The three measures do different jobs. The qualitative measure is the grade point average the policy specifies at each evaluation, or a comparable assessment against a norm where a grade point average is not appropriate. On top of that sits a rule with a fixed federal content: a student enrolled in a program longer than two academic years must, at the end of the second academic year, have a grade point average of at least a "C" or its equivalent, or academic standing consistent with the institution's graduation requirements. The quantitative measure is pace, calculated either by dividing cumulative hours successfully completed by cumulative hours attempted, or by working out how many hours the student should have completed by that point to finish in time. Remedial courses need not be included. Sitting over both is the maximum timeframe, which for an undergraduate program measured in credit hours or clock hours is a period no longer than 150 percent of the program's published length; for a graduate program the institution defines it, based on the program's length.
Withdrawals, incompletes, repeats and transfer credits are where the pace calculation bites. The policy must describe how each of them affects the grade point average and the pace, and one item is dictated federally: credit hours accepted from another institution toward the student's program "must count as both attempted and completed hours." A student who transfers in 45 credits therefore arrives with those credits helping the pace ratio, while courses dropped after the census date, failed, or repeated typically count as attempted without counting as completed. This is the route by which a student with a perfectly respectable grade point average loses aid: nothing is wrong with the grades, but attempted hours have run ahead of completed ones.
Warning and probation are not synonyms, and which one is available depends on how often the school evaluates. A school that evaluates at the end of each payment period may place a student who failed the standard on financial aid warning and keep disbursing aid for one payment period, and the regulation says in terms that warning status "may be assigned without an appeal or other action by the student." If the student fails again at the end of that payment period, the school may move them to financial aid probation only if the student appeals and the school either determines the student can meet the standard by the end of the next payment period or develops an academic plan that will get them there by a specific point in time. A school that evaluates annually has no warning status to offer. Its students go straight to probation, which means the appeal is required at the first failure rather than the second. Probation lasts one payment period, during which the school may impose conditions such as a reduced course load or specific courses, and at the end of it the student must either meet the standard or meet the requirements of the academic plan.
The satisfactory academic progress appeal is a real appeal, which is not true of everything called an appeal in a financial aid office. 34 CFR 668.34(b) defines it as "a process by which a student who is not meeting the institution's satisfactory academic progress standards petitions the institution for reconsideration of the student's eligibility for title IV, HEA program assistance." The regulation tells schools that permit appeals to publish the basis on which one may be filed, naming "the death of a relative, an injury or illness of the student, or other special circumstances," and to require the student to explain why they failed and what has changed that will let them succeed at the next evaluation. What families usually mean by "financial aid appeal", asking the aid office to adjust the figures behind an award because of changed circumstances, is a different authority entirely and belongs to professional judgment. A school is not obliged to permit satisfactory academic progress appeals at all; one that does not must instead describe how a student can reestablish eligibility, and every policy must provide for notifying students of the results of an evaluation that affects their eligibility.