I have the SD of my outcome. How many respondents do I need?
Source:vignettes/from-sd-find-n.Rmd
from-sd-find-n.RmdSuppose we know the standard deviation of our outcome in a reference population and we have a specific treatment effect in mind. We want to find the sample size required for the study to have adequate power.
from_sd() piped into find_n() answers this
question.
Without control variables
Imagine we are planning to replicate Ahler and Sood (2018), who find that correcting respondents’ misperceptions of their out-party reduces affective polarization. In their experiment, respondents first report their perceptions of the percent of out-party members with certain demographic attributes, then receive the correct information. Compared to a control group, the treatment group evaluated supporters of the out-party more favorably on a 101-point feeling thermometer. Ahler and Sood estimate a treatment effect of 6.4 points with a 95% confidence interval of [3, 10]. Broockman, Kalla, and Westwood (2022) closely replicate this result, estimating a treatment effect of 3.9 with a 90% confidence interval of [1.1, 6.6].
To predict the standard error using Rule 3 of Rainey (2026), we need the standard deviation of the outcome in a reference population. The 2020 American National Elections Study (ANES) asks a similar question: respondents report their feelings toward the Democratic or Republican party on a 101-point feeling thermometer. Ahler and Sood ask about supporters of the party rather than the party itself, so the measures are not identical, but the ANES provides a reasonable approximation. The standard deviation of the ANES feeling thermometer responses is 20.8.
The lower bound of Ahler and Sood’s 95% confidence interval is 3 points. We use this as our assumed effect, tau = 3. We want 95% power.
library(powerrules)
from_sd(sd_y = 20.8) |>
find_n(tau = 3, power = 0.95)#> -- Power Analysis ------------------------------------------------------
#> Design: balanced, between-subjects
#> Source: reference population SD
#> CI level: 90% (size-0.05 test of directional hypothesis)
#>
#> Inputs:
#> SD(Y) = 20.8
#> tau = 3
#> power = 95%
#>
#> MDE factor = qnorm(0.95) + qnorm(0.95) = 3.29 [Table 2]
#> n (planned) = 2 * (3.29 * 20.8 / 3)^2
#> = 1,041 per condition (2,082 total) [Rule 6]
#>
#> -- Manuscript sentence (edit as needed) --------------------------------
#> For a balanced, between-subjects design, assuming a standard deviation
#> of 20.8, the experiment requires 1,041 respondents per condition
#> (2,082 total) for 95% power to detect a treatment effect of 3 units,
#> using a one-sided test at the 0.05 level.
#>
#> Note: The paper rounds the MDE factor to 3.3 for 95% power. This
#> software uses the exact value (3.29), so results differ slightly from
#> hand calculations using the rounded factor.The output computes the MDE factor for the requested power level, then solves for the sample size (Rule 6). The study requires 1,041 respondents per condition (2,082 total) for 95% power to detect a treatment effect of 3 points.
With control variables
Broockman, Kalla, and Westwood (2022) control for a seven-point party identification scale and partisan strength. In the 2020 ANES, these two variables have an R2 of 5% for the feeling thermometer toward the Democratic and Republican parties (rather than “supporters of” those parties). For R2 = 5%, regression adjustment shrinks the standard error by about 2.5%.
#> -- Power Analysis ------------------------------------------------------
#> Design: balanced, between-subjects
#> Source: reference population SD
#> CI level: 90% (size-0.05 test of directional hypothesis)
#>
#> Inputs:
#> SD(Y) = 20.8
#> R^2 = 0.05
#> tau = 3
#> power = 95%
#>
#> MDE factor = qnorm(0.95) + qnorm(0.95) = 3.29 [Table 2]
#> n (planned) = 2 * (3.29 * 20.8 * sqrt(1 - 0.05) / 3)^2
#> = 989 per condition (1,978 total) [Rule 6]
#>
#> -- Manuscript sentence (edit as needed) --------------------------------
#> For a balanced, between-subjects design, assuming a standard deviation
#> of 20.8 and control variables that explain 5% of the variance in the
#> outcome, the experiment requires 989 respondents per condition (1,978
#> total) for 95% power to detect a treatment effect of 3 units, using a
#> one-sided test at the 0.05 level.
#>
#> Note: The paper rounds the MDE factor to 3.3 for 95% power. This
#> software uses the exact value (3.29), so results differ slightly from
#> hand calculations using the rounded factor.With control variables, the required sample drops from 1,041 to 989 per condition. These controls help, but modestly.
How to choose tau and power
The tau argument is the treatment effect assumed for
computing power. The lower bound of a confidence interval from a closely
related study is a useful starting point (Rainey
2026). In the Ahler and Sood example, the lower bound of their
95% confidence interval is 3 points.
The power argument (default 0.80) sets the desired power
level. Higher power requires more respondents. Rainey (2026) recommends 80% power as a baseline
and 95% power as a stretch goal. Both pipelines above use 95% power; at
80% power (the default), the required sample sizes would be smaller.