The advice for applicants is generally to apply to about 9 schools at a proportion of 3 reach schools, 3 slight reach, and 3 easy acceptance (is there really such a thing for graduate schools?). In reality, I think it is very worth it to apply to more in the reach and slight reach category.

The PhD application acceptance has both a non-random component (objective applicant quality measures) and a random component (maybe someone on the committee really likes your writing sample, had a great experience meeting your letter writer at a conference, or was really impressed with the last person they met from your undergrad program).

In my mind, there are at least three reasons for an applicant to apply to more than the conventional 9 schools. First, if you think the random component is large, then, given the returns from getting into a higher ranked school, it is quite advantageous for the applicant with a marginal application to try quite a few reach schools to see if they have good draw at one of them. Another rationale for applying to more schools is if an applicant doesn't understand clearly where they sit on the distribution of applicants. It might not be clear what schools are reach and which are ridiculous reaches. Finally, if the applicant doesn't have time to "learn" about the process (and take a second try at it the next year) then there is a big cost to taking the risk of getting too many negative draws.

Even with these reasons, 25 applications sounds like a lot. But I would work through it by helping the applicant understand their chances and think through costs and benefits, rather than focusing on a specific number.

Possibly related story: My husband applied to 9 schools and got into the best and worst of the 9. The fact that the best school accepted him was the result of a good draw and has had many benefits. If he hadn't had that good draw, I would have encouraged him to apply to more than 9 the next time.

The advice for applicants is generally to apply to about 9 schools at a proportion of 3 reach schools, 3 slight reach, and 3 easy acceptance (is there really such a thing for graduate schools?). In reality, I think it is very worth it to apply to more in the reach and slight reach category.

The PhD application acceptance has both a non-random component (objective applicant quality measures) and a random component (maybe someone on the committee really likes your writing sample, had a great experience meeting your letter writer at a conference, or was really impressed with the last person they met from your undergrad program).

In my mind, there are at least three reasons for an applicant to apply to more than the conventional 9 schools. First, if you think the random component is large, then, given the returns from getting into a higher ranked school, it is quite advantageous for the applicant with a marginal application to try quite a few reach schools to see if they have good draw at one of them. Another rationale for applying to more schools is if an applicant doesn't understand clearly where they sit on the distribution of applicants. It might not be clear what schools are reach and which are ridiculous reaches. Finally, if the applicant doesn't have a good outside option, there will be a big cost to "learn" about their chances with the process via this year's application. While some would be happy to take a second try at it the next year, for others there would be a big cost to taking the risk of getting too many negative draws.

Even with these reasons, 25 applications sounds like a lot. But I would work through it by helping the applicant understand their chances and think through costs and benefits, rather than focusing on a specific number.

Possibly related story: My husband applied to 9 schools and got into the best and worst of the 9. The fact that the best school accepted him was the result of a good draw and has had many benefits. If he hadn't had that good draw, I would have encouraged him to apply to more than 9 the next time.