# How can I pass a statistics exam that overly relies on memorizing formulas?

Long story short, I struggle with a statistics exam at the university. Although I know how to solve all types of exercises and the rationale behind each of them, I still can't pass this exam, because we are supposed to know ninety formulas by heart. I am not complaining about the volume of work that has to be put in, because I did all that had to be done. Still, when practicing, I was looking every time on a sheet where I had all formulas needed.

During the exam, though, we were not allowed to use any help. I find it extremely hard to memorize ninety formulas by heart. I literally don't know how to memorize them; it would take me years to learn them by heart. How can I do this?

(Assume for the sake of this question that it’s too late to change the exam mode to one which does not require so much memorizing.)

• The ability to memorize ninety formulas seems like a really weird thing to be testing on a university exam. OP, are you sure you haven't misunderstood what the professor expects you to do? Commented Sep 3, 2023 at 10:14
• @DanielHatton: Sadly, I don’t find this surprising at all and have observed many exams that are similar, though not quite this extreme. The reasons for this are usually: 1) The professor thinks that memorizing formulas is relevant and a good proxy for understanding (after all, they have memorised all of them). 2) Exams that can just quiz you on this are easier to create. 3) In certain disciplines, most students are happy if they can pass an exam by rote memorisation instead of actually understanding the subject. E.g., I would expect this if this is Statistics for Pharmacists. Commented Sep 3, 2023 at 12:58
• @Wrzlprmft Indeed. What I had in mind, though, was that the prof might have created exam questions that can be solved either by students memorizing the formulas or by students deriving them from first principles as needed, but with the prof having a preference for the first principles approach, which leads the prof not to be concerned if the memorization task is really difficult. Commented Sep 3, 2023 at 14:24
• My guess is that the professor doesn't expect the students to know 90 formulas by hearth, but know the concepts and derive the formulas from a basic set of formulas, but some students have tried to simplify the work by using a 90 formulas sheet. For example, you don't need the formula for mean and median if you know what mean and median are.
– Pere
Commented Sep 3, 2023 at 17:29
• @Pete: This was my thought also. For example, in extremely elementary treatments of electricity, Ohm's law V = IR is sometimes stated in 3 different ways (the other two being I = V/R and R = V/I), and to present the topic without assuming students know any algebra, students are sometimes told (or very weak students in other situations take it upon themselves) to become familiar with all 3 versions. Commented Sep 3, 2023 at 22:18

In addition to @Buffy suggestions: A lot of formulas in statistics are variations on the same theme. If you understand the relationship between the formulas you can learn for each "block of formulas" the main formula and the relationship with the others in that block.

It does not help you, but for the statistics exam I make I allow a list of formulas. If the student needs or wants to use statistics later, I would prefer that he/she/they look the required formulas up rather than rely on a memory from a course they took years ago. If that is the way I want them to use statistics, then the exam should reflect that.

• +1 I'm even wondering what 90 different formulas can be included on a basic statistics course. To get to 90 formulas I would need to put a lot of variations of the same formulas.
– Pere
Commented Sep 3, 2023 at 17:26
• @Pere: Here's the formula card that comes packaged with the Weiss Introductory Statistics text: math1312.weebly.com/uploads/2/1/2/0/2120132/… Commented Sep 3, 2023 at 23:57
• ... on which there just happens to be exactly 100 bullet-point formulas (ignoring the statistical tables at the back). Of those, I think maybe about a dozen may be duplicates, algebraic rewrites, or analogs of other formulas. Commented Sep 4, 2023 at 3:13
• @DanielR.Collins And as predicted in the answer, these are variations of significantly less than 100 formulas. But even if you distill this down to somewhere between 10 and 20 meaningfully distinct formulas, teaching statistics as 'memorize all of them and then apply the correct one' is a terrible way to teach statistics almost guaranteed to produce garbage when applied in real world setting. Commented Sep 4, 2023 at 12:32
• @quarague: That's not remotely a correct accounting, as noted in my prior comment. Commented Sep 4, 2023 at 13:58

One way to memorize a lot of things is to write each one on a separate index card and carry them about with you at all times. Perhaps write a name on one side and the formula on the other.

Never leave home without your card deck.

Then, frequently during the day, whenever you have a stray moment, pull out the deck and refresh your memory with a few cards. Perhaps put a check mark on each card when you review it.

If you master a formula, you can leave its card behind - or not.

Another help is to write the formulae down when you review it. The coordination between hand and brain helps memory.

• Or you can use a flashcard app on your phone, for example Anki, which algorithmically determines when to show you each card based on how well you remembered it the last time you saw that card. Commented Sep 3, 2023 at 12:04
• I'm not sure why you would expect writing flashcards on a keyboard and then reading them on a screen, to be less effective than writing them on paper and then reading them on paper. What the electronic version adds is spaced repetition, which is one of the most effective learning techniques for domains in which it's applicable (and it's definitely applicable for memorising formulae). Commented Sep 3, 2023 at 12:20
• It's strange that from this experience, you conclude that the learning effect was attributable to the cards being physically made of paper. It is much more reasonable to suppose that your valedictorian's success was a result of what he did with those cards. Digital flashcards are used exactly the same way, but with the added benefit that the application keeps track of which cards you need to review and when. Commented Sep 3, 2023 at 12:59
• @kaya3, no, it was the fact that he wrote them on paper himself. It was his physical act of writing that engaged the brain. Typing has much less of an effect in my experience. It isn't the paper, it is the writing. And the key words (to me) in the paper you cited earlier is "for similar problems". Commented Sep 3, 2023 at 13:06
• I think that's an interesting hypothesis but I wonder if there is any evidence for it. Regardless, the point of flashcards is not really the writing them in the first place (though writing your own flashcards is better than using someone else's, because you can build a deck that is specifically relevant for what you need to learn), rather it's that you periodically review them. Commented Sep 3, 2023 at 13:07

1. Understand the basic intuition.

My best method of memorizing formulas is to simply understand the rationale (or rather "intuition") behind them.

Let me explain it on the basis of the Gaussian: Perhaps you remember how the "Bell Curve" looks like:

• It is curved upwards at the center similar to the f(x)=-x^2 (parabola)
• It has two asymptotes y=0 at lim x->infinity and lim x->-infinity.
• The function is even (have an axis of symmetry)

That 2 facts make me suppose that it has something to do with f(x)=-x^2 and f(x)=e^-x functions. Let combine them into: f(x)=e^-x^2. Of course what we got is some kind of ideal "Unit Gaussian Function". The last thing we need to do is to shape it with some custom parameters in order to place it in a proper place:

• Standard derivation (small sigma) to determine its "width". This is our 1/2sigma.
• Shift it along X axis to place its mean value/center at a proper place. This is our (x-mi) term.
• Multiply it by some factor in order to give it a desired max value or the most often case: to normalize it so the probability of x taking any value from X is equal 1. This is our: 1/sqrt(2pi)sigma term.

2. Learn how to derive

Other thing I usually do is to try to learn how the formula was derived from more fundamental theorems or formulas. When I sometimes forget the exact formula, I usually try to take some time to make a very quick "proof" for the formula. For instance: https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution

Unfortunately it does not work well for formulas derived using very complex integrals or differential equations. Fortunately there is not so many of them. They usually result in functions that are not elementary anyway and must be published only in a form of tables or value sheets.

## Write

I didn't have to memorize a set block of formulas, but I had to memorize a lot of raw material for my qualifying exams. (I assume everyone does.)

Writing is widely reputed to aid in memorization and in my anecdotal case it certainly did: My studying consisted of the creation of dozens of pages of handwritten notes. Then condensing it into a second set of notes maybe 30% shorter. And then a third set, 30% shorter again. (Alongside more traditional attempts at study, understanding and memorization, of course.)

The reduction in length was partly an indication of how much I felt I'd already memorized, but the sensation of actually writing those notes by hand, multiple times, helped. By the time of the exams, I was able to call up key information by imagining all those times I had already written it.

In your case, if discrete formulas are your downfall, I would make flashcards out of them, and as the act of study I would start by writing new cards down from memory.

## Practice

If these formulas are simple enough that you can also work problems by hand, then definitely do that, too. And when I say work them by hand, I mean it literally. Start with the problem, write down the applicable formula(s), and work the problem with nothing more than a hand calculator to handle the mechanics of the arithmetic.

(Again, I am assuming the formulas and problems would be simple enough that this is viable.)

Categorize them as much as possible, perhaps by topic.

Then order them in some reasonable way within each category, perhaps by similarity. Try to remember the similarities/small differences between them and why they exist.

Then write them down in some organized way on a sheet, and color code them by category.

The idea is you want to break this gigantic pile down into easily digestible little chunks.

Finally it's just a matter of revisiting and reviewing this colorful result regularly: when you wake up, before you go to sleep, waiting in a queue somewhere, when you're on a train ...

For memorising a set of formulae (or words in a language, names of people, or other simple facts), a good choice is spaced repetition. This is implemented by applications such as Anki, which you can use on a desktop computer or a smartphone; you write the facts you want to memorise on "cards", typically with two "sides" (formula name on one side, formula on the other). You review those cards periodically (e.g. see one side and try to remember the other), and after reviewing a card you can tell the app whether it was easy or difficult for you to recall. That feeds back into how frequently the app asks you to review that card.

This is very similar to Buffy's answer, but I've written it as a separate answer because Buffy specifically promotes the use of physical paper flashcards over digital ones. The upside of using an app is the spaced repetition: it keeps track of how well you have learned each card so far, and then gets you to review those cards at (in theory) the optimal times for increasing your memory retention.

• I don't have any scientific proof for that but some people believe that having a physical piece of written paper helps our brain better remember its content. as an "anegdotical proof" I noticed that it works for me as well. Commented Sep 3, 2023 at 15:15

I once took a quantum mechanics course where the professor thought it was really important that we memorize all the relevant formulas. The only way I managed to get through it was to, right before the test, sit and copy the formulas over and over again on a piece of paper for 15-30 minutes. Then as soon as I was given the test, I would write them down on my scratch paper. Good luck.

• This is the technique I used. It is much easier to remember them all in one go at the start of the exam, perhaps before the timer starts, than one at a time during the exam as you need them. Commented Sep 6, 2023 at 7:05

This is an ideal example of a problem that can be solved by using a mind palace: https://www.coursera.org/articles/memory-palace

The answer by @csstudent1418 How to pass a statistics exam that overly relies on memorizing formulas? is basically a "memory palace lite".

I don't know how much time you have. But I would recommend you learn how to use memory palaces, and then apply them to this exam and any similar memorization problems in the future

In stats, we talk about cookbooks (lots of how-to procedures to remember, without background theory) and spookbooks (lots of fundamental theory that tends to frighten rather than enlighten the students).

Actually, I find the cookbooks rather spooky, with all their mysterious "you must never" and "you must always" and I find I can cook just fine working from spookbooks.

Now to the advice: if you study a good maths prob book such as Bain and Engelhardt, you might be able to work out the logic behind the formulas, which makes them easier to memorise.

The standard way is to keep your sheet by your side while you do the exercise. When you need to look at it, don't. Instead make an active recall effort. Try writing down the formula by heart. When you fail, look up the sheet, transcribe the formula and finish the exercise. Do enough exercises and eventually you will know the formula.

Additionally, a key thing is how you do your sheet. Write it by hand, make it neat and organized by topics or type of formula or whatever sounds like a good categorisation to you. Key thing is you need to struggle to organise the sheet, for me that acts as sorting things in what feels logical to me and massively help recall.