0Quick Answer

1What Is Simple Random Sampling?

Simple random sampling (SRS) is a probability sampling method in which n units are selected from a population of N units in such a way that every unit has the same inclusion probability, and every possible sample of size n is equally likely. It is the reference design that all other sampling methods are compared against.

The method has two forms. Without replacement means a unit can be drawn only once, which is the normal choice in field and survey work. With replacement allows the same unit to be drawn twice, which simplifies the variance formula but wastes effort. This tool draws without replacement using a partial Fisher-Yates shuffle, so a duplicate can never appear.

Researchers use SRS because the estimator is unbiased, the variance formula is simple, and reviewers accept it without argument. Its weakness is that it ignores structure: if the population contains distinct strata or clusters, a stratified or cluster design will usually give a smaller standard error for the same cost.

Figure 1.1 Every unit in the frame has the same inclusion probability under simple random sampling.

2Sampling Setup: Enter Your Data and Design

Measurement data (one group per box)

3Results

4Interpretation of Results in Detail

How to read each number

The mean (x̄). Under simple random sampling the sample mean is an unbiased estimator of the population mean. That does not mean your particular sample mean equals the true value. It means that if you repeated the same draw many times, the average of all those sample means would land on the population mean. So never report the mean alone. Report it with its standard error or confidence interval so the reader can see how much sampling noise sits behind it.

The standard deviation (s). This describes how spread out the individual units are. It is a property of the population, not of your effort. Collecting more data does not shrink s. If s is large relative to the mean, the variable itself is patchy, and you will need a bigger sample to pin down the mean.

The standard error (SE). This describes how much the sample mean itself would bounce around across repeated samples. SE = s / √n. Because of the square root, halving the SE needs four times the sample. This is the single most useful sentence to remember when a supervisor asks whether to add 10 more plots: going from n = 40 to n = 50 improves precision by only about 11 percent.

The margin of error (MOE). MOE = t × SE, adjusted by the finite population correction when needed. It is the half width of the confidence interval, expressed in the same unit as your measurement. If your mean tree height is 18.4 m with a MOE of 1.2 m, you are saying the plausible range is roughly 17.2 to 19.6 m.

The 95 percent confidence interval. The correct reading is procedural: if the whole study were repeated many times, about 95 percent of the intervals built this way would contain the true population mean. It is not a 95 percent probability that the true mean lies in your one interval. Practically, a narrow interval means your design was adequate; a wide interval means the estimate is too vague to support a management decision.

The relative standard error (RSE). RSE = 100 × SE / x̄. This is the fastest quality flag because it is unit free. Most statistical agencies treat RSE below 10 percent as good, 10 to 25 percent as usable with a caution note, and above 25 percent as too unreliable to publish as a standalone figure. This tool shows a green, amber or red badge on exactly those cut points.

The sampling fraction and FPC. The fraction f = n/N tells you what share of the population you measured. When f is above 0.05 the finite population correction √(1 − f) meaningfully shrinks the standard error, because you are running out of population left to be uncertain about. When you census the whole population, f = 1 and the correction drives the SE to zero, which is correct: there is no sampling error in a complete count, only measurement error.

The required sample size (n required). This is what the tool solves backwards from your observed variability and your target margin of error. If n required is much larger than the n you drew, your pilot has told you the design is under-powered and you should either add units, accept a wider interval, or switch to a stratified design that removes some of the variance.

Comparing groups. If two group intervals do not overlap at all, the difference is almost certainly real. If they overlap slightly, the difference may still be significant, because overlapping confidence intervals are a conservative test. Run a formal two-sample test rather than eyeballing the bars when the overlap is small.

What the numbers cannot tell you. None of these statistics detect a bad sampling frame. If your list of households missed every seasonal migrant, the confidence interval will still look tight and reassuring while the estimate itself is biased. Precision statistics only describe sampling noise, never frame coverage error or measurement error.

5How to Write Your Results in Research

Use the templates below. Every one of them names the design, the sample size, the seed, the sampling fraction and the uncertainty, which is exactly what reviewers check first.

Rules that make a results paragraph pass review

1. Name the design in the first sentence. Write "units were selected by simple random sampling without replacement", not "units were selected randomly". The word randomly on its own is ambiguous and reviewers flag it.
2. Give N before n. The reader needs the population size to judge the sampling fraction. "40 of 400 quadrats (10 percent)" is complete; "40 quadrats" is not.
3. State the randomisation instrument and seed. One clause is enough: "drawn with a Mulberry32 pseudo-random generator, seed 20250607". This single detail converts a claim into a reproducible procedure.
4. Report the mean with a dispersion measure, always. Choose SD when describing the units, SE or CI when describing the estimate of the mean. Say which one you used, because M = 52.4 ± 1.3 is meaningless without a label.
5. Match decimal places to precision. If the SE is 1.3, reporting the mean as 52.4382 is false precision. One more decimal than the SE is the usual convention.
6. Say whether the FPC was applied. If f > 0.05 and you did not apply it, your intervals are too wide; if you did, say so, because the reader cannot reproduce your SE otherwise.
7. Report non-response and replacements. "Three drawn units were inaccessible and were replaced from the pre-drawn reserve list in the order drawn" protects you from an accusation of convenience substitution.
8. Put precision in the abstract. A mean without an interval in the abstract is the most common minor revision request in ecology and survey journals.
9. Do not confuse SD and SE in figures. State in the caption which one the error bars show, and whether they are 1 SE or a 95 percent CI.
10. Archive the frame and the drawn list. Deposit the sampling frame, the seed and the drawn IDs as supplementary data. This is now expected by most journals with an open data policy.

Common wording mistakes and the fix

6Formulas Used

7How to Use This Tool

1. Type your study area or project name so it appears in every template and export.
2. Pick a sample dataset preset if you want to see how the tool behaves, or skip straight to your own data.
3. Enter the population size N, that is the total number of units in your sampling frame.
4. Enter the sample size n you plan to draw, or use the required n the tool reports after a pilot run.
5. Set a random seed. Write this number in your field notebook; it is what makes the draw reproducible.
6. Paste your measurements as comma separated numbers, one box per group, and rename each group.
7. Or switch to the upload tab, choose a CSV or Excel file, then click the columns that should each become a cluster.
8. Choose the confidence level and the reserve list percentage, then click Draw Sample and Calculate.
9. Read the summary cards, the per group table, the drawn sample and the four plots.
10. Download the Field Pack for the field day, and copy the ready made methods paragraph into your manuscript.

8Detailed Reference Tables

Table 8.1 Critical values

Table 8.2 Sample size for a proportion at 95 percent confidence (p = 0.5, worst case)

Table 8.3 Finite population correction factor √(1 − f)

Table 8.4 Precision verdict thresholds used by this tool

9Example Results (8 Worked Cards)

10How to Collect Raw Data in the Field

10a. The 18 point field protocol for simple random sampling

Plan

1. Build the sampling frame first. Write every unit once, with a unique ID, in a spreadsheet. A frame with duplicates or gaps breaks the equal probability property before you set foot outside.
2. Fix the unit definition in writing. A quadrat is 10 m by 10 m with the peg at the south west corner; a household is everyone who ate from one kitchen last night. Ambiguous units create measurement noise you cannot fix later.
3. Decide n from a target margin of error, not from habit. Run a 10 unit pilot, feed the pilot SD into this tool, and read the required n.
4. Draw the simple random sample before travelling and record the seed in the notebook, not only on the phone.
5. Draw a reserve list of 20 percent extra units in draw order. Replacements must come from this list, in order, never from whatever is nearby.

Figure 10.1 The drawn sample and the reserve list, plotted before the field day.

Kit

6. Carry the printed Field Pack from this tool. Phones die; paper does not.
7. Carry a GPS or a phone with offline maps and check its accuracy reading before you accept a point.
8. Carry the measuring kit matched to the unit: 30 m tape and pegs for quadrats, calibrated balance for weights, a DO meter with a fresh calibration for water.
9. Carry two pencils and a waterproof sheet cover. Ink runs in rain and a lost sheet is a lost field day.

Locate

10. Go to the drawn unit, not the convenient one. This is the single most common way a random sample quietly becomes a convenience sample.
11. Navigate to the coordinate, then place the unit by a fixed rule, for example the peg goes at the coordinate and the quadrat extends north east. Use the same rule everywhere.
12. If a unit is genuinely inaccessible, mark it Not found, write the reason, and take the next reserve unit in order.

Figure 10.2 Substituting an easy unit for a drawn unit destroys the equal probability property.

Collect

13. Fill one datasheet row per unit, on the spot. Never write from memory at the end of the day.
14. Record a true zero as 0, never as a blank. A blank means not measured; a zero means measured and none present. Confusing the two is the most damaging data entry error in field work.
15. Record effort as well as outcome: start time, end time, observer, weather. Effort lets you check whether tired observers produced different numbers.

Figure 10.3 One completed datasheet row, including a true zero.

Figure 10.4 Three different records that must never be written the same way.

Check

16. Before leaving each unit, re-read the row aloud against the ground. Two seconds of checking saves a week of cleaning.
17. At the end of the day, count rows against the drawn list. Rows plus Not found entries must equal n exactly.
18. Photograph every sheet the same evening and enter the numbers within 48 hours, while the handwriting still makes sense.

Figure 10.5 The effort block that turns a raw count into a comparable rate.

10b. Datasheet column specification

Zero rule: write 0 for measured and none present; leave blank only when the unit was not measured at all. No renumbering rule: keep the unit IDs exactly as drawn, even if you visit them out of order.

10c. Filled worked datasheet

What this sheet shows:

• Row 3 is a true zero, written as 0, so it still contributes to the mean.
• Row 4 is a non-response with the reason recorded, so the loss is auditable.
• Row 5 uses the first reserve unit in draw order, which keeps the design defensible.
• Row 8 flags an uncertain count in the notes rather than silently rounding it.
• The observer column changes at row 6, which lets you test for an observer effect later.

10d. Blank print ready datasheet

Download the blank sheet as a .csv file, open it in Excel, Google Sheets or LibreOffice, then print it. Rows are sized to stay readable after a black and white photocopy, and the same columns match the Field Pack export.

11Which Sampling Method Should You Use?

Decision tree. Answer these in order and stop at the first yes.

1. Do you have a complete list of every unit, and is the population fairly uniform? Use simple random sampling.
2. Do you have a list, but the population splits into subgroups that clearly differ, such as habitat types or income bands? Use stratified random sampling.
3. Do you have a list in a fixed order and want an easy field rule? Use systematic sampling with a random start, unless the list has a repeating cycle.
4. Is there no list, but the population comes in natural groups you can list, such as villages or plots? Use cluster sampling.
5. Is the population huge and spread over long distances? Use multistage sampling.
6. Do units differ hugely in size, so bigger units should have a bigger chance? Use probability proportional to size.
7. Is the target rare and clumped? Use adaptive cluster sampling.
8. Is the population hidden or hard to reach, with no frame at all? Use respondent driven or snowball sampling, and accept that it is non-probability.

12Troubleshooting and Common Sampling Errors

13Assumptions, Bias and Limitations

• Complete frame assumption. Every unit must appear exactly once. Units missing from the frame have zero selection probability, which produces coverage bias no statistic can detect.
• Equal probability assumption. Any substitution of a convenient unit for a drawn unit breaks this and turns the design into convenience sampling.
• Independence assumption. Drawing without replacement makes units slightly dependent, which is exactly what the finite population correction accounts for.
• Approximate normality of the sample mean. The t interval relies on the central limit theorem. For strongly skewed variables with n below about 30, use a bootstrap interval instead.
• Non-response bias. If missing units differ systematically from found units, the estimate is biased even though the interval looks fine.
• Measurement error. These formulas describe sampling noise only. A miscalibrated instrument shifts every value and no confidence interval will warn you.
• Efficiency limitation. Simple random sampling ignores structure, so it is often the least precise probability design per unit of cost in a heterogeneous population.
• Travel cost limitation. Randomly scattered points mean long walks. Systematic or cluster designs cover ground far more cheaply.

14Conclusion

What simple random sampling gives you

Simple random sampling is the design that needs the least defending. Because every unit carries the same inclusion probability, the sample mean is unbiased, the variance has a closed form, and no reviewer will ask you to justify a weighting scheme. That simplicity is why it remains the reference point for every other sampling method, and why an SRS estimate with a stated seed and a stated sampling fraction is accepted almost anywhere.

What it costs you

Simplicity is paid for in the field. Random points scatter, so travel time rises, and in a heterogeneous population the standard error will be larger than a stratified design would deliver for the same n. If your pilot shows a large standard deviation driven by an obvious subgroup difference, moving to stratified sampling is usually the cheapest precision gain available.

What to check before you publish

Confirm four things: the frame was complete and dated, the seed and drawn list are archived, the finite population correction was applied whenever the fraction exceeded 5 percent, and every mean is reported with a standard error or a confidence interval. Those four lines are what separate a design a reviewer trusts from one they query.

What to do next

Use the required sample size from your pilot to plan the full survey, print the Field Pack with the reserve list, and keep the same seed convention across seasons so repeat surveys stay comparable. If your relative standard error is still above 10 percent after increasing n, the variable itself is patchy and the answer is a better design, not more walking.

15Test Yourself

16Frequently Asked Questions

17Cite This Tool

18Related Tools

• Stratified Random Sampling Calculator for proportional, equal and Neyman allocation.
• Systematic Sampling Calculator for interval k with a random start.
• Sample Size Calculator using Cochran, Yamane and Krejcie-Morgan.

19Glossary of Terms

20References

1. Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Wiley. Publisher page
2. Lohr, S. L. (2021). Sampling: Design and Analysis (3rd ed.). CRC Press. https://doi.org/10.1201/9780429298899
3. Thompson, S. K. (2012). Sampling (3rd ed.). Wiley. https://doi.org/10.1002/9781118162934
4. Kish, L. (1965). Survey Sampling. Wiley. Archive record
5. Krejcie, R. V., & Morgan, D. W. (1970). Determining sample size for research activities. Educational and Psychological Measurement, 30(3), 607-610. https://doi.org/10.1177/001316447003000308
6. Yamane, T. (1967). Statistics: An Introductory Analysis (2nd ed.). Harper and Row. WorldCat record
7. Horvitz, D. G., & Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47(260), 663-685. https://doi.org/10.1080/01621459.1952.10483446
8. Neyman, J. (1934). On the two different aspects of the representative method. Journal of the Royal Statistical Society, 97(4), 558-625. https://doi.org/10.2307/2342192
9. Hansen, M. H., & Hurwitz, W. N. (1943). On the theory of sampling from finite populations. Annals of Mathematical Statistics, 14(4), 333-362. https://doi.org/10.1214/aoms/1177731356
10. Elzinga, C. L., Salzer, D. W., & Willoughby, J. W. (1998). Measuring and Monitoring Plant Populations. BLM Technical Reference 1730-1. BLM library
11. Krebs, C. J. (1999). Ecological Methodology (2nd ed.). Benjamin Cummings. WorldCat record
12. Buckland, S. T., et al. (2001). Introduction to Distance Sampling. Oxford University Press. Publisher page
13. Gregoire, T. G., & Valentine, H. T. (2007). Sampling Strategies for Natural Resources and the Environment. Chapman and Hall. https://doi.org/10.1201/9780203498880
14. Bartlett, J. E., Kotrlik, J. W., & Higgins, C. C. (2001). Organizational research: determining appropriate sample size in survey research. Information Technology, Learning, and Performance Journal, 19(1), 43-50. ERIC record
15. Levy, P. S., & Lemeshow, S. (2008). Sampling of Populations: Methods and Applications (4th ed.). Wiley. https://doi.org/10.1002/9780470374597
16. Sarndal, C. E., Swensson, B., & Wretman, J. (1992). Model Assisted Survey Sampling. Springer. https://doi.org/10.1007/978-1-4612-4378-6
17. Rao, J. N. K., & Molina, I. (2015). Small Area Estimation (2nd ed.). Wiley. https://doi.org/10.1002/9781118735855
18. Groves, R. M., et al. (2009). Survey Methodology (2nd ed.). Wiley. Publisher page
19. Marsaglia, G. (2003). Xorshift RNGs. Journal of Statistical Software, 8(14), 1-6. https://doi.org/10.18637/jss.v008.i14
20. Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman and Hall. https://doi.org/10.1201/9780429246593