Comparing Two Treatments
Aligned to S-IC.B.5 — Common Core State Standards for Mathematics.
What this lesson teaches
In a randomized experiment, subjects are randomly assigned to two treatments and the results compared. To judge whether a difference is real or due to chance, you simulate re-shuffling the data many times and see how often chance alone gives a difference that large.
Worked examples
Example 1
A drug group's pain drops 8 points; the placebo group's drops 3, a difference of 5. To test it, pool all results and randomly re-split into two groups many times. If a gap of 5 rarely appears by chance, the treatment difference is significant.
Example 2
Two fertilizers are compared on 40 plots, randomly assigned 20 to each. Fertilizer A averages 62 bushels, B averages 55, a gap of 7. Re-shuffle the 40 results into two groups of 20 at random, 1,000 times, and record each simulated gap. If only 12 of the 1,000 reach 7 or more, the p-value is about 0.012 and the difference is unlikely to be chance alone.
Example 3
Common mistake: drawing a cause-and-effect conclusion from an observational study. If people who take a vitamin report fewer colds, that is not evidence the vitamin works — the people who choose vitamins may also sleep more or exercise more. Only RANDOM ASSIGNMENT to treatment groups balances the unknown differences, which is what lets an experiment claim causation and an observational study claim only association.
Practice questions
- State the observed difference: group A gains 12, group B gains 7.
- Why does random assignment strengthen the comparison?
- In a simulation, a difference this large happened 2% of the time: is it significant?
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
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Comparar Dos Tratamientos
En un experimento aleatorizado, los sujetos se asignan al azar a dos tratamientos y se comparan los resultados. Para juzgar si una diferencia es real o por azar, simulas volver a mezclar los datos muchas veces y observas con qué frecuencia el azar por sí solo produce una diferencia tan grande.
Ejemplo 1: El grupo con el medicamento reduce el dolor 8 puntos; el grupo con placebo lo reduce 3, una diferencia de 5. Para probarlo, junta todos los resultados y vuelve a dividirlos al azar en dos grupos muchas veces. Si una diferencia de 5 rara vez aparece por azar, la diferencia del tratamiento es significativa.
Ejemplo 2: Se comparan dos fertilizantes en 40 parcelas, asignadas al azar 20 a cada uno. El fertilizante A promedia 62 fanegas, B promedia 55, una diferencia de 7. Vuelve a repartir los 40 resultados en dos grupos de 20 al azar, 1,000 veces, y anota cada diferencia simulada. Si solo 12 de las 1,000 llegan a 7 o más, el valor p es de unos 0.012 y es poco probable que la diferencia sea solo azar.
Ejemplo 3: Error común: sacar una conclusión de causa y efecto de un estudio observacional. Si las personas que toman una vitamina reportan menos resfriados, eso no prueba que la vitamina funcione — quienes eligen tomar vitaminas quizá también duerman o hagan más ejercicio. Solo la ASIGNACIÓN ALEATORIA a los grupos de tratamiento equilibra las diferencias desconocidas, y eso es lo que permite a un experimento afirmar causalidad y a un estudio observacional solo asociación.
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