Why Random Sampling Matters in Data Analysis: A Polymarket Case Study

Full population summary

The mean was 36.7 times the median. The largest market was more than 12,600 times the mean. Those two comparisons immediately warn that volume is not distributed symmetrically.

The distribution has a long right tail: most markets are relatively small, while a tiny number are enormous. NIST notes that extreme values can distort the mean and standard deviation in skewed distributions. That is exactly what made this population such a useful stress test for random sampling.

Polymarket volume is concentrated in very few markets

Dividing the population into lifetime-volume bands shows how little the typical market resembles the average market.

Two comparisons make the imbalance concrete:

• 244,737 markets, or 63.78% of the population, recorded less than $10,000 in volume or no volume at all. Together, they generated only 0.91% of total volume.

• 5,278 markets with at least $1 million in volume represented just 1.38% of the population but generated 70.15% of all volume.

The grouped bars compare each volume band's share of markets with its share of total volume. The reversal from the low-volume bands to the high-volume tail is the central structural fact behind the sampling results.

The ten highest-volume Polymarket markets

The concentration is visible even at the very top. The ten largest closed markets in the archive generated $4.77 billion, equal to 10.26% of total historical volume.

The two presidential-winner markets alone represented 5.52% of all historical volume and 53.82% of the top ten's volume.

Rectangle area represents each market's share of volume among the ten highest-volume markets. The treemap makes the dominance of the first two markets visible without flattening the rest of the ranking into equal categories.

How sample size changed the estimate

For each sample, we calculated the sample mean and its signed percentage error relative to the known population mean of $121,216.48:

Signed mean error (%) = ((sample mean - population mean) / population mean) × 100

A negative value is an underestimate. A positive value is an overestimate. Absolute error ignores direction and measures distance from the benchmark.

Summary across the five seeded draws

The average absolute error at 1,000 markets looks worse than at 100 because one extreme draw dominates the average. The median absolute error tells the more typical story: it declined at every sample size.

This is itself an important statistical lesson. The metric used to summarize sampling error can be affected by the same heavy-tail problem as the data being sampled.

All 15 reproducible results

Four of the five 10,000-market samples landed within 25% of the population mean. None of the 100-market samples did. Yet the best individual result in the experiment was a 10,000-market underestimate of only 6.52%, not a perfect match.

Each point is one fixed-seed sample. The zero line represents the population mean; points above it overestimate average volume and points below it underestimate average volume.

Why the 1,000-market sample produced a 252.93% error

The most dramatic draw had a mean of $427,814.31—more than three and a half times the population mean.

That sample included the market “Fed increases interest rates by 25+ bps after January 2026 meeting?”, which had $216,455,743.15 in volume.

Inside a 1,000-row sample, that single row contributed roughly $216,456 to the sample mean:

$216,455,743 / 1,000 ≈ $216,456

The contribution from that one market was already greater than the full population mean of $121,216.48. The other 999 markets then raised the final estimate further.

This is the leverage of an extreme observation. A large sample usually reduces sampling variability, but a rare observation can still produce a large miss when the distribution is heavy-tailed.

At 10,000 rows, the same $216.46 million market would contribute about $21,646 to the mean. It would still matter, but it would have one-tenth as much leverage as it had in the 1,000-row sample.

How likely is a random sample to capture a whale market?

Only 36 of the 383,707 eligible markets had at least $100 million in volume.

For sampling without replacement, the exact probability of selecting at least one of those 36 markets is:

P(at least one whale) = 1 - C(N - K, n) / C(N, n)

Here, N is the total population, K is the number of whale markets, n is the sample size, and C(a, b) is the number of ways to choose b items from a.

The probabilities are calculated exactly without replacement from 383,707 eligible markets and 36 markets with at least $100 million in volume.

Larger samples better cover the population, but they are also more likely to encounter the rare markets that dominate its total volume. Those forces are not contradictory. They are part of the same sampling process.

Random sample vs representative sample

These terms describe different things.

A sample can be random but not look representative because of ordinary sampling variation. This is especially common with small samples, rare subgroups, and heavy-tailed variables.

A sample can also look representative on one dimension and fail on another. A set of markets might match the population's topic mix while missing its high-volume tail. Representativeness is therefore not a permanent label attached to a dataset; it is a judgment tied to a variable and research purpose.

Random selection provides a defensible design. Sample size and sampling strategy affect how often that design produces estimates close to the target.

Mean, median, and the meaning of “typical”

The population mean was $121,216.48, while the median was only $3,299.94.

Neither statistic is wrong. They answer different questions:

• The mean describes volume per market after total volume is shared equally across every market. It is essential for totals, expected contributions, and analyses in which every dollar matters.

• The median describes the middle market after all markets are ordered by volume. It is more informative when the question is what a typical market looks like.

When the maximum exceeds $1.53 billion and the mean is 36.7 times the median, reporting only “average volume” hides the structure of the population. A strong analysis should show both measures and disclose the tail.

How to choose a sample size

There is no universal sample size that makes every random sample representative.

The right choice depends on:

• the statistic being estimated;
• the variability and shape of the population;
• how rare the observations of interest are;
• the precision required;
• whether subgroup estimates are needed;
• the sampling design; and
• the cost of processing more data.

If the target is a rare category, begin with its prevalence rather than a generic sample-size rule. If the target is a heavy-tailed mean, simulate or repeat samples and inspect the sampling distribution. If every exact dollar matters and the full dataset is available, calculate the population total instead of estimating it from a sample.

When simple random sampling is not enough

For cross-platform work, start with the Polymarket Historical and Kalshi Historical archives, then make sure the unit of analysis and filters mean the same thing before comparing samples.

How to select a reproducible random sample in Lychee

Lychee places random sampling inside the same no-code workflow used for filtering, aggregation, data sheets, charts, and published dashboards.

1. Open a historical dataset

Open Polymarket Historical or Kalshi Historical, then select the table you want to study.

2. Define the eligible population

Apply the filters that determine which rows can enter the sample. For this experiment, the eligible population was closed Polymarket markets with non-null volume.

The order matters: filters define the population, and random sampling selects from that population.

3. Enable Random Sample

Open the Research Tools section under Refine Query, enable Random Sample, and enter the number of rows.

4. Set a seed

Add a seed when you need the result to be reproducible. Reusing the same seed with the same eligible population and sample size recreates the same draw.

5. Run and analyze the sample

Calculate the count, mean, median, total, standard deviation, minimum, and maximum. Then create a chart, save the sheet, export the data, or assemble the evidence in a dashboard.

Random sampling is currently designed for Lychee's historical-data workflows and selects distinct rows without replacement. The practical advantage is that sample selection, descriptive statistics, visualization, and publication happen in one research environment.

What this experiment does and does not prove

This case study is deliberately transparent, but its scope matters.

1. There were five samples at each size. Fifteen runs are enough to demonstrate variability, not to map the complete sampling distribution.

2. The samples were seeded. The reported rows and statistics are reproducible, but a different seed can produce a different valid result.

3. The analysis focused on market volume. Prices, outcomes, categories, duration, traders, and trades can have different distributions.

4. Polymarket volume is unusually skewed. The observed error rates are not universal rules for other datasets.

5. The archive has a fixed scope. This analysis used closed markets with non-null volume from launch through December 2025 and was run on August 30, 2026.

6. A market is not a dollar or a trade. Sampling market rows equally answers a different question from sampling trades, traders, dollars, or time periods.

7. The population benchmark was known. In many real studies, the true population mean is unavailable, so uncertainty must be estimated rather than measured directly.

The defensible conclusion is not that 10,000 rows are always enough. It is that larger samples produced lower typical error in these reproducible draws, while the heavy tail continued to matter.

Frequently asked questions about random sampling

What is simple random sampling?

Simple random sampling is a probability-sampling method in which every unit in the eligible population has the same chance of selection and every possible fixed-size sample has the same probability of being chosen.

What is a simple random sampling example?

This Polymarket analysis is a simple random sampling example: we defined a population of 383,707 eligible markets and selected distinct groups of 100, 1,000, and 10,000 markets with equal inclusion probabilities.

Is random sampling the same as simple random sampling?

Not always. Simple random sampling is one type of random sampling. Stratified, cluster, systematic, and weighted probability designs also use randomness but structure the selection process differently.

What is the difference between a random sample and a representative sample?

A random sample is defined by how observations were selected. A representative sample is judged by how closely it reflects population characteristics relevant to the research question. A correctly selected random sample can be unrepresentative by chance.

What is the difference between random sampling and random assignment?

Random sampling selects observations from a population. Random assignment places already-selected participants or units into experimental groups. The first supports population inference; the second helps isolate treatment effects.

What does sampling without replacement mean?

It means a selected observation cannot be selected again in the same sample. Each market in a Lychee random sample is therefore distinct.

Does random sampling eliminate bias?

No. It reduces bias from subjective row selection, but it cannot fix incomplete coverage, bad measurements, inappropriate filters, nonresponse, or a poorly defined target population.

Does a larger sample always produce a more accurate result?

No individual larger sample is guaranteed to beat every smaller sample. Larger samples generally reduce expected sampling variability, but rare extreme observations can still cause large errors. In this experiment, median absolute error improved with size even though one 1,000-row draw was the largest miss.

How large should a random sample be?

There is no universal answer. Sample size depends on population variability, the statistic being estimated, required precision, rare-group prevalence, subgroup needs, sampling design, and available resources.

Why are the mean and median so different here?

Polymarket market volume has a long right tail. A small number of enormous markets pull the mean upward, while the median remains tied to the middle-ranked market.

Can I create a random sample without Python or Excel?

Yes. Lychee can draw a seeded random sample from supported historical datasets after the eligible population is defined. The sample can then be summarized, charted, saved, and published in the same workflow.

The real lesson from random sampling

Random sampling gave every eligible market a fair route into the analysis. It did not make the population simple.

The experiment exposed a market structure in which 1.38% of rows generated 70.15% of volume and one rare market could transform an otherwise ordinary sample mean. Increasing the sample size improved the typical result, but it did not remove chance from the process.

That leads to a better research habit:

Do not ask only whether a sample was random. Ask what population it came from, which statistic you are estimating, how the population is distributed, and how much uncertainty remains.

You can reproduce the workflow in Lychee's Polymarket Historical workspace, change the sample size or seed, redefine the eligible population, and test another market characteristic. From there, save the data sheet, build an interactive visualization, and publish the evidence so other researchers can inspect the path from data to conclusion.