Simple Random Sampling
A plain-English guide to simple random sampling — what it is, the formula, two worked examples, and a 10-step field protocol you can follow today.
Quick answer
Simple random sampling is a probability sampling method where every unit in a population has an equal and independent chance of being chosen, so every possible sample of size n is equally likely. You list and number every unit, then draw n of them at random — usually without replacement. It removes selection bias and lets you generalise your results to the whole population.
- Each unit's chance of selection equals n / N (sample size divided by population size).
- Works best when you already have a complete list of the population — a sampling frame.
- Almost always done without replacement, so no unit is picked twice.
Picture a bag of 100 numbered marbles. Close your eyes, shake the bag, and pull out 10. That is simple random sampling in one sentence. Every marble had the same chance of landing in your hand, and the 10 you pulled were not chosen because they looked interesting or were easy to reach.
What is simple random sampling?
Simple random sampling is a probability sampling method in which every unit in a population has an equal and independent chance of being selected, so every possible sample of a given size is equally likely. It is the baseline that every other sampling design is measured against.
Two conditions must hold for a draw to count as a true simple random sample. First, every unit needs the same selection probability. Second, the selections must not depend on each other beyond the no-repeat rule. When both hold, your sample is, on average, a small mirror of the population, and the usual maths for confidence intervals and significance tests applies cleanly. This idea traces back to Neyman’s 1934 work on representative sampling, which set out why random selection beats hand-picking "typical" units.
The opposite of this is convenience sampling, where you take whoever or whatever is easiest to reach. That is fast, but almost always biased, because easy-to-reach units are rarely typical units. The simple random sampling definition exists precisely to rule that bias out.
When to use simple random sampling (and when not to)
Use simple random sampling when you have a complete list of your population and no strong reason to treat any subgroup differently. It is the safe default for small to medium studies on a fairly uniform population.
It starts to struggle in three situations. If the population is spread across a huge area, a random draw can scatter your sites and eat your travel budget. If an important subgroup is rare, a pure random draw might miss it completely. And if there is no list to draw from in the first place, you cannot do it at all. Here is how simple random sampling in research methodology compares with the common alternatives.
| Method | How it works | Best when | Main weakness |
|---|---|---|---|
| Simple random | Number every unit, draw at random | You have a full list and a fairly uniform population | Can miss small subgroups; scatters field sites |
| Systematic | Pick every k-th unit after a random start | Units are already in a list or a line (transects, files) | Bias if the list has a hidden pattern |
| Stratified | Split into groups (strata), sample within each | Clear subgroups you want represented | Needs group information up front |
| Cluster | Randomly pick whole groups, then sample inside them | Population is spread in natural clumps | Less precise per unit sampled |
| Convenience | Take whatever is easiest to reach | Quick pilots and rough checks only | Biased; results cannot be generalised |
When you compare simple random sampling vs systematic sampling, the key question is whether your list has a repeating pattern. If it does, systematic sampling can lock onto that pattern and mislead you — a problem Madow and Madow described in their 1944 work on systematic sampling. When subgroups matter, stratified sampling is usually the better choice, and Royall’s 1970 paper on finite population sampling theory explains why a model-based view can sometimes beat pure randomisation when you have good auxiliary information.
How to do simple random sampling: step by step
To select a simple random sample, you build a complete list, number it, generate random numbers, and pick the matching units without repeats. Five steps cover the whole process.
- Define the population. Write down exactly who or what is eligible. A vague population gives a vague result.
- Build the sampling frame. Make a complete list of every unit. This is the step people skip, and it is where most studies go wrong.
- Number every unit. Assign 1 to N, with no gaps and no duplicates.
- Decide the sample size n. Use a precision or power calculation, not a guess.
- Draw the numbers at random. Use a random number generator or a random number table. Discard any number outside 1 to N, and discard repeats, until you have n unique units.
That is the whole simple random sampling method. The discipline lives in steps 1 and 2; the randomness in step 5 is the easy part once a computer or a free tool does it.
The simple random sampling formula, in plain English
The core simple random sampling formula is the selection probability: every unit is chosen with probability n / N. Everything else — the sampling fraction, the standard error, and the finite population correction — follows from that one fact, a result formalised in Horvitz and Thompson’s 1952 paper on sampling without replacement.
Selection probability and sampling fraction
Where: n = sample size, N = population size, f = the sampling fraction (the share of the population you measured). The inclusion probability for a simple random sample without replacement is the same n / N for every unit.
The number of distinct samples you could possibly draw without replacement is the binomial coefficient C(N, n) = N! / (n!(N − n)!), and a true simple random sample gives each of them an equal chance — the property Hansen and Hurwitz built their 1943 sampling theory around.
The sample mean and the finite population correction
The sample mean is an unbiased estimate of the population mean. Its standard error shrinks as n grows, and shrinks a little further when your sample covers a large slice of a small population. That second effect is the finite population correction, or FPC, and it is the part most quick guides skip — even though Cochran’s 1946 comparison of sampling designs showed it can matter for precision.
Where: x̄ = sample mean, s = sample standard deviation, n = sample size, N = population size, and √(1 − n/N) is the finite population correction. A common rule of thumb is to drop the FPC when the sampling fraction f is below about 5%, since it barely changes the answer.
Two worked examples of simple random sampling
Numbers make this method click faster than any definition. Here are two full simple random sampling examples — a small classroom case using the lottery method, and a field-survey case that needs the finite population correction.
Example 1: the lottery method (small population)
Setup. A wildlife biology class has N = 40 students, and only n = 8 seats are free on a field trip to a sloth bear release site. The instructor wants a fair, unbiased way to choose who goes.
Step 1 – number everyone. Each student gets a unique number from 01 to 40.
Step 2 – read random numbers. Using a random number generator (or table), the instructor reads two-digit numbers one at a time, keeping a number only if it is between 01 and 40 and has not already been kept:
07, 52 (skip — over 40), 19, 33, 02, 71 (skip — over 40), 28, 19 (skip — repeat), 41 (skip — over 40), 15, 09, 36
Step 3 – stop at n = 8. The kept numbers, in order, are 07, 19, 33, 02, 28, 15, 09, 36 — eight unique students, each with selection probability n / N = 8 / 40 = 0.20 (20%).
Example 2: estimating a mean with the finite population correction
Setup. Across N = 500 villages bordering six forest ranges, a survey team draws a simple random sample of n = 50 villages and records the number of livestock-depredation incidents reported in the last year.
Results. The sample mean is x̄ = 4.20 incidents per village, with sample standard deviation s = 2.10.
Step 1 – sampling fraction. f = n / N = 50 / 500 = 0.10 (10%).
Step 2 – standard error without the FPC. SE = s / √n = 2.10 / √50 = 0.297.
Step 3 – apply the FPC. √(1 − f) = √(1 − 0.10) = √0.90 = 0.949, so SEFPC = 0.297 × 0.949 = 0.282 — about 5.1% smaller.
Planning tools and data collection
In practice, you do the random draw with a spreadsheet's random-number function, a statistics package, or a small purpose-built tool — the maths above never changes by hand. A dedicated Simple Random Sampling Calculator for drawing your sample and checking the finite population correction is in development on StatsUnlock.
Once your sample is drawn and the data are in, the next job is summarising it. If you need the centre of your sample's distribution, the free Median Calculator on StatsUnlock gives you the median with a step-by-step solution. If you want a robust measure of spread that is not thrown off by a few extreme villages or plots, the free Mean Absolute Deviation Calculator on StatsUnlock walks through MAD, standard deviation, and the confidence interval in one go — useful for checking the SE and FPC numbers from Example 2 above.
Sample results: what a typical output looks like
A finished simple random sampling draw should leave you with a short, clean record like the one below, built from Example 2.
| Quantity | Symbol | Value |
|---|---|---|
| Population size | N | 500 villages |
| Sample size | n | 50 villages |
| Sampling fraction | f = n/N | 0.10 (10%) |
| Sample mean | x̄ | 4.20 incidents/year |
| Sample standard deviation | s | 2.10 |
| Standard error (with FPC) | SE | 0.282 |
| 95% confidence interval | x̄ ± 1.96 SE | 3.65 to 4.75 |
Keep this table — or its equivalent — with your raw data. It is the record a reviewer or a future you will look for first.
How to use simple random sampling in the field: a 10-step protocol
Use this 10-step protocol to run a simple random sample end to end, from defining your population to archiving your results. It is written so a student or field team can follow it on site without extra reading.
Step 1: Define population & question
Start by writing one sentence that names your population and your question, for example: "All forest-edge villages in the six study ranges, to estimate average livestock-depredation incidents per year." If you cannot write that sentence cleanly, the rest of the study will be hard to defend. A vague population — "villages near the forest" — gives a vague result, because nobody, including you, can say afterwards exactly who was eligible to be sampled. Decide on a clear boundary (a forest range map, a list of administrative units, a date range) and write it down before you do anything else. This single sentence becomes the first line of your methods section, so get it right now.
Step 2: Permits & sampling frame
Apply for any permits you need from the forest department, ethics committee, or local administration before you set foot in the field — field seasons stall fast without this; while you wait, build the sampling frame: a complete list of every village, plot, transect, or camera-trap point that fits your population definition from step 1. Use revenue records, forest range maps, or a GIS layer of grid cells. This list is the part people skip, and it is where most studies quietly go wrong, because anything missing from the frame has zero chance of ever being sampled, no matter how random your draw is.
Step 3: Number every unit
Go down your sampling frame and assign a unique number from 1 to N to every unit, in any order, whether alphabetical, geographic, or the order they appear in your record. Then check your work: does the highest number equal the total count N? Are there any duplicate numbers, or any unit with no number at all? A spreadsheet's COUNTIF and a sorted list catch almost every mistake here in under five minutes. This step feels mechanical, but an off-by-one error or a duplicated number quietly shifts every selection probability away from the clean n/N you are relying on.
Step 4: Decide sample size n
Decide n from how precise you need your estimate to be, not from what feels achievable. A simple starting point is the standard margin-of-error formula for a proportion or mean, adjusted with the finite population correction once n gets close to N. If you are testing a hypothesis instead of estimating a value, run a power calculation for the comparison you plan to make. Either way, write down the target precision or power before you compute n, so the number is defensible later. Round up, not down. A slightly larger n costs a little more time but protects your confidence interval.
Step 5: Draw the random sample
Open a random number generator (a spreadsheet's RAND function, a statistics package, or a printed random number table) and read numbers one at a time. Keep a number only if it falls between 1 and N and has not already been kept. Discard out-of-range numbers and repeats, exactly as in Example 1 earlier in this guide. Stop the moment you have n unique numbers. Write the full sequence down, including the discards, with a timestamp or seed value. This record is what makes your sample reproducible: anyone with the same seed can regenerate the identical draw.
Step 6: Plan field logistics
Mark the n selected units on a map or GIS layer and group them by accessibility, such as road distance, forest-range boundary, or season. Random selection does not guarantee a convenient route, so this step is about logistics, not randomness; you are free to choose the order you visit units in, as long as you visit all of them. Estimate travel time between clusters of points, check which units fall inside protected areas needing extra clearance, and build a realistic day-by-day visit schedule. Build in 10-15% slack for weather, vehicle breakdowns, or a unit that turns out to be unreachable.
Step 7: Prepare equipment & brief team
Pack a GPS unit (or a phone with offline maps) loaded with the coordinates of every selected unit, plus printed or digital data sheets that match the variables in your sample-results table. Before heading out, brief every team member on the protocol: how to confirm they are at the correct unit, what counts as a valid observation, and what to do if a unit cannot be reached. Five minutes of briefing here prevents hours of cleanup later, because two field workers who interpret "livestock-depredation incident" differently will quietly corrupt your sample mean before you ever see the data.
Step 8: Visit units & record data
At each unit, follow the same recording protocol every time: same questions, same units of measurement, same rounding rules. Before leaving, re-read the data sheet out loud or have a second team member check it: are all fields filled in, do the numbers look plausible, and does the GPS point match the planned location? This in-field QA step catches most errors: a missed field, a transposed digit, a unit visited by mistake — while you can still fix them on the spot. Fixing an error in the office, weeks later, usually means a wasted return trip.
Step 9: Handle problem units
Some selected units will be inaccessible (flooded road, active landslide), refuse participation (a household survey), or turn out not to exist (a map error). Decide your replacement rule before the field season starts: the cleanest option is to draw a small reserve list of extra random numbers at the same time as step 5, and substitute the next reserve unit in order when a problem occurs. Log every substitution with a reason. This matters for your methods section, because a high replacement rate concentrated in one area (say, all the hardest-to-reach villages) can reintroduce the bias simple random sampling was meant to remove.
Step 10: Enter data, calculate & archive
Enter the data into a spreadsheet or R/Python script, compute the sample mean, standard error, and finite population correction as in Example 2, and check the numbers against a calculator such as the Mean Absolute Deviation Calculator on StatsUnlock for a second opinion. Then archive four things together in one folder: the raw data, the cleaned data, the analysis script, and the random number seed (or the full discard-and-keep sequence from step 5). This bundle is what lets a co-author, a reviewer, or future-you regenerate the exact same sample and the exact same numbers a year from now — the final, often-skipped piece of a reproducible simple random sample.
Tips and notes for simple random sampling
These are the small habits that separate a clean random sample from one that quietly falls apart under review. Split into two groups: things to do, and things to watch for.
Tips
- Generate more random numbers than you need (about 20% extra) so you have ready replacements for duplicates, out-of-range draws, and units you cannot reach.
- Always record the seed for your random number generator. Without it, no one (including future you) can reproduce the draw.
- If your sampling frame has gaps or duplicate entries, clean it before you draw. A messy frame is the single biggest source of hidden bias.
- For small populations (under a few hundred), apply the finite population correction. It is one extra line of arithmetic and it changes your confidence interval.
- Pilot your sample size on last season's data if you have it. A quick variance estimate from 20-30 old records will sharpen your n far better than a guess.
Notes
- Simple random sampling needs a complete list of the population. If you cannot build that list, look at cluster or systematic sampling instead.
- "Random" does not mean "haphazard". Picking units that happen to be convenient is not simple random sampling, even if it feels unbiased.
- In Excel,
=RAND()next to each row, then sort, gives you a quick random order. In R,sample()with a fixed seed does the same job and is easier to document. - If your study involves people, animals, or protected sites, get the right ethics and forest-department permits before you draw any sample — the sampling design itself rarely needs approval, but access to the units almost always does.
- A random sample protects against selection bias, not against measurement error. Careful data collection still matters just as much.
How to report simple random sampling in a research paper
State three things in your methods section: the population and frame, the sample size with its justification, and the exact draw method including the seed. Here are templates you can adapt directly.
Methods section (APA-style)
sample(), seed = 2026). Each plot had a selection probability of 0.10."Thesis or dissertation phrasing
Plain-language summary
Whichever version you use, always state the seed or the exact source of your random numbers. That single detail is what lets another researcher reproduce your sample exactly.
Common mistakes with simple random sampling
- Treating a convenience list as a sampling frame. A list of "easy to reach" units (roadside plots, willing respondents) is not the same as a list of all units. Build the frame from the full population, not from access.
- Skipping the finite population correction for small N. When the sampling fraction is above about 5%, leaving out the FPC overstates your standard error and makes your confidence interval wider than it needs to be.
- Reusing the same random numbers across studies. Each draw needs its own seed and its own sequence. Recycling an old random number list quietly turns a random sample into a fixed, repeated one.
- Replacing "hard to reach" units with the nearest convenient one. Swapping unit 47 for unit 48 because 47 is across a river reintroduces the bias that random sampling was meant to remove. Replace using more random draws, not proximity.
- Confusing "random sampling" with "random assignment". Simple random sampling is about how you choose units to study. Random assignment is about how you allocate those units to treatment groups in an experiment. The two answer different questions.
- Calculating sample size after the data are in. Sample size needs to be set in advance, from your target precision and an estimate of variance — not adjusted afterwards to make a result look stronger.
Conclusion: why simple random sampling still earns its place
Simple random sampling is the baseline every other probability sampling method is judged against. Give every unit in your population a number, decide how many you need, and let a random number generator — not your judgement — choose which ones you study. Done correctly, it gives every unit an equal and known chance of selection, which is exactly what makes the statistics that follow (means, proportions, confidence intervals) valid.
A few things to carry forward from this guide:
- You need a complete, clean sampling frame before you can draw a true simple random sample. If you cannot list every unit, this is not the right method — look at cluster or multi-stage sampling instead.
- Sample size and the finite population correction are not optional extras. They decide how precise your estimate is and how honest your confidence interval looks.
- Always record your seed and draw method. Reproducibility is what separates a defensible sample from a lucky one.
- Use simple random sampling when your population is small to moderate and a full list exists; reach for systematic, stratified, or cluster designs when the population is large, spread out, or naturally grouped.
Once you have your sample data in hand, the next step is usually summarising it. The free Median Calculator and Mean Absolute Deviation Calculator on StatsUnlock are good places to start turning your random sample into the numbers your report or paper actually needs.
Frequently asked questions
Q1. What is simple random sampling?
Simple random sampling is a method of choosing a sample where every unit in the population has an equal and known chance of being selected. You number every unit in your sampling frame, then use a random number generator or a lottery draw to pick the units for your sample. It is the baseline probability sampling method that all the others (systematic, stratified, cluster) are compared against.
Q2. What are the four main types of probability sampling?
The four main probability sampling methods are simple random sampling (every unit has an equal chance), systematic sampling (pick every k-th unit after a random start), stratified sampling (split the population into groups, then sample randomly within each group), and cluster sampling (randomly select whole groups, then study every unit inside the chosen groups).
Q3. How do you do simple random sampling step by step?
First, define your population and build a complete sampling frame. Second, give every unit a unique number from 1 to N. Third, decide your sample size n. Fourth, use a random number generator (or seeded software like R's sample()) to draw n numbers without repeats. Fifth, the units matching those numbers are your sample. Always record the seed so the draw can be reproduced.
Q4. What is the formula for simple random sampling?
The core formula is the selection probability, p = n / N, where n is your sample size and N is the population size. For example, if N = 40 and n = 8, every unit has a p = 8/40 = 0.20 (20%) chance of being chosen. When estimating a mean from a small population, the finite population correction, the square root of (1 - n/N), is applied to the standard error.
Q5. What is the difference between simple random sampling and stratified sampling?
Simple random sampling draws units directly from the whole population with no grouping. Stratified sampling first splits the population into meaningful subgroups (strata), then draws a random sample from each one. Stratified sampling usually gives more precise estimates when the subgroups differ a lot from each other, but it needs extra information to define the strata. Simple random sampling is simpler and works well when the population is fairly uniform.
Q6. What is the difference between random sampling and simple random sampling?
"Random sampling" is the broad family of methods that use chance to select a sample, including simple random, systematic, stratified, and cluster sampling. "Simple random sampling" is one specific method within that family, where every unit is drawn directly and independently from the full population with an equal chance. So simple random sampling is a type of random sampling, but not every random sampling method is "simple."
Q7. Is simple random sampling biased?
No, simple random sampling is designed to be unbiased: every unit has the same known chance of selection, so the sample should reflect the population on average. Bias can still creep in through a poor sampling frame (missing or duplicate units), non-response, or replacing hard-to-reach units with convenient ones. The method itself is fair; how it is carried out determines whether bias appears in practice.
Q8. What are the advantages and disadvantages of simple random sampling?
Advantages: it is free of selection bias, easy and cheap to run for small to moderate populations, and every unit has a known, equal chance of selection, which makes the statistics straightforward. Disadvantages: it needs a complete sampling frame, which is often hard to build, and it can become impractical or expensive for very large or widely spread populations, where systematic, stratified, or cluster sampling tend to work better.
Q9. How do you calculate sample size for simple random sampling?
You start with how precise you need your estimate to be (your margin of error), a confidence level (commonly 95%), and an estimate of the population's variability (from a pilot study or past data). Plug these into the standard sample-size formula for a mean or proportion. If your sampling fraction (n/N) is above about 5%, apply the finite population correction to refine the result for small populations.
Q10. What is the difference between simple random sampling with and without replacement?
"Without replacement" means once a unit is selected, it is removed from the pool so it cannot be picked again — this is the standard approach and gives n distinct units. "With replacement" means a selected unit goes back into the pool and could be picked more than once, which is less common in field studies but is the assumption behind some statistical formulas. Without replacement is more efficient for a fixed population size.
Q11. How do you do simple random sampling in Excel?
List your population in a column, then add a helper column with =RAND() next to each row. Copy this formula down for every unit, then sort the whole table by the helper column from smallest to largest. The first n rows after sorting are your simple random sample. Convert the RAND values to fixed numbers first (copy, then "Paste Values") so they do not recalculate and shuffle your sample again.
Q12. What is a real-life example of simple random sampling?
A wildlife survey team numbers all 40 forest plots in a reserve from 1 to 40, then uses a random number generator to draw 8 plot numbers without repeats. Those 8 plots, and only those, are surveyed for the study. Large national surveys, such as household health or population surveys, also use simple random sampling (often combined with other designs) to choose individuals or addresses from a complete list.
Q13. How do you report simple random sampling in a research paper?
State the population size (N), the sample size (n), the sampling fraction (n/N), whether the draw was with or without replacement, and the exact method and software used to generate the random numbers, including the seed. For example: "A simple random sample of n = 50 was drawn without replacement from N = 500 using R 4.3 (sample(), seed = 2026)." This lets another researcher reproduce your exact sample.
References
The following peer-reviewed studies support the simple random sampling theory, formulas, and field-survey applications covered in this guide, from the founding statistical papers to applied wildlife and ecological monitoring studies.
