Mean & SD to % & number

Here is a calculator to impute the proportion and number of responders (or remitters) from the mean and standard deviation, assuming the normal distribution. This method has been developed using depression and anxiety meta-analyses (Furukawa TA et al., 2005) and validated with a schizophrenia dataset (Samara et al., 2013) and an insomnia dataset (Furukawa Y et al., 2024).

How to use

The calculator is a single form with four toggles, so the same page covers what used to require four separate calculators:

• Score type — whether you are entering endpoint scores or change scores from baseline.
• Dispersion — whether the paper reports an SD or an SE. If you choose SE, it is converted internally (SD = SE × √N), so no hand conversion is needed.
• Responders are below / above the threshold — below for negatively scored outcomes (HAM-D, PHQ-9, PANSS, the Insomnia Severity Index), above for positively scored ones (WHO-5, GAF, sleep efficiency). You can also switch this by clicking either side of the distribution plot.
• Threshold — enter it directly as an absolute value (typical for remission criteria), or let the calculator derive it from the baseline mean as a percentage reduction or a point reduction (typical for response criteria). When you work with change scores, an absolute threshold is read on the endpoint scale and converted to the change scale via the baseline mean, so no hand conversion is needed there either. The derived threshold is displayed with its formula so you can check what is being computed.

Enter the group size, the mean and the SD (or SE) from the trial report, define the threshold, and the calculator returns the percentage of the group beyond the threshold under the normal distribution assumption, the corresponding number of patients, and a plot of the assumed distribution with the counted area shaded.

For example, if the threshold is 12 (say, half the baseline HAM-D mean), the number of patients in the group is 98 and the endpoint mean and SD of HAM-D are 15 and 6, the calculator returns 30.9% and 30 patients below 12.

One practical point: the threshold must be expressed on the same scale and at the same time point as the mean and SD you enter. The percentage- and point-reduction modes handle the usual case — a response definition anchored to the baseline mean — but the underlying logic is unchanged: everything is converted to a single cut-off on the scale you entered. If neither an SD nor an SE is reported — when all you have is a confidence interval, a median with quartiles, a range or a p-value — derive an SD first with the SD calculator based on the Cochrane Handbook, then come back here.

Method and primary sources

The calculation is the normal cumulative distribution function applied to the standardised distance between the threshold and the group mean. For a negatively scored outcome the imputed proportion of responders is the area below the threshold; for a positively scored one it is the area above it. Nothing more elaborate happens, and the substance of the method lies in whether the assumption behind it holds. The percentage is displayed to one decimal place, and the number of patients is computed from the unrounded proportion.

The approach was introduced for meta-analyses of depression and anxiety trials, where many reports give only continuous endpoint data while the clinically meaningful summary is the proportion of responders [1]. It was subsequently tested against the observed responder numbers in schizophrenia trials [2] and in trials of cognitive behavioural therapy for insomnia [3], in both cases by comparing imputed with reported counts in the same studies. Imputing dichotomous outcomes this way lets a review include trials that would otherwise be excluded from the responder analysis, and reduces the selective-reporting problem that arises when only the trials that happened to publish responder rates contribute to it.

How to cite

Cite the original method paper, and state in the Methods section which trials had imputed rather than observed responder numbers:

For trials that reported only means and standard deviations, the number of responders was imputed from the endpoint mean, standard deviation and group size assuming a normal distribution, as described by Furukawa et al. [1] and validated against observed responder rates in schizophrenia [2] and insomnia [3] trials, using the calculator at https://yukifurukawa.jp/mean-and-sd-to-percent-and-number/. The response threshold was defined a priori as [definition].

Limitations

• Normality is assumed. Symptom scores are often skewed or floor-limited, particularly at the end of successful treatment, and skewness biases the imputed proportion. The imputation is least reliable where the distribution is most obviously non-normal.
• The result moves with the threshold. A threshold set one or two points differently can change the imputed responder count substantially. Define it in the protocol, apply it identically to every arm and every trial, and consider a sensitivity analysis with an alternative threshold.
• Imputed counts are estimates, not data. Label them as imputed in the forest plot or the data table, and consider a sensitivity analysis restricted to trials that reported observed responder numbers.
• Validation exists for depression and anxiety, schizophrenia and insomnia scales. Applying the method to other populations or to scales with very different distributional behaviour extrapolates beyond what has been checked.
• The mean, the SD and the group size must all refer to the same group at the same time point. Baseline-observation-carried-forward and other imputation schemes used inside the trial are inherited unchanged.
• The number of patients is rounded to a whole number from the unrounded proportion, so the rounding error never exceeds half a patient per group.
• Where individual participant data are available, use them instead of imputing.

References

1. Furukawa TA, Cipriani A, Barbui C, Brambilla P, Watanabe N. Imputing response rates from means and standard deviations in meta-analyses. Int Clin Psychopharmacol. 2005;20(1):49-52. PubMed 15602117
2. Samara MT, Spineli LM, Furukawa TA, et al. Imputation of response rates from means and standard deviations in schizophrenia. Schizophr Res. 2013;151(1-3):209-214. PubMed 24262679
3. Furukawa Y, Sakata M, Yamamoto R, et al. Components and Delivery Formats of Cognitive Behavioral Therapy for Chronic Insomnia in Adults: A Systematic Review and Component Network Meta-Analysis. JAMA Psychiatry. 2024;81(4):357-365. PubMed 38231522