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  4. Level Five: Statistics

Elaborations on Level Five: Statistics

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In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

Statistical investigation

S5-1: Plan and conduct surveys and experiments using the statistical enquiry cycle:

  • determining appropriate variables and measures;
  • considering sources of variation; gathering and cleaning data; using multiple displays, and re-categorising data to find patterns, variations, relationships, and trends in multivariate data sets;
  • comparing sample distributions visually, using measures of centre, spread, and proportion;
  • presenting a report of findings.

This means that students will use the statistical enquiry cycle to plan and conduct investigations. The cycle has five phases that relate to each other. Some enquiries follow these phases in sequence but often new considerations mean that a statistician must go back to previous phases and rethink. The phases are:

 Statistical investigation cycle.

 
At Level Five students should be able to pose suitable questions for data driven inquiry. The questions may be:
  1. Summary, for example, what is the normal height of a 14-year-old female?
  2. Comparitive, for example, do males do more exercise than females?
  3. Relational, for example, is there a relationship between television watching and lack of exercise?
Given a question or assertion, students need to decide on appropriate variables, for example, age, gender or hours of TV viewing, to answer a question or interrogate an assertion. The choice of attribute leads to choices of measures for that attribute, for example measure exercise by both time in minutes and intensity using a 1-10 self-reported scale.
At Level Five students sophistication in data collection and analysis should extend to the need for representative sampling and adequate sample size, avoidance of bias in surveys and sampling techniques,  systematic collection and processing of data that does not narrow potential responses, and appropriate use of technology to sort and display data.
 
Students should use a variety of displays to find patterns or relationships in multivariate data sets. This range of displays should extend to using measures of centrality and spread such as mean or median, range and quartiles. This means that displays such as box and whisker plots and histograms are accessible.

Students should analyse the data by comparing distributions visually using multiple graph types, preferably generated by technology. They should use informal inference to look for differences between distributions, for example, the median of one group is higher than the upper quartile of the other. Students should choose the most appropriate data display to report their findings and draw conclusions from the data related to their investigative question. They should recognise that all findings from the analysis of samples must be interpreted with uncertainty and be cautious in generalizing the results to whole. Supporting teaching resources.

Click to download a PDF of second-tier material relating to Level 5 Statistical Investigations (288KB)

Statistical literacy

S5-2: Evaluate statistical investigations or probability activities undertaken by others, including data collection methods, choice of measures, and validity of findings.

This means that students will evaluate the statistical investigation or probability activity undertaken by others by considering features of the investigation. These features include the appropriateness of sampling methods (for example number, representativeness), quality of the data collection (for example questions asked, accuracy of measurement, fairness of the experiment), choices of measures (types of questions, and responses allowed), data analysis (technology use, choice of displays) and the extent to which claims made are supported by the evidence. Together these factors should be used to evaluate the investigation as a whole. Supporting teaching resources.

Probability

S5-3: Compare and describe the variation between theoretical and experimental distributions in situations that involve elements of chance.

Students at Level Five understand that elements of chance have an effect on the certainty of results from surveys or experiments. Through examples from real life they should understand that statistics usually involves situations where the actual probabilities are not known, for example, probability of catching a disease. They should recognise situations where deterministic theoretical models are not possible, for example chance of a bus being early, and distinguish them from situations where probabilities can be reasoned from all the possibilities.
This means that students will identify the theoretical probabilities for situations involving chance by using proportions of possible outcomes.  For example, they will recognise that the probability of rolling an even number on a standard die is 1/2 because there are 6 possible outcomes and 3 of them are even, 3/6 = 1/2.
They will carry out experiments to test the probability of events and compare their results with theoretical probabilities. They will understand that some variation between experimental estimates of probability and theoretical probabilities is normal, for example, when rolling a die 10 times they will not usually roll an even number 5 of the times.  They will understand that a larger sample is likely to provide a more accurate theoretical probability, proportionally speaking, than a small one. 
Students will also understand that the results of past trials in probability experiments do not impact on future events, for example the fact that an even number has been rolled three times in a row does not make it more likely that the fourth roll will be an odd number. Supporting teaching resources.
 

S5-4: Calculate probabilities, using fractions, percentages, and ratios.

This means that students will calculate probabilities for probability situations that involve two or more events.  These events may be independent (for example rolling two dice, or tossing two coins) or dependent (for example drawing two cards from a deck of cards without replacement, or choosing to students from the class at random).  They will be able to model these situations using models such as tree diagrams, tables and systematic lists and assign theoretical probabilities as proportions using fractions, percentages and ratios, for example the odds of winning the game are 2:3.
Students should also be able to estimate probabilities given results of sufficient trials. Such estimates are always approximate, and require use of equivalent proportions, for example an experiment results in a ratio of 43 blue: 79 red. An estimate of the probability of red is 2/3 or 67%. Supporting teaching resources.