Bound Unique Reach When Channel Totals Cannot Reveal Their Overlap — Free
Two channels can each reach hundreds of people without reaching hundreds of different people. Show the bounds implied by marginal totals before reporting a combined audience or frequency.
The proof surface
Revenue concentration is not audience overlap: marginal channel counts only bound unique reach and combined frequency, never identify cross-channel people.
InputChannel code | unique reached users within channel | impressions within channel; common audience-universe size (users)
Rare deviceRevenue concentration is not audience overlap: marginal channel counts only bound unique reach and combined frequency, never identify cross-channel people.
Output artifactWidth of possible unique-reach interval plus complete labeled working
Cost$0 local core · no account, card, paid key or subscription · filing proposal has no checkout
Sample, not your facts: Invented records: Cedar channel | 600 | 900; Willow channel | 400 | 800. Common audience-universe size (users) = 1000; expected Width of possible unique-reach interval: 400 users of unresolved reach range. These are not quotations, verified observations or personal evidence.
Before using the overlap-bounds range sheet
Use aggregate, non-sensitive channel totals for the same time window and the same defined audience universe. Each channel’s reach must already be deduplicated within that channel. The app does not join identities, fetch analytics or estimate overlaps from impressions. A supplied common universe caps the possible union, but it must actually contain every counted person; different populations or windows invalidate the model. The range describes what remains unknown with marginal counts alone. It is not uncertainty from a sampled confidence interval and does not assign likelihood to any possible overlap arrangement. No conversion, attribution or return-on-spend conclusion is available.
Why the flat version breaks
Marginal reach is simply added
The sum is an upper bound before the universe cap, not a confirmed unique-user count. Complete overlap could leave the union as small as the largest channel. This sheet displays both possibilities so a polished dashboard cannot imply deduplication that never occurred. The result is useful precisely because it refuses to choose an invented midpoint.
Impressions are used to infer overlap
A person may receive multiple impressions in one channel, so impression totals do not reveal cross-channel identities. Dividing one total by an assumed union produces only a conditional average. The app keeps channel frequency and combined frequency bounds distinct, without a reach-estimation model that was not supplied or tested.
Different windows make the bounds look rigorous
Mathematical bounds require a coherent set universe. Mixing a weekly channel count with a monthly count changes the user job and can violate the intended comparison even if all numbers fit. The calculator cannot verify platform definitions or audience membership. Confirm those assumptions with the responsible analyst and label the window in the source record before sharing the range.
How to work the overlap-bounds range sheet
Align the audience definition and window
Check that every channel uses comparable unique-user and impression definitions over one period. Use a neutral channel code rather than campaign identifiers containing customer data. Within-channel unique reach must not exceed impressions or the common universe. A channel with zero reach must have zero impressions in this model. If a platform’s metric differs, do not force it into these cells and call the result reconciled; ask the analyst which counts can be compared.
State the common universe cap
Enter a positive whole audience size up to one billion. This is a reader-supplied population bound, not a measured subscriber list. It affects the upper union bound but does not disclose the pairwise overlap. Channels with positive counts may overlap completely or partly in many ways. Do not use an arbitrary smaller population to make the estimated frequency look stronger; the cap must be defensible from the source definition.
Compute tight marginal union bounds
The lower bound on unique reached users is the largest channel reach. The upper bound is the smaller of the sum of channel reaches and the universe size. Their difference is the unresolved reach range in the headline. Sum all impressions only when the counts refer to distinct channel deliveries. If the lower bound is positive, divide total impressions by the upper and lower reach bounds to obtain minimum and maximum possible combined average frequency.
Report a range rather than a fabricated combined total
Copy both unique-reach bounds, the frequency bounds where defined and the assumptions. A channel’s own impressions-per-reached-user average is printed separately. Ask for consented, approved aggregate overlap evidence if a narrower bound is necessary; do not introduce identity tracking to make this tool more persuasive. The optional filing prototype exports the same disclosed range, not a hidden best estimate or marketing performance guarantee.
What the overlap-bounds range sheet keeps distinct
Question
Before
Check this working
Marginal reach is simply added
The sum is an upper bound before the universe cap, not a confirmed unique-user count. Complete overlap could leave the union as small as the largest channel. This sheet displays both possibilities so a polished dashboard cannot imply deduplication that never occurred. The result is useful precisely because it refuses to choose an invented midpoint.
State the common universe cap
Impressions are used to infer overlap
A person may receive multiple impressions in one channel, so impression totals do not reveal cross-channel identities. Dividing one total by an assumed union produces only a conditional average. The app keeps channel frequency and combined frequency bounds distinct, without a reach-estimation model that was not supplied or tested.
Compute tight marginal union bounds
Different windows make the bounds look rigorous
Mathematical bounds require a coherent set universe. Mixing a weekly channel count with a monthly count changes the user job and can violate the intended comparison even if all numbers fit. The calculator cannot verify platform definitions or audience membership. Confirm those assumptions with the responsible analyst and label the window in the source record before sharing the range.
Report a range rather than a fabricated combined total
The overlap-bounds range sheet replaces this named manual reconciliation, not source verification or the responsible human's decision.
Run it on the samples, right here
FIRST-LOAD
HYPOTHESIS / PROTOTYPE — checkout unavailable. Calculation is local. A draft is saved automatically in this browser profile when storage is available; Reset to sample clears it. Optional Pro history stores only five summaries and has its own deletion control. State links encode your inputs and can remain in browser history, clipboard or recipients’ records; share only non-sensitive rows. Optional external AI formatting leaves this device. The required site analytics beacon reports page activity; shared URLs contain encoded inputs. Do not treat an encoded URL as private. The calculator has no input-collection endpoint.
Aggregate-count bounds only: no identity deduplication, attribution, conversion inference or performance guarantee. Verify a common audience/window and metric definitions with the responsible analyst; use only approved non-sensitive aggregate evidence.
Data note: This overlap-bounds range sheet calculates in the tab from Channel code | unique reached users within channel | impressions within channel. No input-collection endpoint, AI request or file upload is built into it. Drafts may be saved locally; explicit input-state links and optional external formatting can disclose the records. Use non-sensitive codes and clear the draft when finished.
Go deeper: the companion app files the same reading as a plotted bar chart sheet
The article demo above runs without limits. The companion app keeps a local history, exports the rows as CSV, prints the overlap-bounds range sheet reading, and holds your drafts on this device — one complete free app run; the proposed $4 one-time filing layer is not for sale.
Keep the complete overlap-bounds range sheet answer free; optional filing proposes its boundary-preserving print, row-and-summary CSV and five local reading summaries. The $4 one-time prototype is not for sale; another calculation remains free in the article demo.
Aggregate-count bounds only: no identity deduplication, attribution, conversion inference or performance guarantee. Verify a common audience/window and metric definitions with the responsible analyst; use only approved non-sensitive aggregate evidence.
What this is built on
Declared local method: The lower bound on unique reached users is the largest channel reach. The upper bound is the smaller of the sum of channel reaches and the universe size. Their difference is the unresolved reach range in the headline. Sum all impressions only when the counts refer to distinct channel deliveries. If the lower bound is positive, divide total impressions by the upper and lower reach bounds to obtain minimum and maximum possible combined average frequency.
Every sample code, measurement, date, price, fingerprint and scenario is invented. Artifact checks do not verify reader data, policies, actual files, tickets, votes or health/accessibility outcomes.
Google’s official Gemini pricing page, fetched 2026-10-01, lists AI Studio access in its Free section, limited model access and free input/output tokens. Free-tier content may be used to improve products. Optional external formatting may require an account; limits/access can change. Manual local entry needs none. Do not send sensitive records.
Before: summed channel totals masqueraded as deduplicated people. After: marginal counts expose a bounded union and a conditional frequency range without invented overlap.
Setting: Common audience-universe size (users) = 1000. Expected summary: 400 users of unresolved reach range.
At least 600 unique users were reached because the larger channel alone reached that many. At most 1,000 were reached, limited both by the sum 600 + 400 and the supplied universe. The unresolved interval width is 400 users. Combined impressions are 1,700; average impressions per unique reached user could range from 1.7 to about 2.833333. Adding channel reach and calling 1,000 confirmed people would assume no overlap that was never observed.
Setting: Common audience-universe size (users) = 1000. Expected summary: 350 users of unresolved reach range.
The maximum single-channel reach is 300 and the sum is 650, below the 1,000 audience cap. Unique reach therefore lies between 300 and 650, a 350-user unresolved range. The 1,600 impressions imply a frequency interval of about 2.461538 to 5.333333. Neither endpoint is more likely just because it is printed: without overlap evidence these are mathematical possibilities, not a probability distribution.
Sample C — boundary convention
East no-delivery channel | 0 | 0
West no-delivery channel | 0 | 0
Setting: Common audience-universe size (users) = 1000. Expected summary: 0 users of unresolved reach range.
No channel reports reach or impressions, so the unique-reach bounds are both zero and have no width. Frequency has no reached-user denominator and is labeled unavailable, not zero average exposure. A row with impressions but zero reached users is inconsistent under this same-universe definition and is rejected rather than treated as anonymous audience.
Optional AI formatting, never the calculation
Manual entry completes this overlap-bounds range sheet for free without signup. If available to you, the free AI Studio interface linked in the sources may format fictional or non-sensitive notes; external access may require an account. No API key or AI call is built into this tool. Free-tier content may be used to improve products. Review each cell and transcribe it to the labeled row schema; do not paste the JSON object into the row box.
Format only these fictional or non-sensitive notes for a overlap-bounds range sheet. Return strict JSON shaped as {"rows": [{"label": "string", "cells": ["string", "string"]}], "setting": "string"}. The columns are Channel code | unique reached users within channel | impressions within channel; the setting is Common audience-universe size (users). Keep all supplied strings and quantities exactly; do not calculate, infer missing entries, invent dates or add advice. If any required value is missing, return an empty rows array and ask me for it separately. I will verify every cell against my source and manually transcribe rows using vertical bars before running the local calculator.
An AI response is not executed, fetched or trusted as a result. Missing values remain questions; the strict local parser checks the rows you actually enter.