DSC6008 Chap.2 Probability, Sampling and Diagnostic Plots
Probability, Sampling and Diagnostic Plots
Separate observations from statistics
A population distribution describes values among units, while a sampling distribution describes a statistic recalculated across repeated samples. That distinction explains why the standard deviation of observations is not the standard error of a mean.
The central limit theorem concerns the repeated-sample behaviour of the mean under suitable conditions, not a transformation of the raw data into a normal shape.
Track the source of every probability statement
Expected value describes the long-run centre of a random quantity, and variance describes squared spread around that centre. A probability model applies to a named variable under stated conditions.
Before selecting a normal, t or chi-square reference, identify whether the variable is an observation, a sample mean, a standardized estimate or a sum of squared deviations. Similar-looking curves can play very different inferential roles.
Reconstruct the standard error from the design
The spread among experimental units is estimated by the sample standard deviation.
The uncertainty in an independent sample mean divides that scale by the square root of the independent unit count. Repeated instrument readings on one unit may improve its recorded response but do not enter that square-root denominator as new assignments.
The sample size must therefore be justified physically, not copied from the number of spreadsheet rows.
Use reference distributions with their conditions
The t distribution acknowledges that the population variance was estimated when a standardized mean is analysed. Its degrees of freedom depend on independent information remaining after parameter estimation.
The central limit theorem can make a mean approximately normal even when individual observations are skewed, but very small samples, severe dependence or dominant outliers can make the approximation unreliable. A formula is not separate from these conditions.
Read a diagnostic as evidence
Histograms and normal Q-Q plots make different features visible.
Systematic Q-Q curvature, changing residual spread or separate bands can identify a model weakness, but no plot proves an assumption. A useful interpretation names the departure and connects it to the later inferential claim.
Heavy tails affect extreme-reference calculations differently from a mild centre mismatch, and a funnel pattern points toward changing variance rather than nonnormality alone.
Link plots to an analysis decision
Start by describing the visible feature without diagnosing it prematurely. Next name the assumption or omitted structure it could reflect.
Finally state what should be checked or changed: inspect influential observations, add a missing grouping variable, transform the response or use a variance model suited to the scale. This sequence makes the plot part of the reasoning rather than an image attached after the result.
What this chapter covers
- 01
Random variables
- 02
Expected value
- 03
Variance
- 04
Sampling distributions
- 05
Standard errors
- 06
Normal Q-Q plots
Worked example · free
Compute uncertainty at the independent-unit level
- 1Use the independent sample count n = 12.
- 2Compute 3.6 divided by the square root of 12.
- 2Report the standard error as approximately 1.039 in strength units.
Key terms
- Sampling Distribution
- The distribution of a statistic over repeated samples generated by the same design.
- Standard Error
- The estimated standard deviation of a statistic under repeated sampling.
- Quantile Plot
- A plot comparing ordered data with quantiles from a reference distribution.
Probability, Sampling and Diagnostic Plots FAQ
Does the central limit theorem make the data normal?
No. It concerns the distribution of a statistic, usually a sample mean, across repeated independent samples. Individual observations can remain skewed. Confusing those two distributions leads to incorrect claims about what a raw-data histogram or residual plot should look like.
Why does independence matter for a standard error?
The square-root reduction in uncertainty relies on new units contributing independent information. Repeated readings from one unit share the same treatment assignment and often common disturbances, so counting them as independent makes the denominator too large and the standard error too small.
What can a Q-Q plot establish?
It can reveal how ordered observations or residuals depart from a chosen reference distribution. A roughly straight pattern supports that aspect of the model, while systematic curvature or extreme departures prompt investigation. It cannot prove normality or invalidate an entire experiment by itself.
Assessment move
For every probability statement, label whether it concerns raw observations or a statistic. Draw the repeated-sampling story in words before applying a standard error. When reading a diagnostic, describe the visible pattern first, then identify the assumption it bears on and the consequence for the intended inference.
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