Example A: optical illusions

There are two variables in these questions: which picture is seen first and gender of the person looking at the picture(s). Which picture is seen first is of interest on its own. The second question is asking if there is a connection or association between this variable and gender of observer.

Data for this investigation is straightforward and quick to collect to obtain many observations and can be collected from fellow students and from family and friends.
A brief explanation should be given to each subject (for example, 'do you see an old or young man in this picture?') but their first response is what should be noted.

Example B: clasping hands and folding arms

The questions in this example involve at least three categorical variables for each
person questioned:

Which thumb is on top when the subject’s hands are clasped?

What is the gender of the subject?

Which arm is on top when the subject’s arms are folded?

An investigation to explore these questions might include other variables (such as which hand is dominant) and could include design issues (such as which action is performed first and should this be randomised), but the three named variables above are the variables of the original questions. The motivation for these questions comes from observation that most people tend to do these actions the same way each time and it can be surprisingly difficult to clasp hands or fold arms so that the other thumb or arm is on top. There may be a genetic link in these simple actions. See, for example, https://humangenetics.suite101.com/article.cfm/dominant_human_genetic_traits which includes the following statement:

'Clasp your hands together (without thinking about it!). Most people place their left thumb on top of their right and this happens to be the dominant phenotype. Now, for fun, try clasping your hands so that the opposite thumb is on top. Feels strange and unnatural, doesn’t it?'

It should be noted (see General statistical notes below) that many observations need to be collected to assess the statement in the quote above. The data are straightforward and quick to collect for many students, and if these questions are investigated by students collecting their own data, as many observations as possible should be collected.

The questions in this example relate to two variables on their own, namely, which thumb is on top and which arm is on top, and also to possible connections or associations between these variables themselves, and between each of these variables and gender.

Example C: Cats and dogs

Three questions have been expressed in this example in the introduction:

The first is asking about people’s preferences between cats and dogs for pets, and so asking each person in a sample of people about one variable.

The second is asking if this preference is different for boys and girls, so is asking if there is any connection or association between preference and gender.

The third is asking about another variable, namely the source of pets.

However it is asking about the source of both cats and dogs so the source needs to be considered in association with the type of pet.

The first question is referring to people in general but the second is referring to boys and girls. This second question could be generalised to males and females and refer to people in general, or we could decide we are going to focus on young people − on boys and girls. The third question is referring to pets rather than people, so it could be included in a survey of boys and girls, but it would refer only to the pets of families with children and not to cats and dogs as pets of people in general. There is no problem with this provided it is made clear in any reporting. That is, all reports of data investigations must include good information on how the data were collected or obtained, so that any recipient of the report has the information that allows interpretation of who or what the data are representative.

Hence a data investigation could be carried out by a survey of students in the school, recording their gender, their preference for cats or dogs as pets (cat, dog, neither, either), and, if they have a cat or a dog, where did they get it.

Alternatively, or in addition, the possibility of the existence of relevant data collected by others could be investigated − that is, the existence of relevant secondary data could be investigated. In particular the third question might be one for which there are data available. In this case, the data investigation could consist of two parts: a survey of students, and consideration of data collected by others. It must be emphasized that there would be two sets of data in such an investigation, obtained under different conditions and possibly referring to different general situations.

Example D: daylight saving

Although students can collect opinions on the question of daylight saving, this question lends itself to investigating data that have been collected. What surveys have been done and what data are available?

Example E: the effect of the Slip, Slop, SlAp messages

As with example D, this question is one for which there are likely to be secondary sources of data. Students may like to ask their classmates and other students in the school about what they do to protect their skin, but the possible responses are not simple and there is much that can be learned from discovering how such questions are investigated by authorities. Asking what students do may be more useful as background to investigating secondary sources of data and in discussion after such data are found and discussed.

Example F: cycle/pedestrian paths

This example involves two categorical variables, transport type and gender of path user, although an investigation might include more information. Type of transport is of interest by itself, and its possible association with gender is the other question of interest. Students could collect data to investigate these questions, possibly on different paths for further interest, but there are some data collection issues to be considered (see next section). Alternatively, there may be secondary data sources available. However secondary sources of data usually provide only data summaries, and to consider the question of association, such sources must provide summaries that specifically address this question.

Example A: optical illusions

Apart from being consistent in giving a brief explanation to each subject (for example, 'Do you see an old or young man in this picture?') and recording their first response, there are few problems in collecting these data. The first few rows of the raw recording sheet would look like:

Example B: clasping hands and folding arms

If it is decided to ask each person to clasp their hands first and then fold their arms, there are just three variables to be recorded for each person surveyed: gender, top thumb in clasped hands (R/L), and top arm in folded arms (R/L).

If it is decided to randomise the order in which subjects are asked to clasp hands and fold arms, then the order of the actions should also be recorded in case it turns out to be important. There is no way of knowing if it is important or not without carrying out an investigation into this question!

If the same order is used, the first few rows of the recording sheet for the raw data would look like the following, with each row corresponding to a person surveyed, and the three columns named by the three variables, and with observations recorded for each subject:

Note that the column with identifiers of subjects is not necessary, but in practice may help in avoiding duplication and in checking data for mistakes or problems.

If the order of the actions is randomised (for example, by tossing a coin), another column is needed, headed 'first action' with entries of 'clasping hands' or 'folding arms', perhaps abbreviated to C and F.

Example C: cats and dogs

Boys and girls and their preferences between cats and dogs as pets.

Asking fellow students their preferences between cats and dogs as pets and recording if they are boy or girl is fairly straightforward once the categories for responses are decided − for example, are we going to allow the response of 'either' or 'I like both' or are we going to ask that the choice be made.

The recording sheet for this is simple and the first few rows would look like the following table. Notice that recording the names is not necessary for the data set but can be useful in avoiding duplication.

Sources of dogs and cats acquired as pets

As described above, students with dogs and/or cats as pets could be asked where did their pet come from, but this information refers only to pets of families with primary school age children. Also, if a family has more than one of either dog or cat or both, the source of one pet in a family is likely to be affected by the source of another pet.

A report on the results of a survey conducted in Victoria in 2004 is available here

The specific purpose of this survey was to:

• Determine the sources of acquisition of dogs and cats;
• Determine the age of pets at time of acquisition; and
• Identify the existence of health problems in cats and dogs when acquired

The questionnaire was designed in consultation with organisations involved in animal welfare, and the survey was then distributed, via a number of different organisations, to members of the general community to complete. Most of the surveys (76.1%) were distributed through veterinary clinics to animal owners; the next major source (11.4%) was through local council events or council reception desks. We will focus here on the data reporting sources of acquisition of dogs and cats.

The raw data sheet for sources of dogs and cats would have a column named 'type of pet' (cat or dog) and a column named 'source'. Each row would correspond to a single pet − either a dog or a cat. Note that the different possible categories for the observations on 'source' would need thought and possibly some researching.

The report does not consider if there are multiple responses from pet-owners. That is, although the data are representative of a much wider group than of families with primary school age children, the report does not comment on the second consideration above (namely, the extent to which multiple responses from a pet-owner relate to each other).

Example D: why does Queensland not have daylight saving in summer?

A referendum was held on 22nd February in 1992 in Queensland after a three-year trial. Following petitions asking for daylight-saving to be introduced or another referendum held, the Queensland government in 2006 commissioned research into the issues. The report from this can be seen here.

The results of the 1992 referendum by state electoral seat can be found on the internet and were summarised in 2010 here.

The problem for Queensland is that the further south and east you went the higher the ‘Yes’ vote in 1992; the further north and west the higher the ‘No’ vote in 1992.

In gathering data such as these, there are two variables to be observed for each person surveyed: location and opinion. Hence a raw data recording sheet would have two columns, with each row corresponding to a person. Of course only summary data would be reported.

Example E: the Slip, Slop, Slap message

The challenge in this general issue or topic is to investigate people’s behaviour and attitudes to sun protection, including whether people get sunburnt or attempt to tan. The difficulties in asking questions relating to people’s behaviour are that behaviour tends to vary from day to day and to depend on many individual influences. Interpretations of questions can also cause problems. For example, a question such as 'Do you wear a hat when outside in summer?' is too vague for people to answer in a way that is consistent across everyone.

The National Sun Protection Survey of 2006-2007 was the second such survey; the first was conducted in 2003-2004. The study is funded by the Cancer Council Australia and the Australian Government through Cancer Australia. The 2006-2007 survey reached respondents through phone interviews conducted on Monday and Tuesday evenings during summer. The interviews focussed on weekend behaviour in summer, and also recorded whether the person was an adolescent or an adult, and in which state they lived. These questions give data for three categorical variables for each person: sun protection behaviour on the weekend; age group; and location (state where they live). Some of the results are considered below.

Example F: cycle/pedestrian paths

Planning a data investigation on use of a cycle/pedestrian path involves quite a lot of decision-making. Apart from deciding on location and time (day of week, time of day), there is also the vexing question of whether to consider individuals or groups as the subjects − because usually (but not always) members of a group are using the same mode of transport. As always, the most important aspects of such questions are discussion, decision and description of how, when and where the data are collected.

If gender is to be recorded, then the data must be observed per user, whether in groups or not.

Example A: optical illusions

203 students were shown the optical illusion picture and asked whether they could see an old or a young man. Below is a 2 × 2 table summarising the 203 responses by whether the old or young man was seen first and whether the student was a boy or girl. Hence overall there are four categories: boy and see old man; boy and see young man; girl and see old man; girl and see young man. The response of each student surveyed appears in one and only one of these four categories. These categories are called joint categories because they are formed by considering together (or jointly) the gender of the student and which picture they saw first. The total numbers of each of the separate categories − boy, girl for gender, and old man, young man for picture seen − are also often given in such tables.

Table of picture seen first in optical illusion and gender of subject

The column graph below represents the frequencies given in this table. It is called a side-by-side column graph because for each of old and young man picture, it gives the frequencies of boys and girls who saw this picture first, side-by-side.

graph of picture seen and gender

Notice that the observations can be split first by either of the variables. In this example, either of the above graphs clearly show the main features of the data, namely that for both boys and girls in this group, the young man was more often seen first in the optical illusion picture, and that in this group, the girls tended to see the young man rather than the old more often than the boys.

However, the second graph above is the more appropriate, as it better represents the data collection, namely, boys and girls were shown the picture and asked which did they see (old man or young man). The first graph tends to represent looking at those who saw the old man first and asking/classifying them as boy or girl.

If there is association between gender and the picture seen first, we would expect the pattern of frequencies over old and young to be different for boys and girls. In these data, there is an indication that for boys, there’s not much difference between seeing old or young first, but the data indicate that girls may tend to be more likely to see the young man than the old. But, at this stage of development and with only 111 girls asked, all we can say is that in this group the girls tended to see the young man more often than the old.

Example B: Clasping hands and folding arms

The same group of 203 students were also asked to clasp their hands and fold their arms. Which thumb was on top when the hands were clasped and which arm was on top when the arms were folded were recorded along with gender. The two tables below are each
2 × 2 tables: the first looks at which thumb is on top in clasping hands; the second looks at which arm is on top in folding arms.

In this group, we see that there are more with left thumbs on top than with right, but the difference is perhaps not as much as people might think from the impression of the article here.

It is of interest and benefit to encourage students to discuss this without any directions or suggestions. The article says 'most people place their left thumb on top of their right' when clasping hands. So in a group of 203 students how many would the students in your class expect to have their left thumb on top?

Table of thumb on top and gender of subject

In this group of 203 students, (which is approximately 57%) placed their left thumb on top. So this is more than half but not much more. If we survey lots of groups of 200 people (taking 200 for convenience!) what sort of numbers out of 200 would the students in your class expect to see with left thumb on top if the article is correct?

Table of arm on top and gender of subject

So more than half this group placed their left arm on top when they folded their arms.

The column graphs below present the data in these tables graphically.

graph of gender, thumb on top

graph of gender, arm on top

We see from the graphs that in this group, girls had a slightly greater tendency than boys to place their left thumb on top. Boys and girls tended to place their left arm on top in folding their arms, but the boys had a greater tendency towards this than the girls in this group.

A question that arises is, 'Does there tend to be any association between which thumb is on top in clasping hands and which arm is on top in folding arms?' To investigate this we must consider the data for thumb and arm together. The table and graph below present this combination of the data for boys and girls combined.

Table of arm and thumb on top

graph of thumb on top, arm on top

If the left thumb is on top, there is no leaning towards left or right arm on top. If the right thumb is on top, the left arm is more likely to be on top. This may reflect an overall leaning towards having the left arm on top.

This topic naturally leads to more questions that could be investigated such as 'Do people tend to clasp their hands and fold their arms the same way each time they do this?' and 'does it matter whether we ask people to clasp their hands first or fold their arms first?' However such questions require much more thought and more maturity in statistical thinking to investigate.

Example B: Clasping hands and folding arms

The table of gender and thumb on top is revisited below but with approximate percentages instead of frequencies. These percentages are of the 203 students in the different categories.

Table of percentages of thumb on top and gender of subject

Note that we no longer have any information about the numbers of students surveyed.

If we want to compare girls and boys in this group, we can calculate the percentages of each gender who placed their thumb on top. We can present these in the following table. Note that in this table, not only do we have no information about the overall numbers surveyed but we also have no information about the proportions of girls and boys in the group. However, we can easily compare the boys and girls with respect to whether they place their left or right thumb on top.

Table of percentages of girls
and of boys with thumb on top

Another form of column graph that can be useful when the focus is on comparing relative frequencies across different groups is the stacked column graph. The graph below shows this for the boys and girls and whether they placed their thumb on top.

Notice that the actual frequencies are retained so we can see that we have more girls than boys in this group. But we can also readily see that the relative frequency of thumb on top for the girls is greater than the relative frequency of thumb on top for the boys. The graph does not give us the percentages but, on the other hand, it does not hide the numbers of observations. Although it is not easy to get the exact frequencies in each group, it does provide an easy-to-see picture of the comparison between the boys and girls. We see instantly that there are more girls than boys in this group and that the girls had a greater tendency to place left thumb on top than the boys.

Example C: cats and dogs as pets

The following graph is from Pawsey (2005)

percentage of cats and dogs
obtained fRom different sources

This shows a side-by-side column graph of percentages. It is not completely clear from the graph what the percentages refer to, but the text in the paper enables the reader to see that the percentages are of sources of cats and sources of dogs separately. The text also states that information was obtained for 1396 cats and 1756 dogs in Victoria from April to December 2004.

The text in the paper after this graph gives the values of the percentages in the above graph in a summary that compares the sources of dogs and cats as pets.

From graph 2 it can be seen that the major sources for acquisition of dogs and cats vary considerably between the two species. The most common source of dogs was through breeders (30%), while this was one of the smaller sources of cats with only 8% of cats acquired in this manner. Cats were most commonly acquired through three major sources, animal shelters, adoption of a stray cat (both 22%) or from friends (19%). Whilst in contrast only 11%, 4% and 13% respectively of dogs were acquired through these same sources. Pet shops supplied 14% of dogs and 9% of cats to pet owners and along with newspapers (14%) and friends (13%) were the next major sources of dogs after breeders. Interestingly only a small number of cats (4%) are sourced through newspapers. Council pounds and acquisition due to personal tragedy were the least common sources for both cats and dogs accounting for only 1-2% of both types of pets.

Pawsey (2005)

The above is a good example of reporting data on two categorical variables, one of which has natural categories (cat, dog) and the other is a more complex variable (source), whose categories require careful thought and description. How the data were collected is carefully described, and the numbers of observations given as well as the percentages. A criticism is that it should be more clearly stated that the percentages in the graph are of two different groups − one set of percentages of the cats, and the other of the dogs.

Example E: the Slip, Slop, SlAp message

The summary of key findings of the National Sun protection Survey of 2006-2007 may be found here.

The National Sun Protection Survey investigates people’s sun-related knowledge, attitudes and behaviours. The 2006-2007 survey was carried out by telephone interviews on Monday and Tuesday evenings over summer and asked about weekend behaviour. There were 5085 adults (18-69 years) and 652 adolescents (12-17 years) in Australia interviewed. The results of the survey are given in terms of percentages. For example, the report gives percentages of adults and adolescents who, when outdoors in peak UV hours on weekend, used head wear (50% and 29%); used sunscreen (37% of both groups); wore ¾ length or long sleeved tops (27% and 20%); got sunburnt (14% and 24%); and attempted a tan (11% and 22%).

Notice that these are percentages of two separate groups (adults and adolescents) so that one of the categorical variables being considered is age group, but the percentages add to more than 100% for each age group because respondents are asked to say yes to any of the behaviours described. For example, an adult respondent might respond yes to the first three behaviours and hence contributes to all three quoted %’s. The second variable 'sun behaviour when outdoors in peak UV periods on summer weekends' is categorical but the survey is permitting multiple responses to it. There is no way of knowing from the quoted results what the frequencies are for the various combinations of behaviours. This is why the focus of the report is on the % who attempted a tan or the % who got sunburnt − and there is no way of knowing how much these overlap or not without access to the original raw data giving the responses per person.

The survey report also gives percentages of adults in each Australian state and territory who attempted a tan, and who got sunburnt on a summer weekend. But the report also includes a very interesting statement: 'state breakdowns of tanning attempt and sunburn incidence amongst teenagers is not available due to the smaller sample size'. That is, 652 teenagers split by state give numbers considered to be too small for quoting %’s. This gives some idea that many observations are needed to be able to quote %’s that can be regarded as reasonable estimates for populations from which the data are obtained.
The report also estimates the number of adults in each state who got sunburnt on a summer weekend − these are not the number who did get sunburnt. They are simply the percentages obtained from the survey multiplied by the adult population of each state at that time and are quoted in a report for public consumption for impact. Do you think this is a good idea?