atama ga ii 頭がいい

In Japanese, atama ga ii 頭がいい means someone is "smart," "intelligent," has "good brains." It's a phrase containing atama 頭, "head," the body part, and the i-adjective ii いい, "good." So it literally means "head is good."

It's also spelled atama ga ii 頭が良い. A synonymous variant is atama ga yoi 頭がよい.

Manga: Kaguya-sama wa Kokurasetai ~Tensai-Tachi no Ren'ai Zunousen かぐや様は告らせたい～天才たちの恋愛頭脳戦～ (Chapter 9, かぐや様は歩きたい)

Usage

Generally speaking, the phrase atama ga ii is used as a compliment.

It follows the patterns of phrases such as:

sutairu ga ii
スタイルがいい
Style is good.
- Knows to dress well.

In particular, a lot of bishoujo 美少女 characters are atama ga ii, because just being a "beautiful girl" isn't enough for a Mary Sue, she has to be smart, athletic, popular, etc.

Manga: Komi-san wa, Comyushou desu. 古見さんは、コミュ症です。 (Chapter 14, あがり症です)

bijin dashi,
atama mo ii shi,
ninkimono dashi,
美人だし、頭もいいし、人気者だし、
She's beautiful, smart, popular,

Of course, atama ga ii can be used sarcastically too, but it's normally used as a compliment.

Atama ga Warui 頭が悪い

The antonym, calling someone "stupid," would be atama ga warui 頭が悪い, "head is bad," because warui 悪い, means "bad."

A synonymous variant is:

atama ga warii 頭が悪ぃ

The negative form of atama ga ii has similar effect:

atama ga yokunai
頭がよくない
Head isn't good.
Isn't smart.
atama ga warukunai
頭が悪くない
Head isn't bad.
Isn't dumb.

Note that there are various ways of insulting somebody's intelligence in Japanese. See baka 馬鹿 for details.

Grammar

The phrase atama ga ii is the predicative clause of a possessive double subject construction. The small subject, usually marked by the ga が particle, is possessed by the large subject, usually marked as the topic by the wa は particle. For example:

kimi wa {atama ga ii}
君は頭がいい
{Good is true about head} is true about you.
{Head is good} is true about you.
Your {head is good}
You're smart.
kimi wa {atama ga warui}
君は頭が悪い
{Bad is true about head} is true about you.
{Head is bad} is true about you}
Your {head is bad}.
You're dumb.

Atama wa ii 頭はいい

Note that because Japanese hates you, there are less common variants with different particles which would affect the nuance of the phrase.

kimi ga atama ga ii
君が頭がいい
- This is emphasizing "you." Your head is good, not anybody else's.
kimi wa atama wa ii
君は頭はいい
- Here we have a contrastive wa は.
- It means your head is good, unlike the rest, which's bad.
- In the sense you're smart, but you suck at everything else.
- Or maybe it's talking about a guy who is smart, but ugly, so there's a plus, but a minus too.
kimi wa atama wa warui
君は頭は悪い
About you: head is bad.
You're dumb.
- Maybe your STR is high, but your INT stat is low, so you mayn't be the best character is our party to do what we're about to do.

Atama no ii 頭のいい

The phrase can also be used as an adjective, coming before a noun. Technically, the phrase becomes a relative clause given that it includes a case marking particle, the ga が. Furthermore, because it's a relative clause, the ga が can be replaced by the subject marking particle no の.

atama ga ii hito 頭がいい人
atama no ii hito 頭のいい人
atama ga ii ko 頭がいい子
atama no ii ko 頭のいい子
A person [whose] head is good.
A smart person.
atama ga ii hito 頭が悪い人
atama no ii hito 頭の悪い人
atama ga ii ko 頭が悪い子
atama no ii ko 頭の悪い子
A person [whose] head is bad.
A dumb person.

Atama ii 頭いい

Sometimes, the ga が is omitted. This is called a null particle, and is represented in analysis as the Greek letter phi φ.

atama φ ii
頭いい
Smart.
atama φ warui
頭悪い
Dumb.
atama φ ii hito
頭いい人
Smart person.
atama φ warui hito
頭悪い人
Dumb person.

Manga: Kaguya-sama wa Kokurasetai ~Tensai-Tachi no Ren'ai Zunousen かぐや様は告らせたい～天才たちの恋愛頭脳戦～ (Chapter 9, かぐや様は歩きたい)

atama φ ii!!
頭いい！！
[Wow, you're so] smart!!
- atama ga ii - without omission.
oneechan φ tensai!?
お姉ちゃん天才！？
[Are you] a genius, sis!?
- oneechan wa tensai? - without omission.
fufu......
ふふ･･････
*chuckle*
maa ne
まぁね
[Sort of].

Atama no Yosa 頭の良さ

The phrase atama no yosa 頭の良さ refers to how smart someone is. Literally, it's turning atama 頭 into a possessive no-adjective with the no の particle for yosa 良さ, which the sa-form, "-ness" form, of ii いい.

atama no
頭の
Head's.
yosa
良さ
Good-ness.
atama no yosa
頭の良さ
The good-ness of the head.
How good the head is.
How smart [you are].
atama no warusa
頭の悪さ
The bad-ness of the head.
How bad the head is.
How dumb [you are].

Atama ga Yosasou 頭が良さそう

The phrase atama ga yosasou 頭が良さそう refers to how smart someone seems. It uses the sou-form of ii, which is irregularly attached to the sa-form: yosasou 良さそう. For warui, it's a regular, attached to the stem: warusou 悪そう.