mathematics

Running total

Running total is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Running total rather than just read about it. In short: A running total or rolling total is the summation of a sequence of numbers which is updated each time a new number is added to the sequence, by adding the value of the new number to the previous running total. Another term for it is partial sum.

Key takeaways

Running total belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Running total to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Running total from memory before moving on to harder problems.

Reference excerpt

A running total or rolling total is the summation of a sequence of numbers which is updated each time a new number is added to the sequence, by adding the value of the new number to the previous running total. Another term for it is partial sum. The purposes of a running total are twofold. First, it allows the total to be stated at any point in time without having to sum the entire sequence each time. Second, it can save having to record the sequence itself, if the particular numbers are not individually important.

Method Consider the sequence (5, 8, 3, 2). What is the total of this sequence? Answer: 5 + 8 + 3 + 2 = 18. This is arrived at by simple summation of the sequence. Now we insert the number 6 at the end of the sequence to get (5, 8, 3, 2, 6). What is the total of that sequence? Answer: 5 + 8 + 3 + 2 + 6 = 24. This is arrived at by simple summation of the sequence. But if we regarded 18 as the running total, we need only add 6 to 18 to get 24. So, 18 was, and 24 now is, the running total. In fact, we would not even need to know the sequence at all, but simply add 6 to 18 to get the new running total; as each new number is added, we get a new running total. The same method will also work with subtraction, but in that case it is not strictly speaking a total (which implies summation) but a running difference; not to be confused with a delta. This is used, for example, when scoring the game of darts. Similarly one can multiply instead of add to get the running product.

Use The concept of a running total is easily understood. It is common in many everyday uses. For example, most cash registers display a running total of the purchases so far rung in. By the end of the transaction this will, of course, be the total of all the goods. Similarly, the machine may keep a running total of all transactions made, so that at any point in time the total can be checked against the amount in the till, even though the machine has no memory of past transactions. Typically many games of all kinds use running totals for scoring; the actual values of past events in the sequence are not important, only the current score, that is to say, the running total. Tabulating machines had dedicated registers that could [only] accumulate running totals of numbers in preselected columns and display or print them. It is trivial for computers to keep running totals in variables or registers. The central processing unit of computers for many years had one general-purpose register called the accumulator. Most arithmetic operations stored their results into this one register. This accumulator, essentially, kept a running total; that is, it "accumulated" the results of individual calculations. A betting accumulator is the running product of the outcomes of several bets in sequence.

See also Moving average a.k.a. running average Prefix sum

References

External links cumulative at Wiktionary (a related term)

Worked examples

Example 1 — a first encounter with Running total

Start with the simplest possible case. Write down what Running total claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Running total before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Running total ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Running total

In research: Running total appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Running total in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Running total is common in secondary-school and first-year university syllabi. It links to neighbouring topics Operations on numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Running total outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Running total in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Running total means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Running total out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Running total in simple terms?

Why does Running total matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Running total?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Running total.