Systematic sample

Put the population in some order, fix an interval $k$ (the sampling interval), draw a starting position at random from the first $k$ places, then take every $k$-th unit after it. That is the whole procedure, and the only randomness in it is the start.

A club has 500 members and you want 50, so $k = 500/50 = 10$. Draw a random integer from 1 to 10, say 7, and the sample is members 7, 17, 27, and so on up to 497. Two facts follow. Every member sits in exactly one of the 10 possible samples, so each has probability $1/10$ of being chosen, and because those 10 samples split the club into 10 groups of 50, their means average to the club mean exactly, so the method is unbiased for $\mu$ (mu) whatever order the list is in. But only 10 samples are possible, out of roughly $10^{69}$ subsets of size 50, and members 7 and 8 can never appear together, since any two selected members differ by a multiple of 10.

The sentence to correct runs the other way: "picking every 10th customer is not random, because the interval is fixed." The random start is the randomness, and it is enough to give every member a known, equal chance. What it does not give is the property that defines a simple random sample, which is about groups rather than individuals. Systematic sampling is a random sampling method and it is not an SRS.

The failure case is a list whose order repeats on a cycle matching $k$. Survey every 7th day from a random start in the first week and every day in the sample falls on the same weekday, so a Wednesday start gives an all-Wednesday sample. The center does not shift, but your one sample sees a single phase of the cycle and can sit far from the truth. When the ordering has nothing to do with the variable being measured, a systematic sample behaves much like an SRS and is far easier to run at a door or a checkout line.