# Key Ideas: Statistical Investigations

## NZC Level 1

The key idea of statistical investigations at Level 1 is collecting data as evidence to tell a story about a question of interest.

At Level 1 students should be working with survey data that they have collected about themselves and their classmates. As a class with their teacher they should be starting to use the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle in their investigations. This involves posing an investigative question that they want to address, collecting and sorting data to answer the question, displaying the data, making statements about what the data shows and answering the question.

Students are typically interested in questions like “who in our class has the most children in their family?”, or “what is the favourite fruit in our class?”. These pre-summary questions are suitable for students and this level and focus more on an individual than the aggregate of the data.

Data displays are limited only by the students’ imaginations. For example, students might show the different types of shoes in the class by taking one shoe from each student and building a display. Students may well draw individual case plots, for example, they draw a graph of number of children in family, where the students’ names are on the horizontal axis and the number of children in family is on the vertical axis.

Students will be making statements about individuals in their displays. For example, Hemi, Jane and Tiana have four children in their family. These statements need to reflect what the data is showing.

This key idea is extended in the key idea of statistical investigations at NZC Level 2 where students are letting go of the individual’s story and moving towards telling the class story.

## NZC Level 2

The key idea of statistical investigations at level 2 is letting go of the individual’s story and moving towards telling the class story.

At Level 2 students are building on the ideas from level one and refining their understanding of different aspects of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle. A key transition point at this level is moving students’ data display knowledge from individual case plots to frequency plots of the variable of interest.

Investigative questions will be similar to those posed previously and will include categorical and whole number variables.

Data displays become a summary of the individual case plots that they were drawing at the previous level. For example, the number of children in a family is now on the horizontal axis. Students place themselves on the graph according to the number of children in their family (this could be through using sticky notes). As each student adds their sticky note to the display the frequency builds up. The frequency is recorded on the vertical axis.

Students will be making summary statements, for example, three students in our class have four children in their family (read the data), or five students have 1 or 2 children in their family (read between the data).

Teachers should be encouraging students to read beyond the data by asking questions such as: “If a new student joined our class, how many children do you think would be in their family?”

This key idea develops from the key idea of statistical investigations at NZC Level 1 where students are collecting data as evidence to tell a story about a question of interest.

This key idea is extended in the key idea of statistical investigations at NZC Level 3 where students are telling the class story with supporting evidence.

## NZC Level 3

The key idea of statistical investigations at Level 3 is telling the class story with supporting evidence.

Students are building on the ideas from level two and their understanding of different aspects of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle. Key transitions at this level include posing summary investigative questions, and collecting and displaying multivariate and time series data.

Summary or time series investigative questions will be posed and explored. Summary investigative questions need to be about the group of interest and have an aggregate focus. For example, what are typical numbers of children in a family for students in our class? What types of fruit do students in our class like?

Data displays build on the frequency plots from level two and can be formalised into dot plots and bar graphs. Students should be encouraged to show a second variable, for example, by using colour. They may like to look at boys and girls fruit preferences.

Students will be making summary statements, for example, the most common number of children in a family for our class is three, nine students have three children in their family (read the data), or most students (16 students out of the 27 in our class) have between two and four children in their family (read between the data).

Teachers should be encouraging students to read beyond the data by asking questions such as: “If a new student joined our class, how many children do you think would be in their family?”

This key idea develops from the key idea of statistical investigations at NZC Level 2 where students are learning to tell the class story.

This key idea is extended in the key idea of statistical investigations at NZC Level 4 where students are adding detail to their stories.

## NZC Level 4

The key idea of statistical investigations at Level 4 is telling the class story in detail with supporting evidence.

Students are building on the ideas from level three about different aspects of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle. Key transitions at this level include posing comparison and relationship investigative questions, planning investigations including data generation, collecting and displaying measurement data, and comparing distributions visually.

Comparison and relationship investigative questions will be posed and explored. Comparison investigative questions need to be about the group of interest and have an aggregate focus. For example, do the boys in our class tend to be taller than the girls in our class? Is there a relationship between armspan length and height for the students in our class?

Students should be planning to collect their own data for the investigative question they have posed. This includes determining appropriate variables and data collection methods. For example, they need to realise that to answer the first question they will need to measure student’s heights. Along with this they will need to think about what units to measure with and whether students should leave their shoes on or not, and who will take the measures. This is data generation.

Students should be using dot plots and scatter plots to display data. When comparing dot plot distributions visually they can identify the middle group by circling it and reason about the placement of the middle groups (shift) relative to one another. They can compare (approximate) centres and the variation of the data in the middle groups. Students can use tools such as hat plots and any statistical software that is available. For scatterplots students should be looking at features such as the trend of the data points and how close the points are to the trend. Adding a third variable, for example gender, by using colour allows for further exploration.

Students should be writing statistically sound statements about what their displays show. The starter “I notice…” is a useful way to encourage students to write about what their displays show. In addition students should be encouraged to write “I wonder…” statements for further investigation.

This key idea develops from the key idea of statistical investigations at NZC Level 3 where students are telling the class story with supporting evidence.

This key idea is extended in the key idea of statistical investigations at NZC Level 5 where students are telling a story about the wider universe with supporting evidence.

## NZC Level 5

The key idea of statistical investigations at Level 5 is telling a story about the wider universe with supporting evidence.

Students are building on the ideas from level four about different aspects of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle. The key transition at this level is the acknowledgement that samples can be used to answer questions about populations.

Students will be posing investigative questions about populations and using samples to answer these.

Students need to realise that the data collected may have to be cleaned. To help with this they should be familiar with the survey questions posed, who the data was collected from and how the data was collected. Once data is identified as needing cleaning, strategies on how to do this should be discussed, for example, whether the value removed, or “cleaned”.

Students may need to recategorise data into broader categories or smaller categories depending on the question they are trying to answer. They will be looking at patterns and trends in displays and using these to answer their investigative question. Thinking routines such as: “What information can you get from this plot?” and “What evidence do you have for saying that?” will be helpful for students.

Students will be starting to use informal methods to make comparisons between sample distributions using box plots and growing their reasoning about sampling variability, shape, spread, unusual and interesting features, and making a call. See CensusAtSchool 2009 Teachers Day.

This key idea develops from the key idea of statistical investigations at NZC Level 4 where students are telling a class story in detail with supporting evidence.

This key idea is extended in the key idea of statistical investigations at Level 6 where students are telling a story about a wider universe taking variation and uncertainty into account.

## NZC Level 6

The key idea of statistical investigations at Level 6 is telling a story about a wider universe, taking variation and uncertainty into account, with supporting evidence.

Students are consolidating and refining their ideas about different aspects of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle. Key transitions at this level include the integration of statistical and contextual knowledge to answer the investigative questions and making informal inferences about populations from samples.

Students should be justifying variables and measures used in the data collection phase and thinking about the possible underlying population distributions for the variables of interest.

In the analysis phase students should be using multiple displays to show different features of the sample distributions. Key features of the sample distributions should be discussed; integrating statistical and contextual information. Students will confidently be using informal methods to make comparisons about populations using sample distributions including reasoning about shift, overlap, sampling variability and sample size.

Students should be reflecting on their findings and how this fits with real world experiences.

This key idea develops from the key idea of statistical investigations at NZC Level 5 where students are telling a story about a wider universe with supporting evidence.

This key idea is extended in the key idea of statistical investigations at NZC Level 7 where students are telling a story about a wider universe, considering sampling variability and sample size effects.

## NZC Level 7

The key idea of statistical investigations at Level 7 is creating and telling a story about a wider universe, considering sampling variability and sample size effects, with supporting evidence.

Students are using the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle in different data collection contexts. A key focus at this level is the use of surveys and associated random sampling techniques. Key transitions at this level include the integration of statistical and relevant contextual knowledge to answer investigative questions and making informal estimated comparison intervals for population parameters from samples.

At this level students will be aware of how to design a suitable questionnaire specific to a given purpose and they will be using random sampling techniques in the data collection phase. They should be evaluating choice of measures for variables and sampling and data collection methods. Students should be provided with relevant contextual knowledge about the situation under investigation.

Students will confidently be using informal estimated comparison intervals to make comparisons about populations from sample distributions.

This key idea develops from the key idea of statistical investigations at NZC Level 6 where students were telling a story about a wider universe taking variation and uncertainty into account.

This key idea is extended in the key idea of statistical investigations at NZC Level 8 where students are supporting their story with sophisticated statistical techniques and informed contextual knowledge.

## Statistical Investigations: NZC Level 8

The key idea of statistical investigations at level 8 is creating, modelling and telling a story about a wider universe, supporting this with sophisticated statistical techniques and informed contextual knowledge.

Students are using the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle in different data collection contexts. A key focus at this level is the use of experimental design principles. Key transitions at this level include:

• the integration of statistical and informed contextual knowledge to answer investigative questions,
• using appropriate statistical models,
• making statistical inferences about populations or processes from samples using methods such as bootstrapping or randomisation to determine estimates, confidence intervals, forecasts, and strength of evidence, and
• evaluating all stages of the cycle.

Students will be sourcing relevant contextual knowledge about the situation under investigation from places such as the internet, the school or local library, newspapers and magazines.

This key idea develops from the key idea of statistical investigations at NZC Level 7 where students were telling a story about a wider universe, considering sampling variability and sample size effects.

Source: http://www.nzmaths.co.nz/key-mathematical-ideas