the main vine, not the loose one. Kei hopu tōu ringa kei te aka tāepa, engari
kia mau te aka matua
Thoughts on Teaching and Learning of Mathematics
Lesson #6 Probability and Statistics
- to connect probability
with all parts of mathematics - to know how
statistics makes sense of very difficult problems
- to understand "random" as a concept
Ahh.. chance! Perhance! What a wonderfully misleading, often
abused but valuable concept. Einstein said "God does not play with
dice!" challenging claims that "quantum chance" rules the world.
Currently accepted quantum theories abound with probabilistic
ideas and deductions suggesting God might well have "played with
dice'. Experimental observation supports the randomness within the
Here is the summation to infinity, over all space and time, of the
square of the probability density of a particle in one dimension.
I admire the simplicity and complexity in this statement. How
something so obvious (it has to be somewhere) is captured by
something so complex and all to do with probability.
The Deep Understanding of Probability Being enabled to estimate, measure and use chance events
to make sense of the world around us and to solve problems.
Probability is a natural part of our language and the concepts are
components of all human endeavour. We use expressions like "on the
balance of probability", "beyond all reasonable doubt",
"possible", "probable", "almost certain", "always", "never", "in
all likelihood", "hardly ever", "sometimes" to name just a few.
The list is large and when asking people of the numeric
probability that might be assigned to each of these expressions
the variation is wide. I asked a court judge of what probability
might he assign to "beyond all reasonable doubt". His answer was
anything over 50%. That frightened me. I had my mind set around
80% or more.
Task 1 for Years 2 to Year 100 Make a list of all the probability terms you can think of,
encounter or might remember, and then order them from least to
most likely. Try and assign a number to describe the
probability. Compare. Warning!...there will be disagreement.
I have a task for Y 7 to 10 all prepared ready for use on the nzmaths.co.nz
website called https://nzmaths.co.nz/resource/number-probability-1
For young people, playing with dice, gaming, and learning
the language of probability is vital. Roll the dice! Play cards.
Play Monopoly. Normalise and use "Paper, Scissors, Rock to sort
issues", and understand the unpredictability of randomness.
Destroy the misconception that if I throw everything but a 6 on 20
or so throws teh next must be a six. In random, we always
underestimate the long run.
M Steele showed me this experiment with great glee: Imagine you
are a coin and write down the result of throwing yourself 100
times, ten rows of ten outcomes. Now do the experiment and record
your outcomes of tossing a coin 100 times. Do not mark which is
which. I will come around and tell you which is your guess and
which is your coin toss result. It is very easy, and fast, to
tell. I challenge you to repeat this task and see if you can sort
how obvious it is to distinguish the difference and be a
"mindreader" in your math class.
We need to learn to use probability to select a "random sample"
and know why this is a good way to sample. We must learn to use
probabilistic language. We need to know what number might be
assigned to "no chance" and "will always happen".
To establish 1/6 and 1/10 and how different these two
probabilities are use a six sided die (plural dice) and a 10 sided
0-9 die. The task for students is to guess a number and then try
and toss that number. It happens quite often on a six sided die
but on the 10 sided is decidedly harder. I made up a sheet for my
local GP to use with heart risk patients who have little
understanding of the improvement in outcomes when moving from a
1/6 (17%) risk to a 1/10(10%) risk. It does not sound much but I
know which group I want to be in. This is a valuable exercise, try
Interwoven in the ideas underpinning probability are proportional
thinking and fractions. These are difficult concepts and it is
wise to stick to the language development and playing games
until fraction knowledge is secure. We do use fractions to
represent probability and we operate on these fractions in normal
ways as well but all probability is between 0 and, unlike
The PPDAC statistical inquiry cycle or be a Data
Detective applies to probability, of course. See below for
more details. P - Ask a Question - "I wonder I can test a coin to see if
it is fair?" P- Make a Plan - I will toss the coin 200 times and make a
pie graph of the results. If it is fair it will be even, or
symmetrical. D - Do the experiment - I toss the coin 200 times. A - Analyse the outcomes - I draw a pie graph of the
results. I can also put the results into iNZight and get
significance bars. C - Make sense and a conclusion - This coin is pretty close
to being fair. The two parts are almost the same size. Heads show
95, Tails 105. The iNZight error bars overlap so no difference can
be supported. The coin appears to be fair...answer the question!
Try figuring out the probability of randomly breaking a stick in
two places and being able to make a triangle with the three
pieces. This is not something that is an obvious calculation and
an inquiry is needed.
What is the likelihood of Kane Williamson being bowled out for a
duck in the next cricket match? What is the likelihood of
President Trump seeing out his first presidential term?
Playing games using dice like Yahzee, or my variation Numahtzee,
Monopoly, Battleships, Dice Games, Black
Jack, Paper-Scissors-Rock are all essential "hands-on" learning.
The folly of gambling can be learned at this level. More advanced
students can simulate LOTTO draws and make up sample tickets using
XCEL to learn how ridiculous winning anything can be.
The obvious is not always apparent in probability and a wide
experience of problems is important to experience this bizarre
idea. In the "Breaking a Stick" problem mentioned above the
geometrical solution is easiest to comprehend. In the Birthday
Problem of "How many people do you need in a group to have a 50%
or better chance that two of the group share a common birthday" is
easy when you ask the opposite.
The Deep Understanding of Statistics Being enabled to collect, use and use data to make
sense of the world around us and to solve problems.
Statistics has undergone major change in NZ and I
acknowledgement the NZ Statistics Association for their collective
wisdom, guidance and hard work. It took many meetings and emails
by many people to settle on ways of reinventing statistics for the
modern world during 1998 to 2008 period when the new NZC was being
Tukey influenced the future with his work and created the
dot plot and the box and whisker diagram now so common in NZ
schools. The computer has brought about a revolution in the
management and analysis of data, it is up to us to be good
interpreters and make valid conclusions. Chris Wild's team in
Auckland University caused the Data Explorer on the Census@Schools
website and the easy to use iNZight Statistics Software.
The Statistical Inquiry Cycle or what is now called PPDAC guides
all investigations in statistics and probability. The past
fascination with mean, median and mode has become "measures of
middle". Now "the eye's have it" and "shape, middle, spread and
oddities of dot plots and box and whisker graphs rule the
reasoning. Add "WWW" for each of these four features as "What am I
talking about", "Where is it" and "What does it mean" and we have
a worthy guide to become a Data Detective.
Statistical Questions. My colleague Pip Arnold completed a PhD on
this topic. Get the question right and you will be much more
likely to be able to answer it. One of the three types of stats
questions is "comparative". I use the frame of "Variable"; "2
Groups"; "> or <", "Population" in explaining and guiding
students to a good question.
Task In the question "I wonder if in NZ Year 9 boys carry
heavier bags than Year 9 girls?" identify all parts using my
PPDAC. This is actually a version of the inquiry cycle
commonly used in science. P = Problem, P = Plan, D = Data, A =
Analysis and C = Conclusion. A statistical inquiry begins with a
query or Problem. "I wonder if in NZ Year 9 boys carry
heavier bags than Year 9 girls?" I might wonder this because I
have a heavy bag usually as a Year 9 student and I see that most
boys are the same. There might be a case here for getting some
storage at school for some of my gear sorted out! My Plan
is to go to the last Census@Schools
website survey and get random sample Data of 30 boys
and 30 girls and the bag weights they reported. Then I will make a
dot plot of these two groups and a box and whisker graph using the
data Explorer. My results are:-
Year 6/7/8 - The boys box is more to the right or higher than the
girls so I think that typically boys carry heavier bags.
Year 9/10 - Boys bags are typically heavier since half of the boys
bags are heavier than 3/4 of the girls bags. The median is 5kg and
spread upwards to over 9 kg. The girls upper quartile is about
4.5kg and spread lower or lighter to about 1kg.
Year 11 - The gap between the medians is large compared to the
overall spread of the two boxes. The difference in the medians is
5-3.2 = 1.8kg and teh overall middle spread is 5.5 - 2.2 = 3.3kg.
This means the higher boys median is likely to be reflected in the
whole population so we can say "yes; boy's bags are likely to be
heavier than girl's bags".
Year 12 becomes more technical again as use of the informal
interval ± 1.5*IQR/√n is established.
Year 13 is about "bootstrapping" and testing for difference.
Means, medians and modes as measures of the middle have been
replaced by "typical". The Census@Schools
website is the GOTO site for all information. The "dot plot" and
the "box and whisker plot" of sampled data is everything for Y7 to
11. The concepts of survey and sampling prioritised.
software is found by asking the internet "iNZight software".
Other online software like NZGRAPHER work on
iPads and Cellphones. INZ Lite is a smart 21st C software
WARNING! Do not let younger
students make use of software to make dot plots and box and
whisker graphs before they can tell you how to make these things
by hand and explain all features. Show them too soon and they will
never draw one again and miss out on that "learn by doing"
experience. "Learning travels up your arms"; well, I told many
students this. Students should not be introduced to the box and
whisker until they are multiplicative and appreciate that 1/4 of
the data is in each part despite the lines and areas being
Here is a diagnostic test to help know if students are ready to
move on. TASK Which section of this box and whisker container the most
Everyone should study statistics for as long as they can and all
pathways for all students from school should include Year 12 or 13
Statistics knowledge. Today's world has a data infestation and
"making sense of data" is now more vital than ever. Computers can
deal with millions of data points and every time you log on to the
internet or use a credit card you are being measured. Know what is
happening. Go to the GeoNet
website or EBOP
buoy website to see how much data is being collected for
scientific purposes. How much data is Nasa collecting from space
A good grasp of the language of statistics is vital. I describe
statistical language as "floppy" as it includes phrases like
"tends to", "the data suggests" rather than exactness of
mathematics like "x = 3" and "length of the pipe is 3.4±.1cm". We
can not say that the mean weight of a trout in Lake Taupo is
2.34kg because it is a population statistic that we can never
know. We can only use a sample to get some interval in which the
mean or median of the population is likely to be. That is
Statistics and probability walk hand in hand with the concept
random. Random sampling causes all those questions we did not ask
in the data to be evenly likely to turn up in any sample and so
eliminate bias and false inference. The size of a random sample is
typically 30 to 50 for a school situation and I made a .ppt to
help answer this question. You can use this at any level and
should get a class to take random samples of different sizes as an
exercise. Best done on computer!
Hence Lesson #6
Making sense of data is statistics. The PPDAC cycle is vital
with hands-on sampling to learn about variation and what you can
say about a population. This is an excellent subject to help
students learn to write to convey meaning. Making sense of
probability and using random, chance, outcomes is just as vital.
Both areas of knowledge applied to problems becomes the
What are the key competencies opportunities that present
themselves for learning probability and statistics.
How are these part of your main vine of learning?
List all the probability words you can and order these from
never to always.
Investigate and master one computer based stats programme, such
as iNZight or NZGrapher. There are others.
Look up Taupo Trout Resources on my website and use these in
junior programmes. Find Kiwi Kapers on Census at Schools.
What are the three types of statistical questions.
This is to help look around my
pages. I have tried to make it consistent in all chapters.