Nine of ten periods have positive returns. “Profitable 90% of the time” looks compelling in the summary. The last period, however, loses 20%. Did an investor who held through all ten periods actually finish with a profit?
Deleting that row improves the average. Copying the table several times increases its size. Before accepting either improvement, read the complete ten-row record and ask what changed in the experience being described—and what new information, if any, was added.
What you will be able to do
- Recalculate a mean and median and explain why they can suggest different impressions.
- Identify how extreme observations, missing-data decisions, and repeated records affect a result.
- Recognize overlapping windows, time dependence, and common drivers, then document what a sample cannot establish.
Nine small gains and one large loss
These are standalone synthetic teaching inputs, not a real C2 backtest or a replacement for C4's account data. Ten consecutive, equally long periods have simple returns of 1% in each of the first nine and −20% in the tenth. Returns share the same net-cost definition, with no outside cash flows.
The arithmetic mean adds the returns and divides by their count. Nine gains total 9%; adding −20% gives −11%; dividing by ten gives −1.1% per period. This describes the sample's average return, not its entire holding-period return or a promise for the next period.
For the median, sort the ten values from smallest to largest and average the fifth and sixth. Both are 1%, so the median is 1%. NIST distinguishes these measures of location and their sensitivity to extreme values.[1]
Neither calculation is wrong. The median identifies the middle of the sample but says little about the depth of its worst result. The large loss pulls the mean downward, directing attention to a part of the record a typical positive period would conceal.
A coherent summary can retain all of them: “Nine of ten periods were positive; median 1%; mean −1.1%; worst period −20%.” Frequency, central position, and loss magnitude answer different questions. None alone reconstructs the investor's experience.
Begin with $100 and hold through the full sequence. Multiply by 1.01 for each of the first nine periods, then by 0.8. The endpoint is approximately $87.49, a total return of about −12.51%. The mean of −1.1% is not this ten-period return, and the median of 1% does not establish growing wealth throughout the path.
Look at where the observations are
For now, think of a distribution as where outcomes lie, how many occur in each part, and how far the extremes are from the middle. Nine observations cluster at 1%, with one at −20%. “Typical return: 1%” leaves much of that description out.
Plot the individual points on a horizontal line or count them within intervals. With a small sample, raw observations can be more informative than a smooth curve. Binning and sample choice affect the appearance. A bell-shaped outline does not establish that future returns follow a normal distribution.
The tails concern more extreme parts of a distribution. Tail risk does not mean the probabilities are already known and merely small. Ten observations establish that this sample contains one −20% outcome, not that the future must contain exactly one such loss every ten periods. Nor do they exclude a larger loss.
The personal consequences may be worse if the loss coincides with declines in other holdings, interrupted income, or difficult liquidation. A return distribution also omits some payment and execution conditions. Connect descriptive statistics to the risk questions already recorded rather than letting the distribution replace them.
Deleting an observation changes the question
Remove the −20% and the remaining nine returns all equal 1%. Both mean and median become 1%. Those numbers describe the retained rows correctly. They no longer describe the original ten-period experience.
An extreme value is not necessarily an error. Incorrect units, untreated splits, or duplicate events can create false anomalies; real gaps and losses can also create large changes. Check original records, corporate actions, and processing rules. Distance from the mean is a reason to investigate, not sufficient evidence for deletion.
If an error is verified, preserve the original value, supporting evidence, and correction method before recalculating. If the result is real, retain it in the complete report. You may show a separate sensitivity comparison without it, stating why, rather than presenting that comparison as truer performance.
Here the mean flips from −1.1% to 1% because of sample selection. No investment skill was created. The sensitivity tells you which event matters and suggests a mechanism to investigate. Saving only the improved chart would remove that useful warning.
Keep missing data distinct as well. A row excluded because it lacks a reliable observation and a row discarded because the outcome is unpleasant need different explanations. The sample record should show what was eligible, what was retained, and why.
A hundred rows need not contain a hundred observations
Copy the original table ten times and label the copies honestly. There are now one hundred rows, with the same −1.1% mean and 1% median. The original ten observations remain the unique source; ninety new independent events have not occurred.
Some calculations can appear more precise if a program treats the copies as independent. That apparent precision relies on an unsupported independence assumption. Copies can test software's handling of table size, but they cannot supply strategy evidence or satisfy a minimum-observation requirement.
Less obvious duplication occurs when one trade appears through two channels, one issuer appears under multiple identifiers, or several products hold the same exposure. Check identities and sources. A duplicate may leave the mean unchanged while changing how much evidence you think you possess.
Ask what a row represents: a day, an event, a company, or a statistic constructed from other rows. Without that unit, sample size is merely file length. The number needs a description of how observations were generated before it can support an inference.
Distinct rows can still depend on each other
Suppose you calculate the previous twenty days' return every day. Today's window and yesterday's share nineteen days. Two hundred rolling windows are not two hundred separate stretches of market experience. Report the windows and their overlap together.
Trades can also respond to the same market condition. Ten trades in one day, all using the same signal, sector, and direction, are real records. They are not automatically ten unrelated tests of the mechanism. The shared dependency matters when moving from a ledger to a claim about stability.
Autocorrelation describes relationships between values of one series at different times. NIST uses it to examine time-series structure.[2] You do not need to calculate a coefficient here; identify why adjacent observations might be related. Failing to find linear autocorrelation would not establish the absence of every kind of dependence.
There is no universal discount that turns rows into an effective independent count. The answer depends on the process, strategy, dependence structure, and question. G71 develops interval and testing interpretations. An arbitrary trade count cannot establish adequacy in advance.
Two rising lines do not identify a cause
Two companies' returns rise together during a quarter. One might affect the other, both might respond to industry demand or market conditions, or the association might be coincidental in this sample. Correlation describes co-movement without supplying a causal direction.
Comparing two trending price levels can be especially misleading. Similar-looking lines do not establish a relationship that predicts tradable returns. Define whether you compare prices or returns, align dates and availability, and examine a mechanism that could support the relationship.
For a claim that raw-material costs affect profit, seek purchasing terms, price transmission, and profit components, while considering mix and other alternatives. A statistical association can guide the document search. It cannot replace business evidence, just as a plausible business story cannot bypass data and counterexamples.
Repeated selection matters too. Searching many objects and windows makes some similar-looking lines easier to find. Preserve the search scope and distinguish a hypothesis recorded before inspection from exploration prompted by a chart. That distinction affects the independent evaluation still needed.
Rewrite a misleading summary yourself
Explanation question: Why does the original median of 1% fail to show a profitable holding period?
Application question: An analyst deletes −20%, reports a 1% mean, then copies the table ten times and declares the sample sufficient. Identify the change made by each operation and what evidence did not increase. Write answers before checking the guide.
The median describes a middle location, not a complete capital path. The loss and compounding still matter. Deletion changes coverage; copying changes row count. Without evidence of a data error, retain the original event. Copies add no independent observations.
Change the final loss
Use another separate synthetic ten-period series. The first nine returns remain 1%, but the last becomes −5%; all other account assumptions stay the same. Calculate the arithmetic mean, median, and complete holding-period return. Does this rule out a later −20% loss?
The mean is 9% minus 5%, divided by ten: 0.4%. The median remains 1%. Capital becomes $100 multiplied by 1.01 nine times and then by 0.95, or about $103.90. The holding-period return is approximately 3.90%, distinct from the per-period arithmetic mean.
The smaller loss changes the outcome because the input changed. This is a sensitivity exercise, not evidence that the future tail disappeared. Do not combine the two teaching sequences as independent records of real strategy performance.
Save a sample-diagnosis record
Record object, dates, frequency, row meaning, return and cost definitions, source, and selection rule. Preserve the complete observations, then list mean, median, minimum, maximum, and at least one observation with substantial influence on the result.
Make one removal or correction comparison, explaining whether it repairs a verified error or demonstrates sensitivity. Retain the original. Check duplicates, rolling overlap, common events, and market drivers; mark unexamined items rather than assume independence.
Hide the row count for a final check. Can the record still describe the different conditions observed? If it says only “many trades,” add the generation process. If one event determines the conclusion, include that dependency in the summary for subsequent testing.
Inspect the sample before admiring the number
Mean and median describe different aspects, and extreme outcomes can determine the complete path. Verify anomalies, preserve the distribution and compounded result, and distinguish error correction from selecting pleasant observations.
Then ask where additional information comes from. Copies, overlap, and shared drivers limit inference; correlation alone does not identify causation. Recording those limits prepares the ground for intervals and tests rather than asking precise calculations to repair an unexplained sample.
References
- NIST: Measures of Location. Mean, median, and sensitivity to extreme values; the numerical sequences are independent synthetic inputs.
- NIST: Autocorrelation. Time dependence and equally spaced observations; not a universal effective-sample-size formula.
Sources checked October 30, 2026. AI-assisted draft. Independent AI full-text and bilingual review is complete. Human financial review and target-reader testing have not been conducted and are not scheduled for this production round.