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Random start
Definition
A random start is an initial position chosen by a specified random mechanism before a systematic sampling sequence is followed. In the simplest equal-interval design, one start is selected uniformly from the first sampling interval, after which every kth unit is chosen. The start gives the sequence its randomization. This entry uses the survey-sampling meaning; random starting points in numerical optimization and randomized questionnaire order are separate applications of the same everyday phrase.
What it means
Selecting every tenth record is a rule, but starting automatically with the first record is not a random-start design. Choosing the initial position at random allows inclusion probabilities to be defined for the resulting sequence. It does not make every possible group of records selectable, and it does not guarantee a representative-looking realized sample. The order of the sampling frame matters because the chosen start determines which positions are selected together.
How it works
For the simple case where population size N is an integer multiple of desired sample size n, set the interval k = N/n. Select r uniformly from the integers 1 through k. Include positions r, r + k and so on through r + (n - 1)k. Record the frame order and selection method so the sample can be audited. More complicated designs, including non-integer intervals and unequal-probability sampling, need their own rules; the simple formula should not be applied mechanically to every setting.
Three practical examples
Example 1: A quality team chooses a random position among the first dozen items and then inspects every twelfth item. Example 2: A survey organization selects households at regular intervals from an ordered address list after drawing an initial offset. Example 3: A records auditor uses a random start before selecting every kth transaction from a defined frame. In each example, missing units, exclusions and nonresponse can change the final sample and require separate attention.
Think of it this way
Think of a comb placed over a row of records. The distance between its teeth is fixed, while the random start slides the comb into one of several initial positions. Once placed, the entire selected pattern follows. The slide creates randomization, but the fixed spacing means some records always travel together and other combinations can never occur. A repeated pattern in the underlying row can align badly with the comb.
Worked example
Suppose a frame has 24 records and the desired sample contains six, so k = 24/6 = 4. A random start of 3 selects positions 3, 7, 11, 15, 19 and 23. There are only four possible samples, corresponding to starts 1 through 4, and each record has inclusion probability 1/4. Now suppose the values repeat 100, 0, 0, 0 across the frame. The population mean is 25. Start 1 gives sample mean 100; each other start gives mean 0. Averaging over the four equally probable starts gives 25, so the sample mean is unbiased under this design. Nevertheless, every realized sample mean is far from 25. Its design variance is [75 squared + 3 × 25 squared]/4 = 1,875. Randomization does not eliminate harmful periodicity.
Illustrative case study
Fictional case study: Aspen Audits samples regularly spaced transactions using a random start. Before selection, an analyst notices that the ordering repeats a weekly processing pattern aligned with the proposed interval. The team reviews the design and considers a different ordering or sampling approach rather than assuming the random start solves the problem. This fictional case illustrates design review. It does not claim that changing the interval always removes periodicity or that the resulting sample automatically supports every planned inference.
Common misconceptions and mistakes
Common mistakes include using a convenient start and calling it random, treating systematic sampling as simple random sampling, and ignoring frame order. Equal individual inclusion probabilities do not imply equal probabilities for all samples of a given size. A single systematic sample can also create difficulties for estimating variance without additional assumptions or design features. The random-start mechanism should not be confused with the computer's random-number seed, which helps reproduce the draw but is not the selected frame position.
Business use
Business use: A random start makes systematic selection practical for audits, inspections and surveys when an ordered stream or list is available. Define the frame, interval and permitted starts before observing outcomes. Check for periodic structure and document departures from the selected sequence. Use estimation methods suited to the actual design. If the work requires uncertainty estimates, plan how they will be obtained rather than assuming the ordinary simple-random-sample formula applies unchanged.
People also ask
Is choosing the first convenient record a random start?
No; the starting position must follow a stated random selection rule. Q: Does every record have the same chance in the simple design? A: Yes, when the specified equal-interval conditions hold. Q: Can every set of six records be selected in the example? A: No; only four structured samples are possible.
Check your understanding
Why is the sample mean unbiased in the periodic example despite poor individual samples?
Answer: Its average across all four equally probable starts equals the population mean.
What does the example warn against?
Answer: Confusing unbiasedness over repeated sampling with a guarantee that one realized sample closely resembles the population.
Related terms
- Systematic sampling selects units at regular intervals
Learn next
Enumerate every possible start for a small frame and calculate unit inclusion probabilities. Compare the resulting possible samples with simple random sampling. Then reorder the same values and examine how the distribution of sample means changes. Study variance estimation and non-integer-interval designs before extending the basic method to a more complex survey.
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Sources
Sources: Statistics Canada, Probability sampling, systematic sampling section: https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm ; Survey Methods and Practices: https://www150.statcan.gc.ca/n1/en/pub/12-587-x/12-587-x2003001-eng.pdf . These primary statistical-agency sources describe random starts, intervals and periodicity concerns. The 24-record calculation and fictional audit case are original illustrations.
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