Statistics and Data
Statistics and Data covers summarising data, measures of centre and spread, correlation and regression, the normal distribution, and introductory inference. Higher Level questions expect interpretation of results, not just calculation.
Core rules
- Centre: the mean uses every value and is affected by outliers; the median is resistant to outliers; the mode is the most frequent value.
- Spread: range, interquartile range (IQR=Q_3-Q_1), and standard deviation. For a data set, (SD=\sqrt{\frac{\sum(x-\bar{x})^2}{n}}).
- Correlation coefficient (r) lies between (-1) and (1). Correlation does not imply causation. A regression line is used to predict within the observed data range.
- Normal distribution: (z=\frac{x-\mu}{\sigma}). The empirical rule gives approximately (68%), (95%), and (99.7%) within 1, 2, and 3 standard deviations.
- Inference for a proportion: (\hat p) is the sample proportion. A 95% confidence interval is (\hat p\pm1.96\sqrt{\frac{\hat p(1-\hat p)}{n}}).
Worked example
In a sample of 100, 40 have a characteristic, so (\hat p=0.4). The standard error is (\sqrt{0.4(0.6)/100}=\sqrt{0.0024}\approx0.049). The 95% margin of error is (1.96(0.049)\approx0.096), giving an interval from about 0.304 to 0.496.
Traps
Using the mean for skewed data without comment; confusing standard deviation with variance; claiming causation from correlation; extrapolating a regression line beyond the data; interpreting a confidence interval as a probability about a fixed parameter; forgetting the square root in a standard error.