informal statistical inference: years 10 to 12
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Informal statistical inference: Years 10 to 12. Maxine Pfannkuch and Chris Wild The University of Auckland. Outline. A story from my research Brief introduction to comparative reasoning and informal inference Experiencing sampling variability Two Activities. A story from my research. - PowerPoint PPT PresentationTRANSCRIPT
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Informal statistical inference: Years 10 to 12
Maxine Pfannkuch and Chris WildThe University of Auckland
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Outline• A story from my research
• Brief introduction to comparative reasoning and informal inference
• Experiencing sampling variability
• Two Activities
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A story from my research• Year 11: Drawing conclusions from the comparison of
two distributions (e.g., Which battery is better, A or B?) (Pfannkuch, 2005, 2006, 2007)
• How was the teacher reasoning from the plots?
• Was the teacher drawing an inference about the population from the sample?
• On what grounds were students drawing conclusions?
• Students - no concepts of sample, population, sampling variability
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Solving these problems• Comparative reasoning
– Elements of reasoning (Pfannkuch, 2006; 2007)– Differentiating between descriptive and inferential
thoughts and contextual (Pfannkuch et al., 2008)• Sampling reasoning
– Web applications and by hand (Pfannkuch, 2008)– Automated web applications (Wild et al., 2008)
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Boy
Girl
1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2
F o o t L e n g t h _ c m
C o l l e c t i o n 1
D o t P l o t
A g e " Y e a r 1 3 "=( )
How would we like students to reason?
Boy
Girl
1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2
F o o t L e n g t h _ c m
C o l l e c t i o n 1
B o x P l o t
A g e " Y e a r 1 3 "=( )
Investigative QuestionDo 13 year-old NZ boys tend to have bigger right foot lengths than 13 year-old NZ girls?
Two randomSamples fromCensusAtSchool
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Overall visual non-numerical comparisons
Boy
Girl
1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2
F o o t L e n g t h _ c m
C o l l e c t i o n 1
D o t P l o t
A g e " Y e a r 1 3 "=( )
Boy
Girl
1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2
F o o t L e n g t h _ c m
C o l l e c t i o n 1
B o x P l o t
A g e " Y e a r 1 3 "=( )
• Overlap• Shift • Unusual features
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Then consider 8 elements: spread, shape, summary statistics, explanatory etc. (Pfannkuch, 2006; 2007)
See “A Teacher’s Guide” attached
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Shape
Boy
Girl
1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2
F o o t L e n g t h _ c m
C o l l e c t i o n 1
D o t P l o t
A g e " Y e a r 1 3 "=( )
Boy
Girl
1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2
F o o t L e n g t h _ c m
C o l l e c t i o n 1
B o x P l o t
A g e " Y e a r 1 3 "=( )
I notice (descriptive): :• the sample distribution for the boys’ foot lengths is roughly
symmetrical with a mound around 24 to 27cm, i.e., unimodal• the sample distribution for the girls’ foot lengths shows a large
mound around 22 to 24 cm and a hint of a small mound around 27cm, i.e., a hint of bimodality
I wonder (inferential): :• if boys’ and girls’ foot length distributions back in the two
populations are roughly symmetric and unimodal. – I expect so for a body measurement such as foot length for both girls and
boys. (contextual)
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Spread
Boy
Girl
1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2
F o o t L e n g t h _ c m
C o l l e c t i o n 1
D o t P l o t
A g e " Y e a r 1 3 "=( )
Boy
Girl
1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2
F o o t L e n g t h _ c m
C o l l e c t i o n 1
B o x P l o t
A g e " Y e a r 1 3 "=( )
I notice (descriptive):• the middle 50% of the boys have a right foot measuring
between 24 and 27cm (IQR = 3cm) whereas the middle 50% of the girls are between 22 and 25cm (IQR = 3cm). – This means that the foot lengths for these boys vary by about the
same amount as these girls’ do. I wonder (inferential): :• if boys’ and girls’ foot length distributions back in the two
populations have similar variability. – I expect so. (contextual)
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Research Lunchtime task -sampling variability
n=30n=10 n=50
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“What I see is not quite the way it really is”
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Some student quotes• I found it interesting when you only need a sample to
work out problems from a large population. It’s just really weird and cool at the same time that you just take any sample that is representative and it gives you near enough accurate answers.
• Comparing distributions with different sized sample graphs because the variation of the graph changing each time when the sample was small but having less variation when the sample size got larger.
• The part where we compares and saw all the different medians from different sample sizes. Coz it was really interesting to see it change. Its buzzy…
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How were students “making a call”?
• Using the MEDIANS (Pfannkuch, 2008)
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Experiencing sampling variability• Hands-on • Computer generated simulations• Keep a history of the variability
– Build up concepts of sample, population, sample size effect, distribution, confidence interval etc.
• Making a call: Which is bigger?– From class activities and simulations get
students to create “decision rules” to make a claim or informal inference
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Whenever I see …
I remember …
Mine could even be like this …
Or even this …
I must take this uncertaintyabout where it really is
into account when I make comparisons!
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Kiwikapers Activities (Source: Pip Arnold)
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DataCards
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Investigation Question• I wonder if …• North Island Brown kiwi tend to be heavier
than South Island Tokoeka kiwi?
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ACTIVITY ONE - COMPARATIVE REASONING
In groups of 2
– Sample one species (n=30) from kiwi population – Do Datacard Plot on sheet provided (round
weights to 1 d.p)
– Look at SHAPE - discuss your descriptive, inferential & contextual thoughts (see teacher guide)
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ACTIVITY TWO - LEARNING TO MAKE A CALL
• Remove datacards and draw a Dot plot & Box plot• Transfer box plot onto transparency• Learning to “Make a call”
– As a class put NI Browns • Side-by-side • On top of each other
– View automated pdf “movie”– Create a decision rule for making a call
• Test on comparison of two samples of NI Browns• Test on comparison of NI Browns and Tokoekas• Test on comaprison of NI Browns an Great Spotteds
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Automated Phase• See Chris’s “movies” of pure effect of
sampling 1samp.pdf & 2samp.pdf– (last frame of each below)
NI Brown
NI Brown
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Developing a method to “make a call”
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Informal statistical inference• Building statistical concepts - provide a pathway
through the curriculum for students to understand the logic of inference
• For more information and “movies” see: www.censusatschool.org.nz/2008/informal-inference/