People and their pets
Lena wants to find out if homes in Ashvale usually have more pets than homes in Moananui.
Y8 (10/2010)  
a)  D  moderate 
b) 
Any 1 of:
[Same] Because they have an even spread of pets throughout the numbers.
[Same] Because … they are spread out fairly evenly. [Same] Because both of the graphs are nearly the same. [Same] Because you can see. [Cannot tell which town has more] because they look pretty even to me.
[No students in the sample did this.]
[Same] Because there is only 1 pet difference between the two towns.
[Same] Because they are one only out from each other. [bold added for emphasis] [Same] Because one is 65 (sic) and the other is 69 (sic) so than mean they nearly have the same amount.
[Same] I added up the amount of pets in each town and divided it by the number of homes. The average for each was about 3. Although Moananui had
a fraction more it was about the same.

very difficult 
This resource is about comparing two statistical graphs which show the distribution of pets in two different towns. The key idea is to recognise that while the graphs differ, their overall shape (distribution) is approximately the same.
Part b) is "very difficult' partly because many students look at detailed features of each graph (such as the total number of pets in each town) and draw conclusions based upon them. This is a more numerical/algebraic approach, where students compare exact numbers or graphical features. Statistical graphics incorporate the idea of variation where small differences are of far less importance than the overall picture. Many students (and teachers too most probably) do not have this mindset that things do not have to be exactly the same to still be the same statistically.
 Makes invalid statements about the graph (e.g., there are 21 pets in each town).
 Gives a valid statement comparing individual features of the graph rather than a holistic statistical comparison (e.g., they have different numbers of pets (63 vs.64), different modes, different numbers at the maximum or minimums, the same range, etc.)
 Attempts an argument about the distribution of the data.
 Identifies that the distributions (graphs) are very similar and that the differences are of little importance.
Common incorrect responses  
b) 
Uses the number of homes (dots  21) on each graph to make a comparison
Uses the number of homes to make a comparison, ignoring homes with no pets (16 vs 18)
Uses the number of categories with no dots to make a comparison
[Same] They both have two homes with no pets.

Common nonstatistical responses  
b) 
Uses specific features of the graph Uses the most or least common category to make a comparison, e.g.,
Uses just the minimum or maximum to make a comparison, e.g.,
Sees a small difference in the number of pets in each sample as indicating that town has more pets (often 63 vs 64) – i.e., ignores small statistical variation, e.g.,
Miscounts the number of pets as being equal (often 63 in each town) , e.g.,

b) 
Attempts at a distributional or statistical statement, e.g.,

Uses the number of homes (dots) on each graph to make a comparison or
Uses the number of homes (dots) to make a comparison, ignoring homes with no pets or
Uses the number of categories with no dots to make a comparison
Some of these students see each dot as representing 1 pet. They may need to construct some dot plots of their own. Use the ARB resources Marine fish or Waiting time.
Other students see that each dot represents the number of homes in each town who were asked how many pets they had. Ask them "What if the 21 homes in Moananui each have between 5 and 20 pets?"
Uses specific features of the graph (e.g., mode, maximum, minimum, total number of pets etc.)
These students need to make the paradigm shift between numerical, algebraic thinking to statistical thinking. The former, referred to as deterministic thinking, looks at exact outcomes, whereas statistical thinking involves variation. This is a big shift. Lots of exposure to statistical investigations is essential. They could also be exposed to numbers of probability investigations. Each of these will help them to view things more statistically.
Click on the link for the probability concept map for a discussion of variation and distribution.
Attempts a distributional or statistical statement
These students have crossed or are crossing from exact (deterministic) thinking to statistical thinking. Get these students to expound upon their statements. This may indicate if statistical or deterministic thinking predominates for them. Encourage visual responses, and discourage those which calculate the exact number of pets. Students often fall back on exact methods as the most reliable, for example the student who said "[Moananui] because you can see that Moananui homes have more pets by just looking at it, but I still counted just to check my answer."
Figure it out
For resources that look at the distribution (shape) of graphs visually:
 Population pyramid (Statistics, Book 2 L4, p. 10); and
 Uniform changes (Statistics L34, p. 2), particularly the first graph.