# Elaborations on Level Four: Statistics

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

S4-1: Plan and conduct investigations using the statistical enquiry cycle:

• determining appropriate variables and data collection methods
• gathering, sorting, and displaying multivariate category, measurement, and time-series data to detect patterns, variations, relationships, and trends
• comparing distributions visually
• communicating findings, using appropriate displays.

This means 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: At Level Four students should be able to pose questions that they want to investigate, consider the appropriate data they need to collect, gather and sort the data in order to develop an answer to their question. The data involved should be multivariate (include many variables, for example gender, age, height, eye colour, bedtime, etc.) so that relationships between the variables can be explored. Students should be able to ask summary questions (of a variable), for example what is the usual range in height for 10-year-old students?, comparison questions, for example are girls taller than boys?, and relationship questions, for example do older students go to bed later than younger students? They should be able to decide which variables are important for answering their question, for example quality of a sports player might be determined by points scored, assists, defensive turnovers or other variables. Students should also consider their methods of data collection, considering issues such as manageability, sampling, surveying, data safety, and technology use. Data displays, including tables and graphs, expected at Level Four are tally charts, frequency tables, pictographs, bar graphs, strip graphs, and pie charts for category data, dot plots, stem and leaf graphs and scatterplots for measurement data, and line graphs for time series data. Students should be able to use computer technology to create these displays to find patterns in the data, including differences and similarities between distributions, for example boys’ heights compared to girls, clusters and outliers within distributions, for example middle and spread, associations of variables, for example height with armspan, trends over time, for example cellphone use over a day, as well as to communicate their findings to others. They should be able to justify their choice of display/s with reference to the patterns they wish to highlight. Supporting teaching resources.

Click to download a PDF of second-tier material relating to Level 4 Statistical Investigations (185KB)

### Statistical literacy

S4-2: Evaluate statements made by others about the findings of statistical investigations and probability activities.

This means students will critically evaluate the strength of an argument proposed by others that is supported by statistical information. At Level Four students should consider features of the statistical investigation of others in weighing the strength of the findings. 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), data analysis (technology use, choice of displays) and the extent to which claims made are supported by the evidence. Supporting teaching resources.

### Probability

S4-3: Investigate situations that involve elements of chance by comparing experimental distributions with expectations from models of the possible outcomes, acknowledging variation and independence.

This means students will understand that probability is about the chance of outcomes occurring. At Level Four students should recognise that it is not possible to know the exact probability of something occurring in most everyday situations, for example the probability of someone being left-handed. They should understand that trialling must be used to gain information about the situation and that the results of trial samples vary, for example different samples of 100 people will have different proportions. Contrived chance events are used to highlight the variation between expected outcomes from models, and experimental outcomes from trialling. Level Four students are expected to use systematic methods such as listing, tree or network diagrams with equally likely outcomes, and tables to find all the possible outcomes of simple one or two stage situations such as tossing two coins, drawing counters from a bag, or rolling two dice. Students should compare the distributions they get from trialling with the expectations obtained from models, accepting variation between samples and that the results of one sample do not impact on the next (independence), for example take samples of twenty counters, with replacement, from a bag that has one-half red, one-third blue and one-sixth yellow. Accept that an eight red, seven blue, and five yellow result is natural and that it will not be compensated by the next sample. Supporting teaching resources.

S4-4: Use simple fractions and percentages to describe probabilities.

Simple fractions and percentages in this objective are common benchmarks like one half (50%), thirds (33.3% and 66.6%), quarters (25% and 75%), fifths (20%, 40%, 60%, 80%), tenths (10%, 30%, etc). Students should know that outcomes that are certain are described by fractions equalling one, including 100%, and outcomes that are impossible are described by fractions equalling zero, including 0%. In contrived situations involving elements of chance, for example totalling two dice, students should know that the count of all possible outcomes gives the denominator of a probability fraction, for example 36 possible outcomes, and the number of desired outcomes gives the numerator, for example there are 9 ways to get a total of either 2,4 or 6 so the probability is 9/36 or 1/4 . In realistic situations where probabilities are estimated, for example the chance of a drawing pin landing safe, students are expected to accept variation from an exact fraction, for example 37 out of 100 were safe which is about or 33.3%. Supporting teaching resources.

Click to download a PDF of second-tier material relating to Level 4 Probability (154KB)