Systematic vs simple random sample

Systematic vs simple random sample: the short answer

Both methods use chance, and both can give every individual in the population exactly the same chance of ending up in the sample. Only one of them is a simple random sample.

A simple random sample (SRS) of size $n$ is drawn so that every possible sample of size $n$ is equally likely. The condition is about whole samples, not about individuals.

A systematic random sample puts the population in some order, fixes an interval $k$, draws a starting position at random from the first $k$ places, then takes every $k$-th unit after it. The random start is the only randomness in the procedure. Once it is drawn, every remaining member of the sample is determined by arithmetic.

That is the whole distinction. Because the start decides everything, only $k$ different samples can ever occur, so almost every subset of size $n$ has probability zero. Equal chance for each individual survives. Equal chance for each group does not.

What each procedure actually does

To take an SRS, you need a list. Number the population 1 to $N$, where $N$ is the population size, generate random numbers in that range, ignore repeats, and take the first $n$ distinct labels. Any combination of $n$ individuals can come out, and all of them are equally likely.

To take a systematic sample, you need an order rather than a list. Choose the interval $k$, draw a start at random from the positions 1 through $k$, then take positions start, start $+ k$, start $+ 2k$, and so on to the end. When $N$ is a multiple of $n$ you set $k = N/n$ and every possible sample comes out at exactly size $n$. You never need to know who is on the list in advance, which is why this method can be run at a checkout line or a factory belt where an SRS cannot.

Now compare the chance one named individual has. In an SRS it is $n/N$, the sample size over the population size. In a systematic sample, individual $i$ is chosen exactly when the random start matches its position within the cycle, and there are $k$ equally likely starts, so the chance is $1/k$. With $k = N/n$ those two numbers are the same. Equal individual probability is not what separates the methods.

Enumerate a roster of 12 and the trap is visible

Take an alphabetical roster of 12 students and a sample of 3. For the systematic plan, $k = \frac{12}{3} = 4$, and the random start is one of positions 1, 2, 3, 4.

Four samples, each with probability $\frac{1}{4}$. Every student appears in exactly one of the four, so every student has probability $\frac{1}{4}$ of being selected.

Now the SRS. The number of samples of size 3 from 12 students is $\binom{12}{3} = 220$, and each has probability $\frac{1}{220}$. Any one student sits in $\binom{11}{2} = 55$ of them, so that student's chance is $\frac{55}{220} = \frac{1}{4}$, identical to the systematic figure.

The two methods part company at the group level. The systematic plan can produce 4 of the 220 subsets, so the other 216 have probability zero. Students 1 and 2 can never be sampled together, because any two selected students sit 4 or 8 places apart on the roster. Under the SRS that same pair turns up in 10 of the 220 samples, a probability of $\frac{10}{220} = \frac{1}{22}$. Same chance for each student, wildly different chances for each group of students.

The differences side by side

Read the first two rows together. Every other difference on this page follows from the fact that a systematic sample randomizes once and then stops.

The one real failure mode: a list with a matching period

Systematic sampling is not fragile in general. Its main failure is one specific situation: when the order of the list repeats on a cycle whose length lines up with $k$. Every unit you take then sits at the same phase of that cycle, so the sample sees one slice of a pattern instead of a spread across it. The other thing to watch is arithmetic, because unless $N$ is a multiple of $k$ the starts return samples of different sizes and the sample mean is no longer guaranteed to be unbiased.

A calendar is the standard case. Sample every 7th day and every day in the sample falls on the same weekday. Suppose a shop's daily customer counts run Monday 40, Tuesday 44, Wednesday 46, Thursday 48, Friday 62, Saturday 96, Sunday 84, and two identical weeks give a 14-day population with mean $\frac{840}{14} = 60$ customers per day. With $k = 7$ the sample is two days of the same name, so the sample mean equals that weekday's count. The seven possible sample means are 40, 44, 46, 48, 62, 96, and 84.

Notice what is and is not broken. Those seven means average to $\frac{420}{7} = 60$, exactly the population mean, so the method is still unbiased. What collapses is precision: you get one of those seven and nothing in between. A Saturday start reports 96, which is 60% above the truth, and a Monday start reports 40, a third below it. An SRS of 2 days could land on both Saturdays too, but only with probability $\frac{1}{\binom{14}{2}} = \frac{1}{91}$, while the systematic plan hands you that sample with probability $\frac{1}{7}$, thirteen times as often.

The repair is to know what the order means. If the ordering has nothing to do with the variable being measured, a systematic sample behaves much like an SRS. If it might cycle, randomize the order before you start, or pick an interval that shares no factor with the cycle length and a list long enough to walk the whole cycle. Both halves are needed. Stepping every 7th or every 14th day lands on one weekday however long the list runs, and stepping every 10th day visits all seven weekdays only once the list runs to 70 days, since seven weekdays take seven selections and a start as late as day 10 needs 70 days to make seven of them. On the 14 days above, a step of 10 returns one or two days and can still hand you Saturday's 96 on its own.

When to use which

Use an SRS whenever you hold a complete list of the population and can reach anyone on it. It assumes nothing about order, leaves every subset possible, and is the benchmark the other methods are judged against. A registrar with a numbered roster of 8,000 students can run one straight off the list, and only needs something else when a known trait splits the students in a way that moves the answer.

Use a systematic sample when the list does not exist yet but the individuals come past you in an order: every 20th item off a production line, every 12th shopper through a door, every 50th file in a drawer. You do not need to know $N$ in advance to start, and the sample spreads itself evenly along the list rather than clumping at one end the way a small SRS sometimes does. Before you commit, ask one question: does anything about this order repeat? If the answer is yes and the cycle length shares a factor with $k$, change the interval or randomize the order first. Nearness is not the test: against a 7-day cycle, $k = 6$ and $k = 8$ are the two closest intervals and neither one locks onto it, while $k = 14$ sits further away and locks on completely.

Both are random sampling methods, so both support generalizing to the population, and neither supports a cause-and-effect claim on its own. Choosing among all the named methods, including stratified and cluster, is worked through in how to choose a sampling method.

The classic mix-up and how to avoid it

The mistake the exam is built to catch is the sentence "everyone had an equal chance of being picked, so it was a simple random sample." Equal individual chance is necessary for an SRS and it is not sufficient. The roster above makes that concrete: 1 in 4 for every student under both plans, and 216 subsets that only one of them can produce.

Two habits keep it straight.

1. Ask what the randomness picks. If chance selects each member of the sample, the plan can be an SRS. If chance selects one thing (a starting position, a set of clusters, a coin flip) and that one thing determines the rest, it is not.
2. Try to name a subset that can never happen. Two neighbours on the list can never both be in a systematic sample. One impossible sample of size $n$ is enough to rule out an SRS.

That second test also rules out stratified and cluster samples, which fail the same condition for their own reasons, set side by side in simple random vs stratified sampling.

The error also runs the other way. Calling a systematic sample "not random because the interval is fixed" is wrong too: the random start is genuine randomness and it is what gives every individual a known, equal chance. For any sample size worth taking, a systematic sample is a random sample and it is not an SRS. Both halves of that sentence get tested.