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Is this true?

Systematic sampling is always at least as efficient as SRS

No — it is false.

The counterexample

A population with a periodic pattern of period k, sampled every k-th unit

Every sampled unit falls at the same phase, so the sample can be maximally unrepresentative.

The kind of mistake this is

Execution slip

The idea was right. The computation was not.

Drill statements like this
statisticssampling

Others that fail the same way

From Sampling and Design of Experiments › SRS, stratified and systematic sampling

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