The Cost of Dichotomization

Splitting a continuous variable into two groups discards information, attenuates effect sizes, and reduces statistical power. Move the sliders to see how much you lose.
The scenario

Does stress predict blood pressure?

Imagine a health psychology study where researchers measure perceived stress using a validated 10-item scale (scored 0–40) and systolic blood pressure via ambulatory monitoring over 24 hours. Both variables are continuous, roughly normally distributed, and there is a true underlying correlation of about ρ = 0.30 between them.

Analyzing these variables continuously — correlating the full stress scores with blood pressure readings — uses all available information. But some researchers instead perform a median split: they label participants as "high stress" or "low stress" based on whether their score falls above or below the sample median, then compare blood pressure between the two groups using a t-test.

This feels intuitive — "high" versus "low" stress is easy to interpret. But the split throws away all within-group differences. A person scoring 15 (just above a median of 14, which is about where published PSS-10 samples cluster) is treated identically to someone scoring 34, even though their stress levels are vastly different. The result is a weaker effect size, a less significant p-value, and lower statistical power. In the simulation below, X represents stress and Y represents blood pressure. Adjust the sliders to see how much information is lost.

Controls
ρ 0.30
Population correlation
N 100
Sample size
Cut 50%
Cut percentile on X
Continuous
Dichotomized
Effect size (continuous)
r = 0.343
p-value (continuous)
p < .001
Effect size (dichotomized)
r = 0.266 (d = 0.55)
p-value (dichotomized)
p = 0.007
Effect Size Retained population values, not this sample
Continuous: ρ = 0.30
Dichotomized: r = 0.239 20% of the effect discarded
Statistical Power ▼ 80% threshold
Continuous: 87%
Dichotomized: 68% -18 percentage points
N needed for 80% power: 84 continuous → 131 dichotomized (1.6× the sample)
Why does this happen? Dichotomization replaces each data point's actual value with its group membership, discarding within-group variance. For a median split on a normally distributed variable, this attenuates the correlation by a factor of roughly √(2/π) ≈ 0.798. Dichotomizing both variables multiplies the attenuation: r is reduced by roughly 2/π ≈ 0.637. That multiplier is a small-ρ approximation to the exact result, (2/π)·arcsin(ρ) — close enough below ρ ≈ .3 (1.5% off at ρ = .30) but increasingly optimistic further up: at the slider's maximum of ρ = .80 the true attenuated value is 0.590, not 0.509. The Effect Size Retained bars above track the exact value rather than the rule of thumb, which is why they read 74% retained at ρ = .80 and not 64%. The lost information translates directly into reduced statistical power — you need a larger sample to detect the same underlying effect, which is what the N needed for 80% power line puts a number on.
How much sample do you waste? Dichotomizing one variable at the median is equivalent to discarding roughly 1/3 of your data. Dichotomizing both is like throwing away about 60% of it — you keep only 2/π ≈ 64% of the information each time, and the losses multiply. The N needed to maintain the same power can increase by 50–100% or more. This is not just theory: van Zwet, Harrell & Senn (2026) analyzed 21,435 randomized controlled trials from the Cochrane Database and found that on average only about 60% of information is retained after dichotomization. Despite binary-outcome trials enrolling larger samples to compensate, fewer of them reached statistical significance than trials that kept the outcome continuous. Their recommendation: model continuously but interpret dichotomously — derive "percentage achieving clinically meaningful improvement" from a continuous analysis rather than by splitting the raw data.
A "median split" sounds principled, but nothing stops a researcher from trying the 40th percentile, then the 60th, then an "upper vs. lower tercile" comparison, reporting whichever version crosses p < .05. Each repositioning of the cut is a separate test on the same data, silently inflating the false positive rate.

Multiply this by other common degrees of freedom — which covariates to include, which items to drop, which subsample to analyze — and you start to see why so many published dichotomized effects failed to replicate.

The replication crisis was not just about fraud or incompetence. It was, in large part, about flexible analyses applied to noisy data with low power — exactly the conditions that dichotomization creates.

A researcher degree of freedom hiding in plain sight

CONTROLS

ρ (population correlation) sets the true linear association between X and Y before any data are generated. Higher values make the cost of dichotomization easier to see — but even modest correlations (ρ ≈ .20–.30) lose noticeably once you split.

N (sample size) controls statistical power for both analyses. Try lowering it to 60–80 and watch how quickly the dichotomized test loses significance while the continuous test still holds.

Cut-point percentile determines where the continuous variable is split. The median (50%) is the most common choice, but it is not the only one — and that is the problem. Try dragging the slider back and forth and watch the p-value on the right. On the default sample it runs from p = .0001 at a 61% cut to p = .068 at an 18% cut — from overwhelming to non-significant, out of the same 100 observations and the same underlying effect. That is not a quirk of this particular draw: across simulated samples at these settings, about 92% contain at least one cut point that is significant and at least one that is not. Press New Sample and try. That is not a bug in the simulation — it is a real analytic choice that researchers make, and it is rarely preregistered.

Watch the effect size as you move the cut, and do not be fooled by it. Pushing the split toward either extreme raises Cohen's d while lowering power. At ρ = .30 and N = 100, the median split gives d = 0.49 with 68% power; a 10/90 split gives a larger-looking d = 0.59 with only 42% power. The effect size grows because the two groups are drawn from further apart in the distribution, but the smaller group shrinks to 10 people and the test gets weaker. An extreme cut is the worst of both worlds: it discards more information and flatters the effect size, which is precisely why a reported d from a dichotomized analysis cannot be compared with one from a continuous analysis.

When both power bars hit 100%. Push ρ or N high enough and both analyses become near-certain to return a significant result, so the power bars saturate and stop discriminating. The cost has not disappeared; it has moved. The Effect Size Retained bars still show a median split discarding about 20% of ρ (35% if you split both variables), and the N needed for 80% power line still shows the dichotomized test demanding a substantially larger sample. Statistical power is a property of the design, not of the estimate you report: a study large enough to survive dichotomization still publishes an attenuated effect size, and every meta-analysis that pools that estimate inherits the attenuation.

Dichotomize X / Both toggles whether only the predictor is split (yielding a two-group t-test) or both variables are split (yielding a 2×2 χ² test). Splitting both compounds the information loss.

Citation

Persson, B. N. (2026). The Cost of Dichotomization [Interactive visualization]. https://bjorn-persson.github.io/visualizations/dichotomization/

@misc{Persson2026dichotomization,
  author = {Björn N. Persson},
  year = {2026},
  title = {The Cost of Dichotomization},
  note = {Interactive visualization},
  url = {https://bjorn-persson.github.io/visualizations/dichotomization/}}