Check whether groups describe the same population
Before pooling, ask whether each group uses the same measurement definition. A monthly income median cannot simply be combined with an annual income median, and a household median does not describe the same observation unit as a person median. Convert compatible measurements before ordering them and state exclusions consistently. If one group reports only full-time workers while another includes all workers, the combined result describes that chosen mixture rather than an automatically meaningful overall population.
Even with raw observations available, distinguish an ordinary pooled median from a survey-weighted median. Survey weights can represent different numbers of population members and require cumulative weight rather than unweighted row ranks. A raw-list calculator does not infer those weights from group labels. Likewise, published rounded medians cannot be used as exact central observations. Clarifying which population and which weighting model are intended prevents a mathematically valid list operation from answering the wrong question.
The pooled median is an order statistic
You generally cannot find a combined median by averaging group medians, even if the groups are equally large. A median depends on the middle position after all observations are ordered together. A subgroup median summarizes its own ranks and loses other locations. When two groups are combined, their observations can interleave in ways that change the central pair. Equal group sizes do not restore the information that was discarded.
Work a counterexample with equal groups
Group A is 1, 2, 100, with median 2. Group B is 3, 4, 5, with median 4. Their median average is 3. Combining all observations gives 1, 2, 3, 4, 5, 100. With six observations, the central ranks are 3 and 4, holding values 3 and 4. The pooled median is 3.5. This single example disproves the claim that equal-sized subgroup medians can always be averaged.
Show why counts do not solve the problem
Keep group B unchanged but replace A with 1, 2, 2. Its size is still three and its median is still 2. The combined list becomes 1, 2, 2, 3, 4, 5, whose median is 2.5. Both scenarios have the same subgroup sizes and medians, yet their pooled medians differ. Therefore no formula using only those summaries can always recover the combined median; the missing distribution information genuinely affects the answer.
| A; B | Group medians | Pooled median |
|---|---|---|
| 1,2,100; 3,4,5 | 2 and 4 | 3.5 |
| 1,2,2; 3,4,5 | 2 and 4 | 2.5 |
Each group contains three observations. Both the plain and count-weighted averages of group medians equal 3.
Do not transfer the weighted-mean rule
Group means can be combined using counts: total mean = sum of group count × group mean divided by total count, provided the groups and measurements are compatible. Medians do not obey that identity. Giving a larger group's median more weight may describe a chosen summary of group summaries, but it is not generally the median of individuals. Name such a statistic honestly if you deliberately use it for another purpose.
Pool compatible raw observations
To obtain the exact combined median, concatenate the individual observations, sort the complete list and identify its middle rank or central pair. First check that units, periods and inclusion rules agree. Decide whether the groups are disjoint: a duplicated person's record can unintentionally receive extra weight. Repeated numerical values are valid observations when they represent genuinely separate records; deduplicating identical values would change their frequencies and may change the median.
Use a frequency table when raw rows are large
An exact-value frequency table can preserve the required rank information without listing each observation on paper. Add counts for matching values, form cumulative frequencies and locate the combined central ranks. Interval-grouped counts are weaker: they can locate ranges but may require interpolation rather than an exact median. Do not replace broad salary bands with their midpoints and label the resulting pooled number an exact median salary.
Verify the two pooled lists with the tool
Run the Median and Mode Calculator on 1,2,100,3,4,5 to verify 3.5, and on 1,2,2,3,4,5 to verify 2.5. Input the complete raw values within the tool's supported limit, not just 2 and 4. The latter input returns 3, the median of two subgroup medians. It is a valid calculation on those two numbers but answers a different question from the pooled-person calculation.
Communicate when the exact answer is unavailable
If only group medians and sizes are published, say the exact pooled median cannot generally be identified from them alone. Request raw values or an adequately detailed compatible frequency distribution. An average of medians can coincidentally match the pooled result in a special case, but one match does not establish a reusable method. Preserve group labels and original summaries when reporting uncertainty so readers understand which information is known and which cannot be reconstructed.