In most instances, this is the case. Most quantitative research utilises probability-based sampling techniques - especially if experimental/quasi-experimental. If the intention is larger sample sizes, for generalisation etc, then it is often impractical to adopt non-probability techniques i.e. purposive and snow-balling. If any, convenience is probably the most appropriate. There is the potential, of course, to use such techniques - specifically for approaches such as case design - but depends on the nature and the sample numbers involved in such designs.
If you want to infer from a sample to a population, you generally do that by using probability/design-based methods so that each member of the sample basically represents a number of members of the population equal to the inverse probability of selection. Various designs have been worked out, usually unbiased*, but not always, and variance* formulas have been derived.
* However, bias and variance are defined by what would happen if the sample were taken many times, not once. But this does at least give a basis for inference.
A purposive sample alone will have an unknown variance, an often large bias when you know nothing about the population, and no way to estimate them. (What Dean said though does remind me that snowball sampling - not an area for me - can, I have read, give you some help with inference, but with relatively large samples. You would need to research that to see if that is true and how it works. I don't recall.)
It is often helpful to stratify based on your knowledge of the population, and what groups would vary more from other groups than amongst themselves. I would expect that to be helpful for qualitative surveys as well. Yes?
The exception I have used to not obtaining inference from purposive samples is when you have regressor data for the entire population, such that you may use 'prediction' to basically impute for missing (that is, here, not-in-sample) data. Once again stratification helps. Variance is then measured based on the actual sample drawn. Bias can be large if data are modeled together that do not go together, but with highly skewed data, such as in establishment surveys, the variance can be so relatively small for a cutoff sample compared to a design-based sample of the same size, that the overall result is better. Still, that is a special circumstance that you are not liable to encounter, leaving a probability sample as likely your only viable option.
The following might be interesting and entertaining:
Brewer, K.R.W. (2014), “Three controversies in the history of survey sampling,” Survey Methodology,
(December 2013/January 2014), Vol 39, No 2, pp. 249-262. Statistics Canada, Catalogue No. 12-001-X.
Note there that Ken Brewer proposes that with small populations, model-based estimators might do well. You still might use random sampling with the predictions, or "balanced sampling," where, for example, the mean regressor value associated with the sample is equal to the mean of the population of regressor values. For highly skewed establishment surveys you might use a cutoff sample and trade a little bias added for a much reduced variance.
Quantitative research follows mostly probability sampling but for few studies it may follow non-probability sampling also. Purposive sample may be use for qualitative research based on the feasibility of the content.
All respondents are chosen based on a specific purpose, objective and question to fulfill the research demand. So, whatever, it is qualitative and quantitative, we can use the purposeful sampling for both qualitative and quantitative research as there is no barrier.
You certainly can use purposive sampling in quantitative research, but this will limit how you can analyze that data. In particular, you could still use descriptive statistics, such as means and percentages, but without random sampling, most reviewers would question whether you could do statistical tests (i.e., inferential statistics).
Convenience sampling technique is applicable to both qualitative and quantitative studies, although it is most frequently used in quantitative studies while purposive sampling is typically used in qualitative studies [5].