# Assessment Report

Level 3 Statistics 2018

### Part A: Commentary

Candidates who performed well across the standards were able to display their knowledge of statistical and probability concepts, make accurate calculations and support their answers with statistical reasoning written in context.  They showed a good level of statistical literacy, taking care to answer the particular question being asked.

Successful candidates demonstrated strong skills in working with probability theory and probability distributions as well as sound reasoning with data and models. They understood that true probability is unknown and that it can be estimated by a theoretical probability and/or the result from an experiment.  These stronger candidates were able to analyse a statistical report, describing study designs, identifying confounding and linking these observations to the strength of a claim in context.

Weaker candidates often did not understand how to provide a statistical response, either failing to provide numerical evidence to support their responses, or providing numerical evidence without linking this evidence to the underlying context.

## 91584:  Evaluate statistically based reports

Candidates who were awarded Achievement commonly:

• correctly identified control and treatment groups in an experiment
• identified an explanatory and response variable
• quoted directly from the report using important values
• calculated a margin of error
• knew that the rule of thumb for the MOE is invalid for a proportion outside of the 30%-70% range
• calculated a confidence interval (without explanation)
• identified the difference between observational and experimental studies
• used the basic terminology for this standard (although may have inconsistently applied it)
• recognised random allocation as a key idea in experimental design but often did not explain why it was used or understand that random allocation helps to reduce bias, not eliminate it.

Candidates whose work was assessed as Not Achieved commonly:

• could not identify control and treatment groups in an experiment
• could not identify an explanatory and response variable
• quoted numerical values but not in context
• did not refer to the statistical nature of the reports
• did not use rule of thumb formula to calculate the margin of error
• failed to use the appropriate margin of error to calculate the relevant confidence interval for a comparison confidence interval
• could not identify a proportion outside of the 30%-70% range
• knew very little about statistical terms and used little statistical terminology
• could not identify random allocation as a key idea in experimental design
• identified a key feature of a study design but failed to describe its relevance to the context
• wrote profusely without answering the question being asked
• did not use a “scatterplot” to show a correlation.

Candidates who were awarded Achievement with Merit commonly:

• explained how random allocation in an experiment reduces bias or creates two fair groups
• identified an issue with a survey and related it to representativeness with a specific population
• identified and described one issue with study design as opposed to survey design
• explained that percentages outside the 30-70% range mean that the rule of thumb Margin of error overestimates the actual MOE
• explained that when collecting data for a survey that a representative sample is important
• identified and described a confounding variable with specific relation to one of the original variables
• analysed a statistical report, identifying observational versus experimental studies, distinguishing between correlation and causation (but not necessarily in great depth)
• calculated a comparison confidence interval but could not correctly interpret it within the context or justify why a claim was true
• commented on key features by referring to statistical evidence provided in the reports without specific details
• identified a population in context.

Candidates who were awarded Achievement with Excellence commonly:

• described an issue with study design and directly linked it to the strength of a claim in context
• calculated a comparison confidence interval and interpreted it in context, justifying a statistical claim using correct statistical language, including identifying the population
• identified and described a confounding variable, fully describing the interaction between both variables
• analysed a statistical report, identifying observational versus experimental studies, distinguishing between correlation and causation, and discussing specific language used in the report
• related their findings back to the report
• used and correctly applied statistical language
• wrote precise answers that contained all necessary information.

Standard specific comments

It is important for candidates to understand the effect of sample size on the margin of error. After calculating comparison confidence intervals candidates need to comment on the claim and appropriately discuss the underlying population.

Candidates need to avoid generic, learned answers, for example “old people do not have the internet” without considering the context of the report or providing necessary explanation to relate their observations back to the statistical reports.

Students would benefit from spending more time on statistical inference for experiments, so that they are able to explain that testing for differences between two group means is necessary before making a claim.  Students also need to become familiar with examples of confounding variables and practicing recognising a range of non-sampling errors.

## 91585:  Apply probability concepts in solving problems

Candidates who were awarded Achievement commonly:

• calculated relative frequencies
• understood the context of the question but failed to provide evidence which was sufficient to answer the question at a higher level
• could model a situation with a tree diagram
• could process information provided in a two-way table.

Candidates whose work was assessed as Not Achieved commonly:

• only gave numbers as answers rather than words to support them, and vice versa
• focused on sampling variability or sample size
• were unable to complete a two-way table or Venn Diagram
• gave probabilities greater than one
• believed that true probability was equal to the theoretical or experimental estimate did not understand the concept of mutually and independent events
• struggled with the concept of conditional probability
• only knew the one test for independence P(AB)=P(A)×P(B), not P(AB)=P(A).

Candidates who were awarded Achievement with Merit commonly:

• understood conditional probability but couldn’t always explain their reasoning
• made errors in working but continued consistently to an answer
• were able to carry out complex calculations but struggled to give reasons or justification
• demonstrated a full understanding of mutually exclusive and independent events
• correctly modelled the situation involving three events.

Candidates who were awarded Achievement with Excellence commonly:

• clearly articulated their thoughts and backed up their statements with relevant probabilities
• understood that the true probability was unknown and that it could be estimated by a theoretical probability and/or a result from an experiment
• had a good level of statistical literacy and actually answered the question being asked
• used appropriate rounding at all levels
• stated assumptions clearly when appropriate
• integrated contextual and statistical knowledge
• could recognise when sampling with, and without, replacement was appropriate
• could interpret simulation results in context.

Standard specific comments

Candidates would benefit from improving their use of statistical terminology. In some cases the candidates were held back from obtaining higher grades by their lack of ability to express contexts appropriately.

There was some confusion when working with likelihood ratio (relative risk). A number of candidates used the words “... times more likely …” rather than “… times as likely …”.

## 91586:  Apply probability distributions in solving problems

Candidates who were awarded Achievement commonly:

• selected appropriate probability distribution models
• calculated simple probabilities using probability distribution models
• understood factors that should be kept constant, when modelling practical situations
• understood terms such as ‘at least’ and ‘no more than’
• calculated the expected value of a discrete random variable from a table of probabilities
• could calculate a proportion from a graph
• identified the correct parameters needed to solve a probability distribution problem
• sketched distribution models in comparison to another model
• did not justify their choice of distribution model adequately.

Candidates whose work was assessed as Not Achieved commonly:

• applied an inappropriate distribution model to a problem
• could not draw a normal distribution graph
• could not calculate a given probability for a normal, Poisson or binomial distribution
• misinterpreted inequalities written as text
• could not calculate the expected value from a table
• could not use the correct mode of their graphics calculator or read probability tables correctly
• gave a probability answer greater than one
• could not label a triangular distribution with correct parameters
• made calculation errors or rounded too severely.

Candidates who were awarded Achievement with Merit commonly:

• completed multi-step problems across a range of distributions; linking their responses to the stated context
• explained the limitations of the chosen probability distribution model in terms of context
• correctly calculated expected value, standard deviation, variance and probability and used these to solve problems
• used probability skills e.g. trees, independence, combined and conditional events in conjunction with distributions to solve problems
• understood the concept and purpose of standard deviation and variance in context
• could find inverse probabilities using the normal distribution
• recognised the use of continuity corrections in approximating discrete data with continuous distributions and applied this correctly
• understood the meaning of “either or” and “both” in a probability problem
• communicated their thinking using appropriate statements, e.g. stating probability
• distribution model and parameters, correctly using probability notation and providing evidence or calculations for any general statements they made.

Candidates who were awarded Achievement with Excellence commonly:

• discussed the appropriateness (or inappropriateness) of a probability distribution model by considering features of the probability distribution, statistical evidence and/or the context of the situation
• devised a strategy to solve multi-step probability distribution problems
• clearly explained their reasoning and justified decisions
• used probability skills e.g. trees, independence, combined and conditional events in conjunction with distributions to solve multi-step problems
• could complete expectation algebra calculations (particularly variance).

Standard specific comments

Candidates struggled to draw normal distribution graphs in relation to another normal graph.

The paper highlighted candidate confusion and misconceptions around a number of terms and concepts: in particular, independence, random and continuous. Some candidates used the term continuous when they meant boundedness and were unclear on the idea of randomness as it related to unpredictability.  Some confused independence due to probability and the property for independence to do with variance.

Some candidates were also unclear on the relationship between variance and standard deviation.

Some candidates confused the Poisson distribution condition around rate and proportionality with the binomial condition that the probability remains constant.  They were also unclear as to what a rate was in a particular context and when discussing it frequently confused the term proportional with probability.

Candidates need to be able to explain how the conditions and/or features of a probability distribution model (e.g. randomness, independence, symmetric, upper/lower bounds) are appropriate or inappropriate to the given context.  They also need to be able to support their statements with statistical and/or numerical evidence in sufficient detail that their chain of thinking is evident.

Candidates would benefit from more time spent on continuity corrections as they were unsure how to apply them in context.

Candidates should leave rounding their working until the final step of working.

When given a graph of real data candidates need to consider the information presented in the graph before applying a probability distribution model. Candidates assumed a model of a discrete random variable without considering using the raw data in calculations first.

To gain Merit or Excellence candidates need to demonstrate proficiency in a number of distributions.

### Mathematics and Statistics subject page

Previous years' reports
2016 (PDF, 228KB)

2017 (PDF, 52KB)