IB Mathematics Applications & Interpretation SL 4.1—introduction to Statistics Notes

Statistics Starts Here: Learning About A Whole Group From A Small Slice

1. Suppose you want the mean height of every 16 year old in your country.
2. There are hundreds of thousands of them, spread everywhere, and more turn 16 every day.
3. Measuring all of them is impossible: too many people, too much time, too much money.
4. So instead you measure a smaller group and use it to estimate the whole.
5. Every topic in statistics grows from this one move, learning about a huge group from a small slice of it.

Population Versus Sample

1. A population is the entire group you want a conclusion about.
   1. In the height example the population is every 16 year old in the country.
2. A sample is the smaller part of the population that you actually collect data from.
   1. A group of 200 students chosen to stand in for all 16 year olds is one sample.
3. We sample because measuring the whole population is often impossible, too slow, or too expensive.
4. Sometimes measuring destroys the item, so you physically cannot test everything: a factory cannot crash-test every car it builds.

Random Samples: Why Randomness Matters

1. A random sample is one where every member of the population has an equal chance of being chosen.
   1. Whether a member is picked must not depend on their data value.
2. The danger you are fighting is selection bias, where the way you choose systematically favours certain values.
   1. If you only measure the school basketball team, your height estimate comes out far too high.
3. Randomness reduces selection bias because it removes your choices, and your unconscious preferences, from who gets picked.
   1. Across many members the high and low values tend to balance out into a fair picture.

Discrete Versus Continuous Data

1. Discrete data comes from counting, so it takes separate values with gaps between them.
   1. The number of siblings a student has can be $0$, $1$ or $2$, but never $1.5$.
2. Continuous data comes from measuring, so it can take any value in a range.
   1. A height could be $172\,\mathrm{cm}$, $172.4\,\mathrm{cm}$, or $172.43\,\mathrm{cm}$, limited only by your ruler.
3. Quick test: ask whether a value can always exist between two neighbouring readings.
   1. If yes, the data is continuous; if there is a fixed gap, it is discrete.

Reliability Of Data And Sources Of Bias

1. A data source is reliable when its method is transparent, repeatable, and free of built-in bias.
   1. Ask who collected the data, how they chose the sample, and whether they had a reason to skew the result.
2. Bias creeps in through a few classic routes, and each one quietly distorts your conclusion.
   1. Self-selection bias: people opt in to a survey, so you only hear from those who feel strongly enough to reply.
   2. Non-response bias: chosen people who refuse to answer often differ from those who do, leaving a lopsided sample.
   3. Leading questions: wording that pushes an answer, such as 'do you agree this helpful policy benefits everyone?', skews replies.
   4. Unrepresentative sampling frame: the list you draw from misses part of the population, for example phoning only landlines.
3. A bigger sample is not automatically better, and this is the mistake almost everyone makes.

Five Ways To Choose A Sample

Different situations call for different ways of picking a sample, and each one trades fairness against effort.

1. Simple random sampling
   1. Every member has an equal chance, usually by numbering everyone and drawing random numbers on the GDC.
   2. Example: label $1200$ students $1$ to $1200$ and use $\mathrm{randInt}(1,1200)$ to pick $50$ numbers.
2. Convenience sampling
   1. You simply take whoever is easiest to reach.
   2. Example: surveying your own friends about their screen time.
3. Systematic sampling
   1. Order the population, pick a random start, then take every $k$th member, where $k$ is population size divided by sample size.
   2. Example: with $1200$ students and a sample of $60$, $k=20$, so take every $20$th name.
4. Quota sampling
   1. Split the population into groups and fill a fixed count from each, chosen non-randomly by the interviewer.
   2. Example: interview $20$ males and $20$ females at a mall until each quota is full.
5. Stratified sampling
   1. Split the population into strata, then sample randomly from each in proportion to its size.
   2. Example: if a year group is $30\%$ of the school, it should be $30\%$ of the sample.

Designing A Stratified Sample

1. Stratified sampling keeps each subgroup in the sample at the same fraction it holds in the population.
2. Work out the sampling fraction $\dfrac{\text{sample size}}{\text{population size}}$, then multiply each stratum size by it.
3. Round each count to a whole number and check that the parts add up to the required sample size.

Outliers: Values That Do Not Fit

1. Quartiles and the interquartile range are covered in SL 4.2 and SL 4.3, so here we only use them to flag outliers.
   1. Recall that $\text{IQR}=Q_3-Q_1$, the spread of the middle half of the data; this formula is in the formula booklet under Topic 4.
2. An outlier is a value more than $1.5\times\text{IQR}$ from the nearest quartile.
3. Equivalently, any value below the lower fence or above the upper fence is an outlier.

$$\text{lower fence}=Q_1-1.5\times\text{IQR} \qquad \text{upper fence}=Q_3+1.5\times\text{IQR}$$

Deciding What To Do With A Flagged Value

1. First check the fences arithmetic, then ask whether the value is physically possible.
2. If it is impossible, such as a negative age, treat it as a recording error and correct or remove it with a note.
3. If it is possible but rare, keep it and comment on it, since it may carry real information.