n two separate studies, the actual difference between the means of a treated group and an untreated group is 3 points. However, in one study, the σM1-M2 is very large and so the 3 points is not found to be significant. In the other study, the σM1-M2 is very small and so the 3 points is found to be significant. What might have caused this big difference in the σM1-M2 for the two studies?

It may be due to the sample size. There is a possibility the sample may be too large, which will affect the significance of the outcome.

The symbol σM1-M2 represents the standard deviation or variability of the difference between the means of two groups. In this case, the difference is between a treated group and an untreated group.

The fact that the σM1-M2 is very large in one study and very small in the other study suggests that the variability of the difference between the means is different in each case.

There are several factors that could have caused this difference in variability:

1. Sample Size: The sample size of the two studies may be different. Generally, larger sample sizes lead to a more accurate estimate of the true population variability.

2. Treatment Effect: The effectiveness of the treatment could vary between the two studies. If the treatment has a stronger effect in one study, it could result in a smaller variability in the difference between the means.

3. Control of Other Variables: The level of control over confounding variables (variables that could impact the outcome) may differ between the two studies. An improved control of confounding variables can reduce the variability in the difference between the means.

4. Measurement Error: The accuracy and precision of the measurement instruments used to assess the outcome variable could differ. High measurement error can increase the variability in the difference between the means.

It is important to note that the significance of the 3-point difference between the means is dependent on both the size of the actual difference and the variability of the difference. A smaller variability increases the chances of detecting a significant difference if the actual difference is the same.