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# Statistics 12

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### Big Ideas

### Grandes idées

Statistics plays an integral role in research, decision making, and policy in society.

Statistics

*Sample questions to support inquiry with students:*- Why is statistical thinking important in our lives?
- How do the statistical sciences help us make decisions?
- What is the role of statistics in the scientific process?

The research question and practical and ethical issues determine whether a statistical study should be observational or experimental.

statistical study

*Sample questions to support inquiry with students:*- How do studies obtaining data enable us to explore research questions?
- What features of a study will make it effective, practical, and ethical for exploring a research question?
- How do we conduct an effective observational study?
- How do we conduct an effective designed experiment?

Statistical analysis allows us to explore, describe, model, and explain variation.

Statistical analysis

*Sample questions to support inquiry with students:*- Why is it important to explore and understand variation?
- How can we describe variation graphically?
- What is the role of probabilistic models for describing variation?
- Can we describe the sampling variation of a statistic, such as the sample mean?

We can develop statistical thinking to help make inferences intuitive.

statistical thinking

*Sample questions to support inquiry with students:*- How can we explore the sampling distribution of a statistic?
- What properties of a sample statistic make it a good estimator of a population parameter?
- How can technology help us appreciate the properties of a confidence interval?
- How surprising are the data from a study if the research hypothesis is true?

Statistical findings gain value through effective communication.

effective communication

*Sample questions to support inquiry with students:*- Why is the communication of statistical findings important?
- How can we best communicate statistical findings verbally and in writing?
- What are the roles of context and the target audience in the communication of statistical findings?
- How can technology assist us in the communication of statistical ideas?

## Learning Standards

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### Curricular Competencies

*Students are expected to be able to do the following:*### Reasoning and modelling

Develop thinking strategies to solve puzzles and play games

thinking strategies

- using reason to determine winning strategies
- generalizing and extending

Explore, analyze, and apply statistical ideas using reason, technology, and other tools

analyze

- consider a research problem and determine viable investigation approaches
- critique existing studies, identifying possible flaws and limitations
- draw viable conclusions from a statistical study

reason

- inductive and deductive reasoning
- predictions, generalizations, conclusions drawn from experiences (e.g., with games and simulations)

technology

- software for recording, exploring, and communicating data
- software tools for illustrating and providing information on probability models
- web-based visualisation/simulation tools that give intuition to inferential concepts

other tools

- manipulatives such as dice, coins, spinners, and other concrete materials

Estimate reasonably and demonstrate fluent, flexible, and strategic thinking about number

Estimate reasonably

- be able to justify the use of an estimate in a statistical context
- appreciate that statistical estimators exhibit variation across different samples
- use intuition when sampling distributions via simulations to make inferences

fluent, flexible, and strategic thinking

- includes:
- appreciating the role of variation
- choosing from different ways to investigate a research question (e.g., Which will be the most appropriate?)

Model with statistics in situational contexts

Model

- use mathematical concepts and tools to solve problems and make decisions (e.g., in real-life and/or abstract scenarios)
- take a complex, essentially non-mathematical scenario and figure out what mathematical concepts and tools are needed to make sense of it

situational contexts

- including real-life scenarios and open-ended challenges that connect mathematics with everyday life

Think creatively and with curiosity and wonder when exploring problems

Think creatively

- by:
- being open to trying different strategies
- appreciating that in statistical contexts, there is often no single correct answer
- proposing a viable research question for investigation
- designing a study to explore a research question

- refers to creative and innovative mathematical thinking rather than to representing math in a creative way, such as through art or music

curiosity and wonder

- asking questions to further understanding or to open other avenues of investigation

### Understanding and solving

Develop, demonstrate, and apply conceptual understanding of statistical ideas through play, story, inquiry, and research

inquiry

- includes structured, guided, and open inquiry
- noticing and wondering
- determining what is appropriate to explore in a research question

Visualize to explore and illustrate variation within and between variables

Visualize

- create and use mental images to support understanding
- Visualization can be supported using dynamic materials (e.g., graphical relationships and simulations), concrete materials, drawings, and diagrams.

Apply flexible and strategic approaches to explore statistical questions in abstract and situational contexts

flexible and strategic approaches

- deciding which statistical ideas are useful in addressing a research question or hypothesis
- choosing an effective strategy to address a research question (e.g., observational or experimental study, choice of variable[s] to measure, display method, inferential approaches)

Explore research questions with persistence and a positive disposition

persistence and a positive disposition

- not giving up when facing a challenge
- engaging in research and exploration with vigour and determination

Engage in statistical thinking to answer questions connected with place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures

statistical thinking

- gain deeper understanding through data collected to answer questions about local cultures

connected

- through daily activities, local and traditional practices, popular media and news events, cross-curricular integration
- by posing and solving problems or asking questions about place, stories, and cultural practices.

### Communicating and representing

Explain and justify statistical thinking and decisions in many ways

Explain and justify

- use arguments based on statistical thinking to convince
- includes anticipating consequences

decisions

- Have students explore which of two scenarios they would choose and then defend their choice.

many ways

- including oral, written, visual, use of technology
- communicating effectively according to what is being communicated and to whom

Represent statistical ideas in concrete, pictorial, and symbolic forms

Represent

- using models, simulations, tables, graphs, words, numbers, symbols
- connecting meanings among various representations
- using concrete materials and dynamic statistical software (applets/simulation tools) to explore variation

Use statistical vocabulary and language to contribute to discussions in the classroom

discussions

- partner talks, small-group discussions, teacher-student conferences

Take risks when offering ideas in classroom discourse

discourse

- is valuable for deepening understanding of concepts
- can help clarify students’ thinking, even if they are not sure about an idea or have misconceptions

### Connecting and reflecting

Reflect on statistical thinking

Reflect

- share the statistical thinking of self and others, including evaluating strategies and solutions, extending, posing new problems and questions

Connect statistical concepts with each other, other areas, and personal interests

Connect statistical concepts

- to develop a sense of how statistics helps us understand ourselves and the world around us (e.g., daily activities, local and traditional practices, popular media and news events, social justice, cross-curricular integration)

Use mistakes as opportunities to advance learning

mistakes

- range from calculation errors to misconceptions

opportunities to advance learning

- by:
- analyzing errors to discover misunderstandings
- making adjustments in further attempts
- identifying not only mistakes but also parts of a solution that are correct

Incorporate First Peoples worldviews, perspectives, knowledge, and practices to make connections with statistical concepts

Incorporate

- by:
- collaborating with Elders and knowledge keepers among local First Peoples
- exploring the First Peoples Principles of Learning (http://www.fnesc.ca/wp/wp-content/uploads/2015/09/PUB-LFP-POSTER-Princip... e.g., Learning is holistic, reflexive, reflective, experiential, and relational [focused on connectedness, on reciprocal relationships, and a sense of place]; Learning involves patience and time)
- making explicit connections with learning mathematics
- exploring cultural practices and knowledge of local First Peoples and identifying mathematical connections

knowledge

- local knowledge and cultural practices that are appropriate to share and that are non-appropriated

practices

- Bishop’s cultural practices: counting, measuring, locating, designing, playing, explaining (http://www.csus.edu/indiv/o/oreyd/ACP.htm_files/abishop.htm)
- Aboriginal Education Resources (www.aboriginaleducation.ca)
*Teaching Mathematics in a First Nations Context*, FNESC (http://www.fnesc.ca/resources/math-first-peoples/)

### Content

*Students are expected to know the following:*role of statistical thinking in research and the scientific method

role of statistical thinking

- census versus sample
- identifying research questions and target population
- historical perspective on the development of statistical research and theories
- role of data in helping to answer questions (e.g., Lind study on scurvy, 1753); randomization as a fairly recent development

observational and experimental studies

observational

- Observational studies involve observation of a sample from the target population, without intervention.
- Observational studies can include surveys and questionnaires.
- When are observational studies necessary and appropriate?
- What are the limitations of observation studies?
- Lurking variables can impact conclusions.
- The wording of survey items can incur bias.
- How should we design an observational study to explore an appropriate research question?

experimental

- Experimental studies involve intervention for collection of data.
- Randomization of treatments to experimental units can eliminate issues with lurking variables and bias.
- There may be practical and ethical concerns (e.g., long-duration studies on smoking, effectiveness of medications).
- How should we design an experiment to explore an appropriate research question?

common graphical representations of variation

graphical

- Graphical methods should always be used to explore data.
- Graphical approaches can display data distributions.
- Focus on interpreting data through bar charts, histograms, dot plots, boxplots, scatterplots, tables.
- Graphical approaches can be used to explore the association between variables (e.g., clustered bar charts, scatterplots).
- Software should be used (e.g., Minitab).
- What are the advantages and disadvantages of different representations?

use of summary statistics to describe variation

summary statistics

- measures of centre, spread (range, variance, standard deviation interquartile range), including five-number summary
- use of Chebyshev’s inequality
- use of correlation in measuring association between quantitative variables

association between two variables

association

- categorical variables: contingency tables — clustered, stacked bar charts
- quantitative variables: scatterplots
- correlation and causation

probability models for variation

models

- binomial distribution:
- When is it appropriate?
- What does it model?
- What assumptions can be made?
- Shape of distribution affected by n and p.

- normal (Gaussian) distribution:
- when it is useful
- roles of the mean and standard deviation, 68-95-99.7 rule

- central limit theorem: describing the variation of a sample mean
- use of simulation software to explore sampling distributions

intuition and appreciation of inferential concepts, such as confidence intervals and hypothesis tests

inferential concepts

- making intuitive inferences based on a large number of simulations
- intuition on interval, estimation of means and proportions via simulation
- inference for proportion via simulation (randomization/permutation tests)
- inference for a mean via simulation (randomization/permutation tests)
- two-sample questions via simulation (randomization/permutation tests)

use of software and technology to enhance statistical ideas

software and technology

- Software can assist us in exploring and summarizing data.
- Online simulation-based learning tools can help us gain intuition of inferential concepts, such as sampling distribution, interval estimation, and hypothesis tests.

communication of statistical findings

communication

- communicating statistical findings in context, appropriate to the target audience
- writing a report on a research project involving an observational study
- writing a report on a research project involving a designed experiment
- presenting to an audience on a research project involving an observational study
- presenting to an audience on a research project involving a designed experiment

**Note:**Some of the learning standards in the PHE curriculum address topics that some students and their parents or guardians may feel more comfortable addressing at home. Refer to ministry policy regarding opting for alternative delivery.