South 고등학교 2학년 Practical Statistics
Chapters: 4
1. Statistics and statistical problems
Why statistics matters · Statistical problem-solving cycle · Population and sample
- The Statistical Inquiry Cycle – A statistical inquiry answers a question with data in five steps: Problem (ask a clear question), Plan (decide who to ask, which variable, how), Data (collect and record), Analysis (tables, graphs, averages) and Conclusion (answer the question, state limits, ask new questions). Then the cycle can start again.
- Sampling: Learning About a Population from a Sample – A population is the whole group we want to know about; a sample is a smaller part we actually check. A good sample is chosen at random so that it represents the population. Simple random sampling gives everyone an equal chance; stratified sampling takes the right share from each group; systematic sampling takes every k-th item. Different samples give slightly different answers (sampling variation), but bigger samples wobble less (law of large numbers). A biased sample gives a wrong answer however big it is.
2. Collecting and organising data
Types of data · Data collection methods · Choosing graphs · Centre and spread
- Data Handling: Collect, Organise and Show Data – Data is a set of facts, such as answers, counts or measurements. Data can be qualitative (words) or quantitative (numbers); numbers are discrete (counted) or continuous (measured). We collect data by surveys, observation, experiments or from existing sources, then organise it with tally marks into a frequency table. We show it with the right graph: bar graphs to compare groups, pie charts to show parts of a whole, line graphs for change over time and scatter graphs for links between two variables. Computers store data as structured tables or unstructured text, images and sound.
- Measures of Dispersion: Range, Mean Deviation, Variance and SD – Dispersion means spread: how far the values sit from the centre. Range = largest − smallest. Mean deviation = average distance from the mean (or median). Variance = average of squared distances from the mean. Standard deviation = √variance. The same ideas work for grouped data when every term is multiplied by its frequency.
3. Analysing data
Normal and t distributions · Estimating a population mean · Estimating a population proportion · Hypothesis testing
- Normal Distribution – A normal distribution is a continuous, symmetric, bell-shaped distribution fixed by its mean μ (centre) and standard deviation σ (spread): X ~ N(μ, σ²). Mean = median = mode. About 68% of values lie within 1σ of μ, 95% within 2σ and 99.7% within 3σ. Any normal value is turned into a standard score z = (x − μ)/σ, which follows N(0, 1); probabilities are areas under the curve, read from a table or calculator as Φ(z).
- Confidence Intervals – A confidence interval is a range of believable values for an unknown population number (a mean μ or a proportion p), worked out from one sample. It has the shape estimate ± margin of error, where margin of error = critical value × standard error. A 95% level means the method catches the true value in about 95% of samples. Higher confidence gives a wider interval; a bigger sample gives a narrower one. If a claimed value lies outside the interval, the data give evidence against the claim.
- Hypothesis Testing – A hypothesis test checks a claim about a population using a sample. Start with the null hypothesis H₀ (no change, e.g. p = 0.5) and the alternative H₁ (what we suspect, e.g. p > 0.5). Choose a significance level such as 5%. Work out how likely the sample result (or more extreme) is if H₀ were true: the p-value. If the p-value is below the level, or the result falls in the critical region, reject H₀. Otherwise there is not enough evidence to reject it. Type I error = rejecting a true H₀; Type II error = not rejecting a false H₀.
4. Statistical inquiry
Carrying out a statistical inquiry · Critiquing statistical studies
- The Statistical Inquiry Cycle – A statistical inquiry answers a question with data in five steps: Problem (ask a clear question), Plan (decide who to ask, which variable, how), Data (collect and record), Analysis (tables, graphs, averages) and Conclusion (answer the question, state limits, ask new questions). Then the cycle can start again.