Calculate Mean SD from Median IQR
Convert summary statistics into practical estimates using a polished, research-friendly calculator. Enter the median, first quartile, third quartile, and optional sample size to estimate the mean and standard deviation from the interquartile range.
Estimator Inputs
Use Q1, median, and Q3. Sample size is optional and shown for context in the interpretation.
Results
Estimated values update instantly after calculation.
Distribution Snapshot
This chart visualizes Q1, median, Q3, estimated mean, and one estimated standard deviation around the mean.
How to calculate mean SD from median IQR
Researchers, clinicians, students, and analysts often run into a practical problem: a paper reports the median and the interquartile range (IQR), but your analysis requires the mean and standard deviation (SD). This happens frequently in evidence synthesis, clinical reviews, biostatistics, health economics, and applied social science. Many studies report nonparametric summaries because their data are skewed or because journals prefer medians for certain endpoints, yet meta-analytic methods and comparative models may still need a mean and a measure of spread.
When you need to calculate mean SD from median IQR, you are usually not deriving exact values. Instead, you are creating statistical approximations from limited summary information. The most widely used idea is straightforward: if a distribution is reasonably symmetric or close to normal, the median is often near the mean, and the IQR can be converted into an SD by using the relationship between quartiles and the normal distribution.
The calculator above follows a practical quartile-based estimation framework. A common approximation for the mean uses (Q1 + Median + Q3) / 3. A common approximation for the standard deviation uses SD ≈ (Q3 – Q1) / 1.35. These formulas are popular because they are simple, interpretable, and often adequate when only quartile summaries are available.
Why this conversion is useful
In many real-world datasets, especially in medicine and epidemiology, authors report medians because the data may not be perfectly normal. However, secondary analysis may still require means and SDs. Typical use cases include:
- Preparing continuous outcomes for meta-analysis.
- Comparing studies that report different summary statistics.
- Building models that expect parametric inputs.
- Approximating missing descriptive statistics for internal reporting.
- Transforming published quartile summaries into a more familiar scale.
The core formulas behind the calculator
If you have the first quartile Q1, the median, and the third quartile Q3, a common practical estimate is:
- Estimated Mean: (Q1 + Median + Q3) / 3
- Estimated IQR: Q3 – Q1
- Estimated SD: (Q3 – Q1) / 1.35
Why divide by 1.35? In a normal distribution, the distance between the 25th percentile and the 75th percentile is about 1.349 standard deviations. That means the IQR can be scaled into an approximate SD by dividing by about 1.35. This does not make the estimate exact, but it gives a useful approximation when the underlying data are not extremely skewed.
| Statistic Available | Approximation | Interpretation |
|---|---|---|
| Q1, Median, Q3 | Mean ≈ (Q1 + Median + Q3) / 3 | Balances the center using lower quartile, midpoint, and upper quartile. |
| IQR = Q3 – Q1 | SD ≈ IQR / 1.35 | Uses the normal-distribution relationship between quartiles and spread. |
| Median only | No reliable SD estimate by itself | Additional spread information is needed for a defensible conversion. |
Important assumptions when estimating mean and SD from median and IQR
The phrase calculate mean SD from median IQR can sound more exact than it really is. In practice, you are making assumptions about the underlying distribution. The closer the data are to symmetry, the more stable these approximations tend to be. If the variable is heavily skewed, has floor effects, ceiling effects, or contains strong outliers, the estimated mean and SD may differ meaningfully from the true values.
You should especially be careful in these cases:
- Highly right-skewed biomarkers or cost data.
- Small samples where quartiles are unstable.
- Bounded scales with strong clustering at one end.
- Zero-inflated variables or time-to-event data.
- Outcomes reported on transformed or ordinal scales.
If the source study explicitly says the data are markedly skewed, your estimated mean and SD should be treated as rough approximations rather than substitutes for original raw data. In higher-stakes analyses, it may be better to contact study authors or use sensitivity analyses.
Worked example: converting quartiles into a mean and SD estimate
Suppose a study reports the following summary statistics for hospital length of stay:
- Q1 = 12
- Median = 16
- Q3 = 21
First compute the IQR:
IQR = 21 – 12 = 9
Then estimate the mean:
Mean ≈ (12 + 16 + 21) / 3 = 16.33
Next estimate the standard deviation:
SD ≈ 9 / 1.35 = 6.67
So a practical summary for downstream analysis would be an estimated mean of approximately 16.33 and an estimated SD of approximately 6.67. That is exactly the type of calculation the tool on this page performs.
When sample size matters
The sample size is not always essential for the basic quartile-only approximation, but it still matters in interpretation. In small studies, quartiles can jump noticeably due to random variation. In larger studies, quartiles are usually more stable, and the approximation may be more trustworthy. Some advanced methods incorporate sample size and additional summary measures such as minimum and maximum values. Those methods may perform better when more information is available.
If you only have median and IQR, your conversion options are naturally limited. That is why quartile-based methods remain useful: they are transparent, easy to reproduce, and suitable when reporting is sparse. Still, if a study also provides minimum, maximum, or sample size-specific formulas from methodological papers, it can be worth comparing results across methods.
| Scenario | Likely Reliability | Best Practice |
|---|---|---|
| Moderate to large sample, roughly symmetric data | Generally reasonable | Use quartile-based approximation and document the method. |
| Strongly skewed data | Lower reliability | Use caution, consider sensitivity analyses, and report limitations. |
| Very small sample sizes | Potentially unstable | Avoid overinterpreting converted values and seek original data if possible. |
| Additional min and max available | Potentially improved estimation | Consider broader published methods that incorporate more summary statistics. |
Mean versus median: why the difference matters
The mean and median are both measures of central tendency, but they answer slightly different questions. The median identifies the middle observation, while the mean balances all values arithmetically. In symmetric distributions, they are often close. In skewed distributions, the mean may be pulled toward the longer tail. This is the main reason conversions from median and IQR must be interpreted carefully.
If Q1, median, and Q3 are almost equally spaced, the data may be fairly symmetric. For example, if the distance from Q1 to the median is similar to the distance from the median to Q3, using quartiles to estimate the mean is often sensible. If the spacing is very uneven, that asymmetry can signal skewness, and your converted mean may be less representative of the true arithmetic average.
Standard deviation from IQR: concept and limitations
Standard deviation measures the average spread around the mean, while the IQR captures the middle 50 percent of the data. They are not identical statistics, but they are related under specific distributional assumptions. The conversion SD ≈ IQR / 1.35 is grounded in the normal distribution, where quartiles occur at known z-scores. Because of that, the estimate is often described as a normal-based SD approximation.
This approach works best when:
- The data are continuous and not excessively skewed.
- The quartiles are measured accurately.
- The study sample is not extremely small.
- You need a pragmatic estimate rather than an exact reconstruction.
It works less well when distributions are heavy-tailed, highly asymmetric, or truncated. In those settings, the true SD may be meaningfully larger or smaller than the converted value.
Best practices for reporting converted values
If you use a tool to calculate mean SD from median IQR, transparency is essential. Always state that the values were estimated rather than directly reported. If you are conducting a review or meta-analysis, include the formula or citation in your methods section. If possible, perform a sensitivity analysis using alternative plausible methods and report whether your overall conclusions change.
- Label the results as estimated mean and estimated SD.
- Document the conversion formula used.
- Note assumptions about normality or approximate symmetry.
- Flag any studies with strong skewness or sparse reporting.
- Consider author contact when exact values are important.
Practical interpretation of the calculator output
The calculator on this page is designed to be fast and transparent. Once you enter Q1, the median, and Q3, it computes the IQR and then estimates the mean and SD. It also generates a simple graph so you can visually compare the quartiles, median, and estimated mean. This helps users see whether the center appears balanced or asymmetric. If the estimated mean is close to the median, that often suggests a more symmetric structure. If they diverge noticeably, caution is warranted.
For educational, screening, and many synthesis tasks, this kind of estimator is extremely useful. It turns limited published summaries into actionable numbers without hiding the fact that they are approximations. That balance between practicality and methodological honesty is exactly what makes quartile-based conversion methods so widely used.
Authoritative resources and further reading
If you want to deepen your understanding of descriptive statistics, evidence synthesis, and distributional assumptions, these reputable resources are useful starting points:
- National Center for Biotechnology Information for biomedical literature and methodological papers.
- University of California, Berkeley Statistics for foundational concepts in statistical inference and distribution theory.
- National Institute of Mental Health for examples of research reporting practices in health studies.
Final takeaway
To calculate mean SD from median IQR, the most practical starting point is usually a quartile-based approximation. Use the median and quartiles to estimate the mean, compute the IQR as Q3 minus Q1, and estimate the standard deviation by dividing the IQR by 1.35. This approach is easy to apply, easy to explain, and often good enough for evidence synthesis and exploratory analysis. Just remember the key principle: these values are estimated, not observed directly. The more skewed or unusual the underlying data, the more carefully you should interpret the result.