Mr Barton’s Maths Notes Stats and Probability 6. Stem and Leaf Diagrams www.mrbartonmaths.com

6. Stem and Leaf Diagrams Why do we bother with Statistical Diagrams? The answer to this question is similar to the one for: “why do we bother working out averages and measures of spread?”. We live in a world jam-packed full of statistics, and if we were forced to look at all the facts and figures in their raw, untreated form, not only would we probably not be able to make any sense out of them, but there is also a very good chance our heads would explode. Statistical Diagrams – if they are done properly - present those figures in a clear, concise, visually pleasing way, allowing us to make some sense out of the figures, summarise them, and compare them to other sets of data. 1. What are Stem and Leaf Diagrams To be honest, Stem and Leaf Diagrams are just a fancy way of listing a fairly large group of numbers in order They are seen as a quicker, more convenient, and ultimately more useful way of presenting data than just a long list of numbers. An example of a typical Stem and Leaf Diagram is on the right stem leaves

2. Constructing a Stem and Leaf Diagram Big Example Here are the times (in minutes) that it takes Mr Barton to actually get out of bed after his alarm has sounded on a Monday morning: 12 6 20 24 52 41 3 35 55 32 11 13 2 25 38 39 41 52 13 59 18 22 29 35 Use the data to construct a Stem and Leaf Diagram, and then calculate the Median and Inter-Quartile Range 2. Constructing a Stem and Leaf Diagram 1. Decide on your stems – these are the digits which go down the left hand side of your diagram. You should choose them in a way so that you have between 4 and 10 groups, and so that each of your leaves is only one digit! 2. Go through your data, in the order in which it is written, and add it to the correct stem on your diagram. I would mark each piece of data once it has been entered, so you don’t loose your spot! 3. When completed, this is your un-ordered stem and leaf diagram 4. Now draw yourself another stem and leaf diagram, but this time put the leaves in order! Note: Everyone seems to want to jump straight to the ordered stem and leaf diagram, but I promise you this way is quicker and a lot safer in terms of mistakes!

Our Example 1. For our stems, we only need the first digit of each piece of data, and I think 6 groups should do us! Note: to make sure we can enter the single digit pieces of data, we must make our first stem start with 0 1 2 3 4 5 2. Next we begin to go through our data, creating them leaves of our diagram, marking off each piece of data as we use it… 12 6 20 24 52 41 3 35 55 32 11 13 2 25 38 39 41 52 13 59 18 22 29 35 0 6 0 6 3 2 1 2 3. Continuing like this eventually gives us our un-ordered stem and leaf diagram 1 2 1 3 3 8 2 0 2 0 4 5 2 9 3 3 5 2 8 9 5 4 4 1 1 5 5 2 5 2 9

Things to Notice: For numbers like 20, we must place a 0 as our leaf Single digit numbers are placed on the top stem If we come across a number that we have already recorded, we must record it again! 4. Having made sure that we have 24 digits as the leaves of our diagram (there are 24 pieces of data!), we can now very quickly change our unordered diagram into an ordered one by placing the leaves on each stem… in order! 0 6 3 2 0 2 3 6 1 2 1 3 3 8 1 1 2 3 3 8 Key 2 0 = 20 2 0 4 5 2 9 2 0 2 4 5 9 3 5 2 8 9 5 3 2 5 5 8 9 4 1 1 4 1 1 5 2 5 2 9 5 2 2 5 9 5. And the final thing we must remember to add to our diagram is a KEY! You must let anyone who looks at your diagram know exactly what each of the leaves stands for… so in this case, I have chosen the 2 and the 0 from the 3rd row, and just explained that this is actually 20!

3. Finding the Median and Inter-Quartile Range from a Stem and Leaf Diagram Remember: our Stem and Leaf Diagram is just a group of numbers, written out in order… and so we don’t have to learn any different skills to find the median and inter-quartile range! (a) Finding the Median It’s the usual thing… the median is the middle number… and if there is an even amount of numbers, then you will have two numbers in the middle. Draw a box around the number/numbers you think are in the middle, and make sure you have the same amount of numbers on either side! 0 2 3 6 Check: There are 11 numbers to the left of our box, and 11 numbers to the right! Our median is half way between our two numbers in the box, and so is… 27! So many people would put 7… but remember what the leaves actually stand for! 1 1 2 3 3 8 2 0 2 4 5 9 3 2 5 5 8 9 4 1 1 5 2 2 5 9

(b) Finding the Inter-Quartile Range Remember: Inter-Quartile Range = Upper Quartile – Lower Quartile The way I do this is to think of the Lower Quartile as being the median of the lower half of numbers… and the Upper Quartile as the median of the upper half of numbers. And I just find these values the same old way… using my boxes, and making sure there are the same amount of numbers on either side! 0 2 3 6 Lower Quartile = 13 (5 numbers either side in the lower half) Upper Quartile = 40 (5 numbers either side in the upper half) 1 1 2 3 3 8 2 0 2 4 5 9 3 2 5 5 8 9 4 1 1 5 2 2 5 9 Inter-Quartile Range = 40 – 13 = 27

4. What’s GOOD about Stem and Leaf Diagrams? Well, the major advantage over things like bar charts and histograms, is that no information is lost – the stem and leaf diagram keeps and allows you to see each original piece of data It is also quite an effective way of ordering and displaying relatively small sets of data 5. What’s BAD about Stem and Leaf Diagrams? Well, it’s quite time consuming, and impractical for large data sets. Imagine how long it would take to sort over 300 pieces of data, and how complicated the final diagram would look!

Good luck with your revision!