Number sequence test

What a number sequence test measures

A number series looks like a math question and is really a pattern question. The arithmetic is small: adding, taking away, doubling. What takes thought is noticing which operation is hiding in the list. That is reasoning with new material, the kind of ability Raymond Cattell called fluid intelligence (Wikipedia: Fluid and crystallized intelligence), rather than remembering a method. Our page on numerical reasoning explains the wider skill. This page is about the test above and how to get faster at it.

In the Cattell-Horn-Carroll (CHC) model of intelligence, number series and number matrix tests are classed as quantitative reasoning (RQ), a narrow ability under fluid reasoning (Gf); in one analysis they loaded .39 and .38 on a quantitative factor (McGrew, 2023). CHC keeps that separate from quantitative knowledge (Gq), which is its own broad ability (McGrew, 2023) and covers understanding basic concepts such as addition, subtraction, multiplication and division (Primi et al., 2013). Numerical reasoning subtests are expected to draw on all three: fluid reasoning, quantitative reasoning and quantitative knowledge (Primi et al., 2013).

Number series tasks have historically been among the best indicators of fluid reasoning (McGrew et al., 2023). There is a caveat. In a 2023 network analysis of Woodcock-Johnson IV data, the Number Series test sat closer to the math achievement tests than to the other fluid reasoning tests, so it may partly reflect math knowledge (McGrew et al., 2023). That matches what people feel when they take one: comfort with numbers helps, but the rule-finding is the real work.

Number series are also a classic test of inductive reasoning. Induction is typically measured with series completion tasks found in many standard batteries (Loe et al., 2018), and analogies and series are the most typical formats for testing it (Primi et al., 2013). The way people solve them has been studied for decades. Simon and Kotovsky first set out the rule schemes used to analyze letter series problems in 1963 (Simon & Kotovsky, 1963), and Holzman, Pellegrino and Glaser described number series solving in four steps: detecting relations, discovering the period, describing the pattern and extrapolating (Holzman, Pellegrino & Glaser, 1983). Those four steps are a good checklist for every question above.

What makes one series harder than another? How hard it is to spot the relationship depends on the arithmetic operations involved (Loe et al., 2018). A 2018 study built an automatic number series generator on five rule types (succession, parallel sequences, clusters, non-progressive coefficient patterns and complex progressive patterns); together the rule types accounted for 69% of the variance in item difficulty, performance reflected a single underlying factor, and scores correlated .60 and .66 with a short public-domain ability test (Loe et al., 2018). In short, a well-built series test measures one thing, and its difficulty is mostly predictable from the rules.

How number series are measured

Clinical batteries treat numbers as their own area. The Stanford-Binet 5 measures Quantitative as one of its five factors (PRO-ED: SB5). The WAIS-IV puts its Arithmetic subtest in the Working Memory index, next to Digit Span (Pearson Introducing the WAIS-IV), which tells you something useful: holding numbers in mind is part of the task. The Woodcock-Johnson IV included the Number Series test discussed above. For how these batteries compare overall, see types of IQ tests.

School ability tests use number series directly. CogAT Form 8 measures reasoning in verbal, quantitative and nonverbal domains (Riverside: CogAT Form 8), and at Levels 10 to 17/18 its quantitative tests are Number Analogies, Number Puzzles and Number Series (Riverside CogAT test descriptions). The OLSAT 8 Quantitative Reasoning cluster uses Number Series, Numeric Inference and Number Matrix items (Pearson OLSAT 8 brochure). Even a nonverbal battery can include them: the UNIT2 has a Numerical Series subtest among its six (PRO-ED: UNIT2).

Employment tests lean on numbers too, under tight time limits. The CCAT has 50 questions in 15 minutes covering verbal, math and logic, and spatial reasoning, and calculators are not allowed (Criteria: CCAT prep). The Wonderlic timed cognitive section has 50 questions in 12 minutes (Wonderlic: FAQ), also with no calculator (Wonderlic: sample questions). The PI Cognitive Assessment has 50 questions in 12 minutes covering simple math, basic logic and language, and reports a Numeric subscore (Predictive Index: Cognitive Assessment). In the military, the AFQT score is computed partly from the Arithmetic Reasoning and Mathematics Knowledge subtests of the ASVAB (Official ASVAB: scores).

The ten series above come from the engine behind the Quantitative Reasoning section of the full IQ Octopus test. The first three use a steady gap. The middle four use gaps that grow by a fixed step or two series taking turns. The last three use alternating steps or each number being the sum of the two before. Every question has six numeric options, and the wrong ones are built from plausible slips, such as applying the right step to the wrong term.

Worked examples: easy, medium and hard

Start with the gaps. In the easy example they are not equal, but each one is twice the one before, which is the signature of doubling. Divide each number by the one before it; a constant ratio confirms the rule.

The medium example breaks the gap check on purpose: the gaps jump up and down. When that happens, split the list into odd and even positions. Each half is usually a simple series of its own, and the question is which half the missing number belongs to.

In the hard example, doubling gets close but misses by the same amount every time. That near miss is the clue. Check two steps to be sure the adjustment is constant, then double the last number and apply it once more.

Common mistakes on number series

• Computing before looking. People start adding at once. Ten seconds spent writing the gaps saves more time than it costs, because the gaps usually name the rule family.
• Stopping one level too early. If the gaps are 2, 4, 8, 16, the rule lives in the gaps, not the numbers. Extend the gaps first, then add.
• Missing a second series. When the numbers go up, down, up, down, there are usually two series taking turns. Treating it as one series produces an answer that looks plausible and is wrong.
• Losing your place in an alternation. In series like add 4, then double, the next step depends on where you are. Mark the last step used before you apply the next one.
• Trusting one step. “Double plus one” and “add the two before” can both fit a single pair. Check every pair you can see.
• Picking a distractor that fits the last gap only. Several options are designed to continue the most recent gap. Put your answer at the end of the list and recheck the whole pattern.
• Freezing on the clock. At 20 seconds with no rule, plug the options in. Only one makes the gaps or ratios regular, and a guess scores more than a blank.

How to practice number series

Because the rule families repeat, number series reward practice quickly. A plan using the free library:

1. Read the method. The guide on how to solve number series questions covers every rule family with examples.
2. Build the gap habit. Work through the easy number series writing the gaps every time, even when you can see the answer.
3. Learn the second-level rules. The medium number series mix growing gaps, square numbers and two interleaved series. Name the family before you compute.
4. Take on the hard families. The hard number series use doubling gaps, double-plus-a-constant, alternating steps and the sum of the two before. Do a few slowly, then a few against a 30-second clock.
5. Practice without a calculator. The CCAT and the Wonderlic do not allow one, and the habit of mental arithmetic keeps your attention on the rule.
6. Use it in mixed sets. The Wonderlic practice test, CCAT practice test and cognitive ability test put number series next to other formats, which is how you will meet them on a real test. The full number series library has worked answers for every item.

Set honest expectations. Retaking a cognitive ability test raises scores by about a quarter of a standard deviation on average, more on the same form (Hausknecht et al., 2007). A meta-analysis of 174 samples found that tests with numerical content showed a smaller first retest gain (0.18) than verbal (0.33) or mixed tests (0.41), and that scores stopped rising after the third sitting (Scharfen, Peters & Holling, 2018). So expect your score on fresh series to climb for a few sessions and then level off.

Practice also has limits beyond the format. Klauer's model of inductive reasoning is assessed with item types such as series completion, analogies and matrices (Pásztor et al., 2022), and a meta-analysis of 74 inductive reasoning training experiments with nearly 3,600 children confirmed gains in fluid intelligence and classroom learning (Klauer & Phye, 2008), with an average corrected effect on intelligence tests of d = .52 as reported by a 2022 paper (Pásztor et al., 2022). But reviews of chess, music and working memory training find that effects shrink as study quality rises and that far transfer rarely occurs (Sala & Gobet, 2017). Getting good at number series makes you good at number series; it is not a shortcut to a higher IQ.

Number series on the full IQ Octopus test

The full test has 25 items in five sections: Pattern Recognition (5), Abstract Reasoning (9), Spatial Reasoning (5), Quantitative Reasoning (3) and Logical Reasoning (3). The Quantitative Reasoning section is three number series from the same engine as this test, rising in difficulty. It is the only section with numbers; the other four use shapes. Items are generated fresh for every attempt, so there is nothing to memorize.

Your score above is a count out of ten, not an IQ. Real IQ scores are deviation scores that compare you with people your own age on a scale with a mean of 100 and a standard deviation of 15 (Wikipedia: Intelligence quotient), and ten items in one format cannot place you on that scale. The full test scores all 25 answers with a Rasch model and reports an estimated IQ with a confidence range. When you are ready, take the full IQ test.