The Improving Mathematics Education in Schools (TIMES) Project

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Data Investigation and interpretation 8

Statistics and Probability : Module 5Year : 8

June 2011

PDF Version of module

Assumed background from F-7

It is assumed that in Years F-7, students have had many learning experiences involving choosing and identifying questions or issues from everyday life and familiar situations, planning statistical investigations and collecting or accessing data. It is assumed that students are now familiar with categorical, count and continuous data, have had learning experiences in recording, classifying and exploring individual datasets of each type, and have seen and used tables, picture graphs and column graphs for categorical data and count data with a small number of different counts treated as categories, and dotplots and stem-and-leaf plots for continuous and count data. It is assumed that students are familiar with the use of frequencies and relative frequencies of categories (for categorical data) or of counts (for count data) or of intervals of values (for continuous data). It is assumed that students have used and interpreted averages (or means) and medians of quantitative (that is, count or continuous) data. It is assumed that students have become familiar with the concepts of statistical variables and of subjects of a data investigation. It is assumed that the focus in exploration and comment on continuous and count data has been on each set of data by itself, but that in Year 6, students have become familiar with considering more than one set of categorical data on the same subjects. In doing so, they have understood that they were investigating data on pairs of categorical variables.

Motivation

Statistics and statistical thinking have become increasingly important in a society that relies more and more on information and calls for evidence. Hence the need to develop statistical skills and thinking across all levels of education has grown and is of core importance in a century which will place even greater demands on society for statistical capabilities throughout industry, government and education.

A natural environment for learning statistical thinking is through experiencing the process of carrying out real statistical data investigations from first thoughts, through planning, collecting and exploring data, to reporting on its features. Statistical data investigations also provide ideal conditions for active learning, hands-on experience and problem-solving. No matter how it is described, the elements of the statistical data investigation process are accessible across all educational levels.

Real statistical data investigations involve a number of components: formulating a problem so that it can be tackled statistically; planning, collecting, organising and validating data; exploring and analysing data; and interpreting and presenting information from data in context. No matter how the statistical data investigative process is described, its elements provide a practical framework for demonstrating and learning statistical thinking, as well as experiential learning in which statistical concepts, techniques and tools can be gradually introduced, developed, applied and extended as students move through schooling.

Content

In this module, in the context of statistical data investigations, we build on the content of Years F-7 to focus more closely on whether we can use data to comment on a more general situation or population. We can do this if our data are, or can be considered to have been, a random set of observations obtained in circumstances that are representative of the general situation or population. Because this is a mouthful, we will here call this the random representativeness of data. So in this module we consider more about how data are collected, how to obtain random representative data and of what we can take our data to be representative. We compare the nature of censuses, surveys and observational investigations. Datasets that are not census data are often called samples of data, so this module includes some introductory notions of sampling to obtain random representative data.

The general meaning of the word sample is a portion, piece, or segment that is representative of a whole. In statistics, a sample of data, or a data sample, is a set of observations such that more, sometimes infinitely more, observations could have been taken. We want our sample of data to be randomly representative of some general situation or population so that we can use the data to obtain information about the general situation or population. A particular dataset might be considered to be representative for some questions or issues, but not for others, and these considerations will also be explored.

For example, if we wanted information about people’s opinions on water recycling, in some locality, then the ideal way of obtaining a sample of opinions that can be used to comment on the whole locality, is by taking a randomly chosen set of people from all of those in the locality and asking their opinions. However, even with complete records and resources available, this has many difficulties as we will see in this module. How about surveying people in a shopping centre on weekend days in that locality? This might be considered reasonably representative of people in that locality with respect to opinions on water recycling, but if we wanted information on people’s preferred time for shopping, then such a method would clearly be highly non-representative of people in that locality.

But even if weekend shoppers in a shopping centre in the locality can be considered representative of that locality, we also need to choose people randomly because only a randomly chosen group of people are truly representative of everyone in the locality.

In this module we use the term representative data to mean a set of observations obtained randomly in circumstances that are representative of a more general situation or larger population with respect to the issues of interest.

We build on the notions of variation introduced across Years F-7 to explore this concept more closely, including sources of variation within and across datasets.

In Year 7, we have seen the concept, use and interpretation of the average of a set of quantitative data; the average is often called the sample mean. We have seen the concept, use and interpretation of relative frequency of a category for categorical data; this is often called the sample proportion (for that category). In this module, consideration of variation across datasets leads us to explore the variation of sample means and of sample proportions across datasets collected or obtained under the same or similar circumstances.

This module uses a number of examples involving the different types of data to explore representativeness of data, variation across and within datasets, and the variation of quantities calculated from data, such as averages or sample means, and sample proportions. Such quantities are called summary statistics. The examples and new content are developed within the statistical data investigation process through the following:

• considering initial questions that motivate an investigation;
• identifying issues and planning;
• collecting, handling and checking data;
• exploring and interpreting data in context.

Such phases lend themselves to representation on a diagram, as follows.

The examples consider situations familiar and accessible to Year 8 students and in reports in digital media and elsewhere, and build on the situations considered in F-7. The module uses concepts, graphs and other data summaries considered in F-7, but focuses on the planning and collecting components of the statistical data investigation process to develop understanding of concepts of representativeness, sources of variation, sampling and variation due to sampling.

Revision of types of data and statistical variables

In F-7, we have considered different types of data, and hence different types of statistical variables. When we collect or observe data, the ‘what’ we are going to observe is called a statistical variable. You can think of a statistical variable as a description of an entity that is being observed or is going to be observed. Hence when we consider types of data, we are also considering types of variables. There are three main types of statistical variables: continuous, count and categorical.

Some examples of continuous variables are:

• time in minutes to get to school
• length in cm of right feet of Year 7 girls
• age in years
• amount of weekly allowance

All continuous data need units and observations are recorded in the desired units.

Continuous variables can take any values in intervals. For example, if someone says their height is 149 cm, they mean their height lies between 148.5 cm and 149.5 cm. If they say their height is 148.5 cm, they mean their height is in between 148.45 cm and 148.55 cm. If someone reports their age as 12 years, they (usually) mean their age is in between 12 and 13 years. Note the convention with age is that the interval is from our age in whole number of years up to the next whole number of years. If someone says their age is 12 and a half, is there are a standard way of interpreting the interval they are referring to? Do they mean 12.5 years up to 13 years, or do they mean some interval around 12.5 with the actual interval not completely specified? Notice that our specification of intervals in talking about age is usually not as definite as when we quote someone’s height, but the principle is the same − observations of continuous variables are never exact and correspond to little intervals.

A count variable counts the number of items or people in a specified time or place or occasion or group. Each observation in a set of count data is a count value. Count data occur in considering situations such as:

• the number of children in a family
• the number of people arriving at a central city railway station in a 5 minute interval during peak time
• attendance at football matches
• the number of hits on an internet site per week

We can see that the first example above of a count variable contrasts with the other examples, in which counts will tend to take many different values − that is, in data on the variables in bullet points 2, 3 and 4 above, repetitions of values of observations are not likely. Also, the sizes of the observations will tend to be large, sometimes very large. For these types of count variables, the types of graphs and summaries used for continuous variables are often appropriate.

In categorical data each observation falls into one of a number of distinct categories. Thus a categorical variable has a number of distinct categories. Such data are everywhere in everyday life. Some examples of pairs of categorical variables are:

• gender and pet preference between cat and dog
• favourite TV show and favourite holiday activity
• gender and favourite food
• favourite sport and colour of hair (e.g. redhead, blonde, brown, black)

Sometimes the categories are natural, such as with gender or preference between cat and dog, and sometimes they require choice and careful description, such as favourite holiday activity or favourite food.

Initial questions that can motivate an investigation.

The following are some examples that involve collecting, or accessing, or obtaining, data for which considerations of representativeness and sources of variation are of core importance in planning the data investigation, and, in the final phase of the data investigation, interpretation in context.

A

A school is planning catering for a (free) end-of-year concert and wants to estimate the number of people who will attend. Should they give a survey to all students or should they survey a subset of students?

B

A school would like parents’ opinions on a number of matters that do not lend themselves to simple questions. The school decides to obtain opinions by asking questions in person of a representative group of parents. They are wondering whether to send a message home asking for volunteer parents, or whether to select parents to be interviewed.

C

A group of students are interested in investigating the length of the most popular songs. They decide to investigate the top 25 songs on the annual JJJ charts over a number of years.

D

When you clasp your hands, which thumb is on top? Most people find that they always have either the left or right thumb on top and that it is very difficult to clasp their hands so that the other thumb is on top. There may be a genetic link in these simple actions. See, for example, http://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.'

A group of students are interested in investigating this.

E

How good are people at estimating periods of time? That is, how good are they at estimating a length of time such as 10 seconds?

F

Governments and the Cancer Council and other groups run advertisements, especially in summer, to try to get people to protect their skin from the effects of exposure to sun. For example, in November 2009, the campaign for summer 2009-2010 was launched at Bondi Beach, with towels on the beach representing the Cancer Council’s estimates of the number of Australians who would die from skin cancer in the next year. They would like to know if people tend to heed the Slip, Slop, Slap messages, and about differences such as whether adults are different to teenagers with respect to sun behaviour.

G

How aware are people of environmental issues? How knowledgeable are they
of relevant facts?

H

Do people tend to use the lift or stairs in going up at a bus or train station?
What proportion of those going up use the lift?

I

How long should the green be on pedestrian crossing lights? How long do people tend to take to cross the road at a pedestrian crossing with crossing lights?

The above are examples of just some of the many questions or topics that can arise that involve considering how to collect data, sources of variation, and variation within and across datasets. The examples also involve considerations of summaries of data and how they vary across datasets. Some of these examples are used here to explore the progression of development of learning about data investigation and interpretation. The focus in this module is on planning for data collection, exploring and interpreting variation, and the variation of features of data.

Identifying issues and planning to obtain representative data

In the first part of the data investigative process, one or more questions or issues begin the process of identifying the topic to be investigated. In thinking about how to investigate these, other questions and ideas can tend to arise. Refining and sorting these questions and ideas along with considering how we are going to obtain data that is needed to investigate them, help our planning to take shape. A data investigation is planned through the interaction of the questions:

• ‘What do we want to find out about?’
• ‘What data can we get?’ and
• ‘How do we get the data?’

Planning a data investigation involves identifying its variables, its subjects (that is, on what or who are our observations going to be collected) and how to collect or access relevant and representative data.

Variation within and across datasets

Statistics is the science of variation and uncertainty − the science of investigating, identifying, measuring, estimating, describing, modelling, attributing, interpreting, minimising and allowing for, variation and uncertainty.

In the data strand, we focus more on the variation aspects. We will consider some aspects of variation in the above examples.

Census data

A census aims to obtain information from a whole population. Questions of representativeness do not arise if we have the required information on the whole population. In reporting and using census data, identifying, describing, investigating and attributing variation are all important, but in principle we do not have to consider variation and uncertainty due to sampling − that is, due to having a subset of a whole population, or to having a set of observations that are representative in some way of a more general situation.

This is 'in principle' because in large and complex censuses, such as national censuses, mistakes, omissions, non-responses or even non-contacts must be investigated, modelled, estimated and allowed for. Such issues are very complex and challenging, requiring very advanced statistical expertise and information. Minimising the risk of these difficulties also requires significant government and statistical knowledge and expertise, with associated thorough and high quality planning.

Example A includes, on a small scale, some of these challenges of a census. All families need to be contacted; the question(s) must be clear to avoid unintentional mistakes; allowance must be made for mis-reporting (in the case of Example A, this is essentially changes of plans after returning the form); and non-returns of forms requires estimation of the unknown intentions of non-respondents. In Example A, the effects of these issues are probably not great − after all, there is still estimation required of the amount of catering required even if the attendance is known accurately! But the example does provide at least some idea of the enormous challenges (and expense) of a national census. But the need for, and value of, high quality national census data for every aspect of strategic planning for a country, cannot be sufficiently emphasized.

The lengths vary from 90 seconds up to an unusually long song (compared to the rest) of 570 seconds (to the nearest 10 seconds). The second longest song is 460 seconds. However almost all the songs are between 2 and 6 minutes long, and most are between
3 and 5 minutes.

Which do you think would be larger for these data? The average or the median? Answer: the average because of the few values that are much larger than the rest of the values; in fact, the average length is 234 secs and the median is 229 secs.

Sample data: categorical data

For categorical variables − and hence categorical data − we are interested in relative frequencies or proportions of the different categories. If we have census data, we can simply report percentages or proportions for a country. If we have sample data that are representative of some general situation, we are interested in using the sample data to estimate proportions for the more general situation.

Hence we need some idea of how much variation we could get across different samples of data and hence how much variation we could get in our estimate of the proportion in the more general situation. Involved in these questions are also the questions of:

• whether our sample(s) of data are representative of the general situation − or, from another viewpoint, of what can we consider our sample(s) of data to be representative?
• can any variation we observe be attributed to, or explained by, some other variables?

The examples above are of categorical data with just two categories so that looking at the variation across samples could focus on looking at the variation in the percentages of one of the categories. In Example D, there was one categorical variable; in Example H there were three categorical variables (peak period, direction, choice of lift or stairs) and the interest in Example H is not just in individual %’s but also %’s within different categories and in comparing these %’s over the two peak periods.

For categorical variables with more than two categories, the %’s in each category vary across samples in a similar way to the above examples. A dynamic illustration of this variation for different sample sizes can be found in the Categorical Variables section of http://www.censusatschool.org.nz/2009/informal-inference/WPRH/ . This dynamic illustration is of a barchart of the %’s using different types of transport to school in Auckland. The samples of different sizes are chosen at random (with replacement) from a large dataset obtained through the Census at School project in New Zealand.

Sample data: continuous data

For samples of continuous data we are interested in describing the variation of values within a sample (just as we are if we have continuous data in a census) and in considering how much variation there could be across samples collected in the same circumstances.

The quantities calculated from data that have been considered in the above examples are proportions (or percentages) for categorical data, and averages for continuous data. These are not necessarily the only quantities of interest, as the following example shows.

Can we seek to explain variability?

In some of the examples above, there is interest in investigating comparisons of data on one variable with respect to one or more other variables. For example, do commuters tend to be more likely in the morning or evening peak to use a lift instead of stairs to go up at a bus station? Are males better at guessing lengths of time than females?

This is investigating if a variable is affected by others. Looking at this from another point of view, we are often interested in investigating whether a variable is affected by others, and if so, to what extent. That is, we are often interested in trying to explain at least some of the variability in data by investigating if other variables may be affecting the data. For example, if which thumb is placed on top in clasping hands is claimed to be genetically linked, then we might wonder if males tend to be different to females, or if left-handed tendencies might be associated with this tendency. That is, we might seek to explain some of the variability across individuals by investigating if there are differences between males and females, and left- and right-handers.

For aspects such as measuring reflexes or guessing time periods, we might wish to investigate whether any or all of age, gender or different conditions affect the result. For example, does listening to music affect people’s ability to guess periods of time or their reaction times? Experiments or observational studies or a mixture of observation and experiment could be designed to investigate these issues. There will always be sampling variability, and almost always at least some natural variability due to variability within and across people and/or natural conditions. But statistical methods are developed to ask if some of the observed variability in the data can be attributed to other variables.

Some general comments and
links from F-7 and towards year 9

From F-7, students have gradually developed understanding and familiarity with concepts and usage of the statistical data investigative process, types of data and variables, types of investigations and at least some types of graphical and summary presentations of data. In developing these, considerations have increasingly arisen of the need for representative data, and of describing and/or allowing for variability within and across datasets.

In order to understand how to interpret and report information from data, students need to develop at least some understanding of the effects of sampling variability. This also helps in developing understanding of the need for formal statistical inference, even if the methods, results and processes of statistical inference are not introduced until senior or tertiary levels. Statistical inference also requires that data be representative of a population or general situation with respect to the questions or issues of interest.

Hence this module has discussed the challenges of obtaining representative data, emphasizing the importance of clear reporting of how, when and where data are obtained or collected, and of identifying the issues or questions for which data are desired to be representative. The module has also used real data and simulations, including re-sampling from real data, to illustrate how sample data and data summaries such as sample proportions and averages can vary across samples.

As in Years 4-7, the examples of this module again illustrate the extent of statistical thinking involved in all aspects of a statistical data investigation, in particular in identifying the questions/issues, in planning and in commenting on information obtained from data.

Questions of planning to obtain representative data to investigate everyday questions and issues involving a number of variables are considered further in Year 9, along with developing further skills in exploring, understanding and interpreting data. These build on the greater emphasis in this module on recognising, exploring and interpreting variation within data and across datasets. As in Years F-8, concepts are introduced, developed and demonstrated in contexts that continue the development of experiential learning of the statistical data investigation process.

Appendix 1

To use Excel to generate random data, requires the add-in of Data Analysis under Tools.

To use Data Analysis under Tools to generate a number of random samples of data on a categorical variable with a given probability for the category of interest, choose Random Number Generator. For Number of Variables, enter 1. For Number of Random Numbers, enter the number of different samples you want to generate − for example, 100 has been used in many of the examples of this module. For Distribution, choose Binomial. Under the Parameters that appear for Binomial, the p Value is the proportion you wish to assume as the true (or overall population) proportion, and the Number of trials is the size of the sample you wish to generate (for example, you might wish to consider samples of 20 people). The output range needs to be a single column of the same size as the Random Number you chose. The output will consist of a set of numbers out of the Number of trials, so divide by the Number of trials to obtain the simulated proportions.

To use Data Analysis under Tools to generate a number of random samples of data from a given set of values (that is to re-sample from a given set of data), the original data needs to be in a column, with a second column consisting of 1/(number of observations) in each cell. As this second column must sum to 1, you might need to slightly adjust values you enter to ensure this. Choose Random Number Generator. In Number of Variables, enter the number of samples you wish to generate. In Number of Random Numbers, enter the size of the samples. In Distribution, choose Discrete. In Value and Probability Input Range, give the column in which you placed the original data, and the column in which each value is equal (allowing for perhaps a slight adjustment) and sum to 1. In Output Range, give a range of number of columns being the chosen number of samples, and the size of each column being the chosen sample size.

 

The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations.

The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations.

© The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE-EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
https://creativecommons.org/licenses/by-nc-nd/3.0/