanced topics (all roads lead to calculus?!). Maths begets maths. So do stem and leaf diagrams lead us anywhere? Do they help pupils develop any useful skills and understanding?

I'm going to say (in a sheepish whisper) that they do. Stem and leaf diagrams are one of a set of tools that could be used to develop understanding of the concepts of median, mode, quartiles, spread, skew and outliers. I think they are accessible to low ability pupils. I think that pupils enjoy creating them. I think they lead to interesting discussions when comparing two sets of data. I certainly don't think they should be a compulsory topic at any age, but I see no harm in teachers using them if they like. Isn't it nice to have that freedom?

John Tukey, the inventor of stem and leaf diagrams, wrote "If we are going to make a mark, it may as well be a meaningful one. The simplest - and most useful - meaningful mark is a digit" (source). Excellent point Mr Tukey. Did you know he also invented the box and whisker plot?

John Tukey is also quoted as saying "There is no data that can be displayed in a pie chart, that cannot be displayed better in some other type of chart". I agree. In the same way many of you don't like stem and leaf diagrams, I don't like pie charts (this excellent blog post about the misuse of pie charts is worth a read).

Right, now I've attempted to justify this blog post, here's my lesson ideas:

Starting Point
You could start by presenting your class with a large set of data (like the one below) and asking them what the mode is. It's hard to spot. Ask your pupils for ideas on how could we organise this data so we can easily see the mode.

12, 54, 21, 56, 32, 14, 42, 23, 20, 48, 21, 34, 12, 15, 16, 42, 47, 46, 32, 21, 45, 43, 21, 34, 26, 32, 17, 18, 45, 21, 44, 32, 12, 23, 45, 24, 35, 38, 28, 18, 29, 20, 10, 36, 12, 42, 47, 32, 47, 37, 12, 54, 23, 17, 29, 38, 32, 12, 17, 13, 21, 34, 12, 15, 16, 42, 47, 46, 12, 16, 43, 23, 16, 46, 52

I find stem and leaf plots rather pleasing to the eye. Much nicer to look at than a boring list of unordered data.

An alternative starting point is a 'beat the teacher' game, which I've adapted from an idea by the NCETM. In this activity, pupils think of a number between 10 and 40. Each number is called out in turn, while pupils make a list in their books. Meanwhile the teacher also makes their own list, but organises it as a stem and leaf diagram. Once all the numbers are listed, the teacher asks the class to work out median, mode and range. The teacher will do it much quicker from their stem and leaf diagram, hence he/she wins the race.

Physical Activities
This blog post is about introducing stem and leaf diagrams using post-it notes. It's a lovely idea for an engaging and accessible lesson. Mr Collins shared a similar idea in this blog post, but he uses students and desks instead of post-its. If you like the idea of human stem and leaf diagrams then you might also like these breath-holding activities - one from Don Steward and a similar one from the NCETM.

Data Types
Encourage pupils to think about different types of data - what is suitable for a stem and leaf diagram and what isn't? For example, could the total number of goals scored per game in a season of football matches be displayed in a stem and leaf diagram? Probably not, because the data would mostly be single digit numbers. Try my worksheet for more on this.

Mini Projects
Don Steward has some nice activities involving stem and leaf diagrams, such as this this Presidents Longevity Project. I also like Tim Jefferson's mini-project about bus timetables. Taking this timetable project further, you could ask pupils to think of other ways to present timetables - this post 'Visualizing a train timetable using a stem-and-leaf plot' suggests a nice alternative.

There's opportunities to extend this topic by looking at variations in the way data is presented (for example using 'split stems') and interesting features of the data (eg outliers, bimodal data, skewness).

Back-to-back stem and leaf diagrams are useful for comparing data. Writing comparisons between data sets - whether represented by stem and leaf diagrams or box and whisker plots - is something that pupils (especially EAL pupils) often find difficult.

There's lots of S1 (A level) stem and leaf questions - like the one below - that could be explored at Key Stage 3 (which begs the question, why were they considered suitably challenging for A level?).

Edexcel S1 November 2004, Question 1

I rarely disagree with people or play devil's advocate so this post is a bit risky for me! Be nice. If you do teach stem and leaf diagrams then I hope you find these ideas helpful. Thanks to those on twitter who shared their views and ideas (especially @srcav - who is not a fan of stem and leaf diagrams!).


  1. Right, great post, and I love the activities mentioned, but I'm not sure I'm convinced on the need, or the point, for stem and leaf.

    The timetable use is a decent use for them, but I'm not sure it's any better than standard layout for timetables, which are easier to read if you aren't familiar with stem and leaf diagrams, which the majority if people aren't as they haven't been on GCSE for that long and are easily forgettable.

    I love the quote about pie charts, I feel the exact same way about them, but I'd also argue that there is nothing that a stem and leaf diagram shows that can't be better shown in other ways. Distribution, medians, quartiles, skewness etc can all be easily seen on a bar chart or a box and whisker plot.

    I do take your point that a lot of things in maths,have no practical uses, here http://wp.me/p2z9Lp-9f I wrote in defence of circle theorems on the similar lines. But I'm not sure I feel the same about stem and leaf, I'll think on it some more!

  2. Thanks for your comment. Great post on circle theorems. Gives me food for thought.

    From a stem and leaf plot it's really easy to see if data is normally distributed. http://sites.stat.psu.edu/~lsimon/stat250/sp98/handouts/handout10.pdf. I like box and whisker plots but I think they're less intuitive.

    It's interesting that these topics - stem & leaf, box plots, circle theorems - are all those that I have absolutely no recollection of studying at school. I must see if I can find a maths syllabus from the 90s, it will be interesting to see what else is new.

  3. Just a follow up to this post... @srcav wrote an excellent blog post which is definitely worth a read, to see the other side of the argument.

    Views on Twitter are mixed. Some teachers think stem and leaf diagrams are pointless, others quite like them. Here’s some of the comments:

    @dazmck “Find them useful for a first look at data.”... “I get a bar graph (on its side) that retains the original data - what's not to like?”... “Also useful as an intermediate step if you are asking pupils to draw a box plot for unsorted data where n is reasonably large”

    @mathnqt ”limited use but confess I like them” ... ”they make sense to the pupils, show all the data but visual like a bar chart. Good to show the median in the middle of all the data”

    @runningstitch “I always liked stem&leaf @EdwardTufte would approve of them, all ink is meaningful, tells much more than just a bar chart!”... “because raw data make up the bars so add meaning, can find actual median from graph, graph is data. Back to back best tho”.

    @missradders “For me it's the power of being able to see the distribution & orig data...”

    @TheMathsMagpie "+ves: No loss of raw data & visual of distribution? Easy to teach ;)”

    Notice how I’ve only listed the comments in favour of stem and leaf diagrams? That’s because I’m a statistician, therefore expert at presenting biased information! :)

  4. It's useful to know what the mathematics is used for and whether it serves a purpose or not. I can't help but feel though that if we didn't teach about this, some students may miss out on the opportunity to apply it appropriately. After all, we don't know what jobs our students will have and what maths they'll need. There are aspects of maths that are taught where we can't always give a great reason for it, but you just never know how useful it could be in the future.

  5. Thanks for your comment. I absolutely agree. I studied statistics at uni and assumed that I would never apply most of the stuff I learnt. But then I got a job at the Bank of England after uni and worked on a project involving sampling and testing banknotes - hence I had to use confidence intervals etc. You never know where these statistical tools will be used.

    I'm also interested in 'real world' uses of box and whisker plots - another blog post to come!

  6. Readers of this post may also be interested in this activity: http://mathsclass.net/comments/stem-and-leaf-plots...-an-activity

  7. And... display your students' birthdays using a stem and leaf diagram! http://mrcollinsmaths.blogspot.co.uk/2012/12/stem-leaf-birthday-diagram.html

  8. Used the 'Beat the Teacher' idea very successfully with Y13 today (International School doing S1 with no prior knowledge of S&L diagrams) They are now totally convinced that I am a witch..

    1. Excellent! A nice bit of mathematical magic.

  9. Thank you for this. I recently used the creation of stem and leaf plots to teach how to analyze data. Students had answered 7 questions anonymously without any labels. Answers were distributed to groups and stem and leaf plots were quickly made. As a class we discussed which plot was our height in inches and which couldn't be. Which plot shows the age of an adult in our home and which doesn't? The discussion and math analysis was beautiful!