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• 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

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)

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

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

Then consider 8 elements: spread, shape, summary statistics, explanatory etc. (Pfannkuch, 2006; 2007)

See “A Teacher’s Guide” attached

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)

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)

Research Lunchtime task -sampling variability

n=30n=10 n=50

“What I see is not quite the way it really is”

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…

How were students “making a call”?

• Using the MEDIANS (Pfannkuch, 2008)

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

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!

Kiwikapers Activities (Source: Pip Arnold)

DataCards

Investigation Question

• I wonder if …

• North Island Brown kiwi tend to be heavier than South Island Tokoeka kiwi?

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)

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

Automated Phase• See Chris’s “movies” of pure effect of

sampling 1samp.pdf & 2samp.pdf– (last frame of each below)

NI Brown

NI Brown

Developing a method to “make a call”

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/