Exam Revision

Types of evidence

• Personal testimony

• Reputable research journal

• Reproducible research

• Nature of the data collection

Selection bias

When participants are more likely to be chosen than others

Randomised controlled trial

One group receives the intervention (the experimental group), while the other (the control group) receives a placebo, standard treatment, or no treatment at all. 

Randomised controlled double-blind trial

A research study where neither the participants nor the researchers know who is receiving the intervention or placebo, reducing bias in results.

Consent bias

When participants choose whther or not they take part in the experiment

Survivior bias

• Only happens after the study

• An observed "improvement" may happen because there are dropouts of the sickest subjects

  

Adherer bias

Certain participants (adherers) keep taking treatment (placebo) as opposed to non-adherers = "improvement" in treatment group due to the adherers

3 precautions of observational studies

1. Obsevational studies can’t establish causation only association

2. Observational studies may present as an RCT

3. Confounding variables may influence results if not properly controlled.

Observational studies

An observational study is one in which the investigator cannot use randomisation for allocation to groups. The assignment of subjects is outside the control of the investigator.

Contemporaneous control

A contempoaneous control group occurs at the same time as the treatment group.

What is the simpson's paradox?

It’s when a trend appears in separate groups of data but reverses or disappears when the groups are combined, often due to a confounding variable. It highlights how misleading conclusions can arise if data isn't properly stratified.

What is data in statistics?

Data is information about the subjects being studied, usually referring to a sample rather than the whole population.

What does IDA stand for and what is it?

Initial Data Analysis – a first look at the data before answering research questions. It checks quality, structure, and suggests patterns or new questions.

What are the key steps in IDA?

A variable measures or describes an attribute of the subjects; each column in a tidy dataset is a variable.

What does high dimensional data mean?

There are more variables (p) than subjects (n), common in big data.

What are the two main types of variables?

Quantitative (numerical) and qualitative (categorical)

How is one qualitative variable visualised?

With a single barplot, where categories are on the x-axis.

How are two qualitative variables visualised?

With a double barplot using colour to show the second variable.

What is a histogram used for?

To visualise the distribution of a quantitative variable across class intervals.

What’s the difference between a standard and density histogram?

Standard shows counts; density shows percentages (area = 100%).

What is the rule of thumb for number of histogram intervals?

Use 10–15 class intervals to avoid over/under condensing data.

What is a sliced histogram?

A histogram where a qualitative variable is shown by slicing each bar with colour.

What do the edges of a boxplot represent?

The 25th and 75th percentiles; the box shows the middle 50% of the data.

How are outliers identified in a boxplot?

They are outside the thresholds:
LT = Q1 − 1.5×IQR
UT = Q3 + 1.5×IQR

What is a comparative boxplot?

A boxplot comparing a quantitative variable across levels of a qualitative variable.

What is a filtered scatterplot?

A scatterplot with more variables shown using colour or shape to distinguish them.

Why is age usually treated as quantitative?

Because it's easier to convert from quantitative to qualitative, not the other way around.

What is the purpose of numerical summaries?

To reduce all data to a single statistic, making it easier to communicate and compare key features like centre and spread.

What are the main types of numerical summaries?

Maximum, minimum, centre (mean, median), and spread (standard deviation, range, IQR).

What is the mean?

The balancing point of a distribution, where the sum of deviations on both sides equals zero.

What is the median?

The middle value of an ordered dataset; 50% of values lie above and 50% below it.

When is the median more useful than the mean?

When the data is skewed or contains outliers, because the median is robust and unaffected by extreme values.

When is the mean more useful than the median?

For symmetric data with few outliers; e.g., calculating averages for prediction or reporting.

How do mean and median behave with skewed data?

Left skew: mean < median

• Right skew: mean > median

• Symmetric: mean ≈ median

What is robustness in statistics?

A property of a summary (like the median or IQR) where it remains reliable even with outliers or skewed data.

Why must the mean gap always equal zero?

Because the mean is the balancing point; all positive and negative deviations from the mean cancel out.

What does standard deviation measure?

The average spread or dispersion of data points from the mean.

What is the RMS (root mean square) in standard deviation?

It calculates the square root of the average squared deviations from the mean.

When is it okay to treat a dataset as a population vs. a sample?

If the dataset includes all subjects of interest (e.g. all house sales in one suburb in one month), it's a population; otherwise, it's a sample.

What percentage of data falls within 1, 2, and 3 standard deviations of the mean?

• 68% within 1 SD

• 95% within 2 SDs

• 99.7% within 3 SDs

What is the interquartile range (IQR)?

The range of the middle 50% of the data, calculated as Q3 − Q1; it's robust against outliers.

What is the difference between quartiles and quantiles?

Quartiles divide data into 4 parts, while quantiles divide data into q equal parts.

What are standard units (z-scores)?

The number of standard deviations a data point is from the mean:

What is the normal curve?

A probability density function for a continuous variable.

What are the parameters of a normal curve?

Population mean and population standard deviation.

What does area under a density histogram represent?

Total probability = 1 (or 100%).

What does P(X < x) represent in a normal curve?

The area under the curve to the left of x.

What rule do all normal curves follow?

The 68%-95%-99.7% rule.

How can a general normal curve be rescaled?

By converting to standard normal using z-scores.

What does the pnorm() function do in R?

Calculates area under the normal curve for a given x.

What does the qnorm() function do in R?

Finds the x-value for a given area under the normal curve.

How to check if data is normally distributed?

Use graphical summaries, 68-95-99.7 rule, and QQ plot.

What is measurement error?

Difference between measured and exact value.

What causes chance error?

Random variation when repeating a measurement.

How to estimate chance error?

Replicate the measurement and calculate standard deviation.

What is bias in measurement?

A constant error added/subtracted from each measurement.

What are the 6 steps in linear regression?

Scatterplot, correlation, residual plot, check assumptions, predict.

What does a scatterplot show?

Relationship between two quantitative variables.

What does the correlation coefficient measure?

Strength and direction of a linear relationship.

What does r = +1 or -1 mean?

Perfect linear association.

What is the regression line?

Line that best predicts Y from X.

What is a residual?

Difference between actual and predicted Y value.

What does a good residual plot look like?

Random scatter around horizontal line.

What does homoscedasticity mean?

Equal spread of residuals across the range of X.

What is the equation for a regression line?

Y = a + bX.

When can we use the regression line for prediction?

After checking assumptions and model fit.

What is RMS error?

Root mean square of residuals; measures prediction error.

What is ecological correlation?

Correlation between grouped means; often overestimates true association.

Does correlation imply causation?

No, correlation shows association, not cause.

What are standard units in regression?

Z-scores; number of SDs from the mean.

What does extrapolation mean?

Predicting outside the range of observed data.

What is a vertical strip in a scatterplot?

Data at one value of X; used to assess spread/homoscedasticity.

What is the prosecutor's fallacy?

Mistaking P(DNA match | Innocent) for P(Innocent | DNA match).

What is chance (probability)?

Long-run frequency of an event occurring.

What is a complement in probability?

P(Event) = 1 - P(Complement).

What is conditional probability?

P(Event A | Event B): probability A occurs given B occurred.

What is the multiplication principle?

P(A and B) = P(A) × P(B | A).

What does it mean for two events to be independent?

P(B | A) = P(B); knowing A doesn't affect B.

What ensures independence in sampling?

Drawing with replacement.

What ensures dependence in sampling?

Drawing without replacement.

What are mutually exclusive events?

Events that cannot occur together.

What is the addition rule for mutually exclusive events?

P(A or B) = P(A) + P(B).

What is a binomial trial?

An experiment with two outcomes, fixed n, and constant p.

What does dbinom() do in R?

Calculates exact binomial probabilities.

What does pbinom() do in R?

Calculates cumulative binomial probabilities.

What is chance variability?

Random variation in outcomes from a chance process.

What is the law of large numbers?

As trials increase, observed proportion approaches expected.

What is the gambler's fallacy?

Belief that deviations will 'even out' in the short term.

What is the box model?

A way to simulate random draws from a population.

What is chance error?

Observed Value - Expected Value.

What is the standard error (SE)?

The SD of the chance error.

What is the expected value (EV)?

The average outcome in a chance process.

How does the box model apply to gambling?

Box contains winnings/losses; draws = plays.

How do you model binary outcomes in a box?

Use 1 for success and 0 for failure.

What is the normal approximation to the binomial?

Using the normal curve to estimate binomial probabilities.

What is the central limit theorem?

Sums/means of large random samples follow a normal distribution.

When can you use the normal approximation?

When number of draws > 30 and data is not too skewed.

What is continuity correction?

Adjusting endpoints by 0.5 when approximating discrete data with normal.