按说: by rights, ordinarily speaking
【高28】副词 / adverbs (情态)
Tone colorsā tone 1á tone 2ǎ tone 3à tone 4a neutral
按说 states what should be true "by rights", by plain logic, by the season or by custom, and what follows shows a reality that does not fit: 按说……，可是 / 可 / 但 / 却……. Think "ordinarily… but…" or "by rights… and yet". The sentence voices surprise or complaint at the gap between what was expected and what is found. The first clause often carries 该, 应该 or 会.
It is a spoken sentence adverb (also 按理说 and 照理说). It stands at the head of the clause, with or without a comma, or right after the subject: 这家店按说该赚钱了. The clause that follows nearly always contradicts the expectation; without it the sentence is left hanging.
Do not mix it up with 按照 / 根据 + noun (hsk3-g37, hsk4-g15), "in accordance with", which names a rule to follow: 按说 never takes a noun after it.
Examples
按说这个时候应该下雪了，可今年一场雪还没下。
Ànshuō zhège shíhou yīnggāi xià xuě le, kě jīnnián yì cháng xuě hái méi xià.By now it should have snowed, but this year there has not been a single snowfall.
按说他早就该到了，可是电话一直打不通。
Ànshuō tā zǎo jiù gāi dào le, kěshì diànhuà yìzhí dǎ bu tōng.He should have arrived ages ago, but his phone just will not connect.
按理说，周末的商场应该很热闹，今天却几乎看不到什么人。
Ànlǐ shuō, zhōumò de shāngchǎng yīnggāi hěn rènao, jīntiān què jīhū kàn bu dào shénme rén.A shopping mall ought to be busy at the weekend, yet today there is hardly anyone about.
这家店按说该赚钱了，可账上还是亏着。
Zhè jiā diàn ànshuō gāi zhuànqián le, kě zhàng shang háishi kuī zhe.By rights this shop should be making money by now, but the books still show a loss.
按说现在是蔬菜供应淡季，可是这里的蔬菜种类还是不少。
Ànshuō xiànzài shì shūcài gōngyìng dànjì, kěshì zhèlǐ de shūcài zhǒnglèi háishi bù shǎo.By rights this is the off-season for vegetable supplies, yet there is still plenty of variety here.
按说合同到期前一个月就该开始谈了，可对方到现在还没有任何消息。
Ànshuō hétong dàoqī qián yí ge yuè jiù gāi kāishǐ tán le, kě duìfāng dào xiànzài hái méiyǒu rènhé xiāoxi.Talks should have started a month before the contract runs out, but the other side has still not sent any word.
Common mistakes
✗ 按说他的话，我们应该走了。 → ✓ 按照他的话，我们应该走了 (按说 takes no noun: "according to" is 按照 / 根据).
✗ 按说天气不错，所以我们出去玩了。 → ✓ 天气不错，所以我们出去玩了 (when expectation and facts agree there is no 按说: it prepares a contrast).