Statistical Officer / Government Statistical Service (GSS) Assessment Descriptive Statistics, Data Analysis and Modelling Syllabus

State the formula for the Pearson correlation coefficient $r$.

$$r = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^{2}}\,\sqrt{\sum_{i=1}^{n}(y_i-\bar{y})^{2}}}$$

When should Spearman's rank correlation be used instead of Pearson's correlation?

Spearman's $\rho$ is used for ordinal data or when the relationship is monotonic but non-linear, or when data contain outliers/are non-normal, since it works on ranks rather than raw values.

Explain the maxim "correlation does not imply causation."

A statistical association between two variables does not prove one causes the other; the link may be due to a confounding (lurking) variable, reverse causation, or coincidence.

What is a confounding variable?

A third variable that influences both the explanatory and response variables, creating a spurious or distorted association between them.

Write the equation of a simple linear regression model.

$$y_i = \beta_0 + \beta_1 x_i + \varepsilon_i$$ where $\beta_0$ is the intercept, $\beta_1$ the slope, and $\varepsilon_i$ the random error term.

What does the method of ordinary least squares (OLS) minimise?

OLS chooses $\hat{\beta}_0$ and $\hat{\beta}_1$ to minimise the sum of squared residuals: $$\sum_{i=1}^{n}(y_i - \hat{y}_i)^{2}$$

State the OLS estimator for the slope $\hat{\beta}_1$ in simple linear regression.

$$\hat{\beta}_1 = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^{2}}$$

How do you compute the OLS intercept $\hat{\beta}_0$ once the slope is known?

$$\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}$$ The fitted line always passes through the point $(\bar{x}, \bar{y})$.

Define the coefficient of determination $R^2$ and interpret it.

$$R^{2} = 1 - \frac{\text{SS}_{\text{res}}}{\text{SS}_{\text{tot}}}$$ It is the proportion of variance in the response variable explained by the model, ranging from $0$ to $1$. In simple linear regression $R^2 = r^2$.

What is a residual in regression, and how is it calculated?

A residual is the difference between an observed value and its fitted value: $$e_i = y_i - \hat{y}_i$$ It estimates the unobservable error term.

List the key assumptions of the classical linear regression model (often remembered as LINE).

Linearity of the relationship, Independence of errors, Normality of errors, and Equal variance of errors (homoscedasticity). Also: no perfect multicollinearity among predictors.

What is homoscedasticity, and what is its violation called?

Homoscedasticity means the error variance is constant across all levels of the predictors. Its violation, where error variance changes, is called heteroscedasticity.

How can a residuals-versus-fitted-values plot be used as a diagnostic?

Residuals should scatter randomly around zero with constant spread. A curved pattern suggests non-linearity; a funnel/fan shape suggests heteroscedasticity; clear structure suggests a misspecified model.

What is multicollinearity and how is it commonly detected?

Multicollinearity is high correlation among predictor variables, which inflates standard errors and destabilises coefficient estimates. It is detected using the Variance Inflation Factor (VIF), with values above 5–10 indicating a problem.

What does the Variance Inflation Factor (VIF) measure for a predictor $j$?

$$\text{VIF}_j = \frac{1}{1 - R_j^{2}}$$ where $R_j^2$ is from regressing predictor $j$ on all other predictors. Higher VIF means greater multicollinearity.

How is a Q-Q (quantile-quantile) plot used in regression diagnostics?

It plots the quantiles of the residuals against theoretical normal quantiles. If residuals are normally distributed the points lie close to a straight 45° line; departures indicate non-normality (e.g. heavy tails or skew).

What is an influential observation, and which statistic is commonly used to detect it?

An influential observation substantially changes the fitted regression when removed. Cook's distance ($D_i$) is commonly used; large values flag influential points (a common rule of thumb is $D_i > 1$).

Distinguish a price index (e.g. Laspeyres) from a quantity index.

A price index measures the change in prices of a basket of goods over time (quantities held fixed); a quantity index measures the change in quantities (prices held fixed). Both express the change relative to a base period.

Write the formula for a simple price relative (index) for one item, expressed to base 100.

$$\text{Price index} = \frac{p_t}{p_0} \times 100$$ where $p_t$ is the current price and $p_0$ the base-period price.

Write the formula for the Laspeyres price index and state which weights it uses.

$$L = \frac{\sum p_t q_0}{\sum p_0 q_0} \times 100$$ It uses base-period quantities $q_0$ as weights (a base-weighted index).

Write the formula for the Paasche price index and state which weights it uses.

$$P = \frac{\sum p_t q_t}{\sum p_0 q_t} \times 100$$ It uses current-period quantities $q_t$ as weights (a current-weighted index).

What is the Fisher Ideal price index?

The geometric mean of the Laspeyres and Paasche indices: $$F = \sqrt{L \times P}$$ It is considered "ideal" because it satisfies the time-reversal and factor-reversal tests.

Name the four classical components of a time series.

Trend ($T$), Seasonal ($S$), Cyclical ($C$), and Irregular/random ($I$) components.

Contrast the additive and multiplicative models of a time series.

Additive: $Y_t = T_t + S_t + C_t + I_t$, used when seasonal fluctuations are roughly constant in size. Multiplicative: $Y_t = T_t \times S_t \times C_t \times I_t$, used when seasonal fluctuations grow proportionally with the level of the series.

What is the difference between the seasonal and cyclical components of a time series?

Seasonal variation is a regular, predictable pattern within a fixed period of one year or less (e.g. monthly/quarterly). Cyclical variation is a longer-term, irregular wave-like fluctuation (e.g. business cycles) not of fixed period.

How does a centred moving average help in time series analysis?

It smooths out short-term seasonal and irregular fluctuations to estimate the underlying trend; "centred" alignment (e.g. a 2×4 average for quarterly data) ensures the smoothed value lines up with the correct time point.

What is exponential smoothing, and write its basic updating equation.

A forecasting method that weights recent observations more heavily, decaying exponentially. $$\hat{y}_{t+1} = \alpha y_t + (1-\alpha)\hat{y}_t$$ where $0 < \alpha < 1$ is the smoothing constant.

Define MAE and RMSE as forecast accuracy measures and note a key difference.

$$\text{MAE} = \frac{1}{n}\sum_{i=1}^{n}|y_i - \hat{y}_i|, \quad \text{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^{2}}$$ RMSE penalises large errors more heavily than MAE because errors are squared.

What is MAPE and why is it useful for comparing forecasts across series?

$$\text{MAPE} = \frac{100\%}{n}\sum_{i=1}^{n}\left|\frac{y_i - \hat{y}_i}{y_i}\right|$$ It expresses error as a percentage, making it scale-independent so forecasts on series with different units/magnitudes can be compared. (Undefined when $y_i = 0$.)

What is a naïve forecast, and why is it a useful benchmark?

A naïve forecast sets the next value equal to the most recent observed value, $\hat{y}_{t+1} = y_t$. It is a baseline against which more sophisticated models should improve; a model worse than naïve adds no value.

In quality-control terms, how are common-cause and special-cause variation distinguished?

Common-cause variation is the inherent, random, predictable noise of a stable process. Special-cause (assignable) variation is due to specific, identifiable factors producing unexpected anomalies, signalling the process is out of control.

On a control chart, what do points beyond the $\pm 3\sigma$ control limits indicate?

They signal special-cause variation — an anomaly unlikely to arise from random common-cause variation alone — prompting investigation, since under normality only about $0.3\%$ of points should fall outside $\pm 3\sigma$.

Define an outlier and give two common ways to identify one.

An outlier is an observation that lies abnormally far from the rest of the data. It can be flagged by the IQR rule (beyond $Q_1-1.5\times\text{IQR}$ or $Q_3+1.5\times\text{IQR}$) or by z-scores exceeding a threshold such as $|z|>3$.

What is the z-score of an observation and what does it represent?

$$z = \frac{x - \bar{x}}{s}$$ It represents the number of standard deviations an observation lies above or below the mean, allowing standardised comparison and outlier detection.

Why is statistical significance not the same as practical significance?

A result can be statistically significant (unlikely under the null hypothesis) yet have an effect size too small to matter in practice, especially with large samples. Valid conclusions consider both the p-value and the magnitude/importance of the effect.

What is the ecological fallacy when drawing conclusions from data?

It is the error of inferring relationships about individuals from aggregate (group-level) data; associations seen at the group level may not hold for individuals within those groups.

What is Simpson's paradox?

A trend that appears within several groups of data reverses or disappears when the groups are combined (aggregated), usually due to a confounding variable or unequal group sizes; it warns against ignoring subgroup structure.

List three threats to drawing valid conclusions from a statistical analysis.

Selection/sampling bias, confounding variables, small or unrepresentative samples, multiple comparisons inflating false positives, and conflating correlation with causation. (Any three.)

What does reproducibility mean in statistical analysis, and how does it differ from replicability?

Reproducibility means obtaining the same results from the same data and same code/analysis. Replicability means obtaining consistent results from new data collected in a new study. Reproducibility concerns the analysis pipeline; replicability concerns the scientific finding.

Name three practices that support a reproducible analysis workflow.

Use version control (e.g. Git), write scripted/code-based analysis rather than manual steps, document with literate programming (e.g. R Markdown/Jupyter notebooks), record software/package versions, and set random seeds. (Any three.)

Why is setting a random seed important for reproducible analysis?

Setting a seed fixes the starting state of the pseudo-random number generator, so any stochastic procedure (simulation, bootstrapping, train/test splits) produces identical results each time the code is run, enabling exact reproduction.