You should use a standard approach, such as g*power, for determining the size of your experiment. The number of participants for the qualitative depends to some degree on the design of your experiment -- for example, will you have a control group?

Good day over there dear researcher.

Mixed method is the combination of both quantitative and qualitative not experimental research.

It depends on what is in your mind, it could be quazi-experimental research.

As to sample size, you need to know the total number of people in that setting , setting your inclusion and exclusion criteria, then you can calculate your sample size.

For the qualitative part depending if its checklist or interview and depending on what you are looking for between 4-6 respondents should be adequate.

I hope you will find the above information useful?

All the best in your endeavor.

With regard to the qualitative part of the study, the classic criterion is "saturation," such that further interviews are no longer yielding new information. Unfortunately, it is typically impossible to predict in advance when you will reach the point of saturation.

Seems to me that the quantitative arm of your MM research will need a sample large enough for at least basic statistical analysis to give a stable result; while the qualitative arm needs only a few of those in the quantitative one; 3-6 would do.

In determining the appropriate sample size for the quantitative part of your mixed-methods research depends on various factors, such as the complexity of your experiment, the effect size you expect, and the desired level of statistical power. Then, it's common to base sample sizes on power analysis, considering factors like expected effect size, significance level, and statistical power. A general guideline is to have a larger sample size for the quantitative part to ensure statistical reliability. Usually in quantitative sample size they used Krejcie and morgan http://www.kenpro.org/sample-size-determination-using-krejcie-and-morgan-table/

As for the qualitative part, the number of participants in IPA (Interpretative Phenomenological Analysis) is typically smaller compared to quantitative studies. A range of 3-6 participants is reasonable for in-depth qualitative exploration, as IPA aims for a detailed understanding of individual experiences. Selecting a subset of participants from your overall sample is acceptable, and it allows you to delve deeply into their experiences without needing to include everyone. However, the key is to ensure that the selected participants represent a diverse range of experiences related to the phenomenon you are investigating. This diversity helps in capturing various perspectives and enriching the qualitative findings. Additionally, the qualitative phase of mixed methods is often exploratory and doesn't necessarily aim for generalizability in the same way as the quantitative phase.

I have no idea where people are getting the now common recommendation of 3-6 participants for the the qualitative strand of an explanatory sequential design. Yes, 3-6 is standard for IPA, but explanatory sequential designs are explicitly based on using the quantitative findings as a pre-existing foundation for the qualitative study, so I think that approach would seldom make IPA a good choice.

The classic standard for the size of a qualitative study is to achieve saturation, i.e., the point at which new cases no longer add any information. This is difficult to predict in advance, too planning for too few cases can be a serious problem.