Learn Hardcore German: Die Wissenschaftlerin erklärt dem Publikum die Zahlen im Diagramm. - The scientist female explains the numbers in the diagram to the audience.

The subject is die Wissenschaftlerin (a single person, “she”), so we use erklärt:
Die Wissenschaftlerin erklärt … = The (female) scientist explains …

Why is it die Wissenschaftlerin and not der Wissenschaftler?

German often marks the grammatical gender of job titles:

der Wissenschaftler – the (male) scientist
die Wissenschaftlerin – the (female) scientist

The ending -in usually indicates a female person. Because the sentence refers specifically to a female scientist, it uses Wissenschaftlerin.

Grammatically, die Wissenschaftlerin is:

feminine,
singular,
nominative case (it is the subject of the sentence).

What grammatical role do dem Publikum and die Zahlen have?

The verb erklären typically takes:

a dative object = the person to whom something is explained
an accusative object = the thing that is being explained

In this sentence:

dem Publikum – dative object (indirect object): to the audience
die Zahlen – accusative object (direct object): the numbers

So the structure is:
[Subject] Die Wissenschaftlerin
[Verb] erklärt
[Dative object] dem Publikum
[Accusative object] die Zahlen
[Further detail] im Diagramm

Why is it dem Publikum and not den Publikum or das Publikum?

Publikum is:

neuter in German (das Publikum)
singular (even though it refers to many people, grammatically it’s one collective noun)

The definite article for neuter, singular, dative is dem:

Nominative: das Publikum (the audience)
Accusative: das Publikum
Dative: dem Publikum (to/for the audience)
Genitive: des Publikums

Because the audience is the indirect object (to whom she explains something), we need the dative: dem Publikum.

Why is it die Zahlen in the accusative, even though die is also nominative feminine?

Zahlen is plural here: numbers. The noun die Zahl (number) has these forms:

Singular:
- Nominative: die Zahl
- Accusative: die Zahl
Plural:
- Nominative: die Zahlen
- Accusative: die Zahlen

For all plural nouns with the definite article, nominative and accusative are both die. That’s why die Zahlen in the accusative looks the same as the nominative.

You know die Zahlen is accusative here because:

The subject is already die Wissenschaftlerin.
The verb erklären expects a dative + accusative object; we already have the dative (dem Publikum), so die Zahlen must be the accusative object (the thing being explained).

Why does dem Publikum come before die Zahlen? Could we also say erklärt die Zahlen dem Publikum?

Yes, both word orders are grammatically correct:

Die Wissenschaftlerin erklärt dem Publikum die Zahlen.
Die Wissenschaftlerin erklärt die Zahlen dem Publikum.

However, the standard / neutral word order in German is:

Dative object before accusative object
when both are noun phrases (not pronouns).

So dem Publikum (dative) naturally comes before die Zahlen (accusative).
If you use pronouns, the word order rules change a bit, for example:

Sie erklärt es ihnen. – She explains it to them.
(Accusative pronoun es before dative pronoun ihnen)

Why is it im Diagramm and not in dem Diagramm?

Im is simply a contraction of in dem:

in (in) + dem (dative article for das Diagramm)
→ im

So:

in dem Diagramm = in the diagram
im Diagramm = exactly the same meaning, just shorter and more natural in everyday German.

German very often contracts a preposition plus a definite article:

an dem → am
in dem → im
zu dem → zum
zu der → zur

Why is Diagramm singular and not plural Diagramme?

The sentence refers to one specific chart/diagram that presumably everyone can see:

das Diagramm – the diagram (singular)
die Diagramme – the diagrams (plural)

The phrase im Diagramm means in the diagram (one).
If it were several diagrams, you would need a different structure, for example:

in den Diagrammen – in the diagrams (dative plural)
in diesen Diagrammen – in these diagrams

Why are Wissenschaftlerin, Publikum, Zahlen, and Diagramm capitalized?

In German, all nouns are capitalized, regardless of their position in the sentence. That includes:

concrete things: Tisch, Haus
people/jobs: Wissenschaftlerin, Lehrer
abstract nouns: Freude, Zeit, Publikum
even many words that come from verbs or adjectives when used as nouns.

So Wissenschaftlerin, Publikum, Zahlen, Diagramm are capitalized because they are all nouns. This is a fixed spelling rule in German.

Could you also say für das Publikum instead of dem Publikum?

You can say für das Publikum, but it doesn’t mean exactly the same:

dem Publikum erklären (dative):
Focus on explaining something *to the audience—they are the *recipient of the explanation.
für das Publikum erklären (preposition für
- accusative):
  Focus on doing the explanation *for the benefit of the audience—like doing it *on their behalf or for their sake.

Often the meanings overlap in real life, but erklären + Dativ is the standard pattern for “explain something to someone.” So dem Publikum is the more natural and idiomatic choice here.

Can the sentence start with Im Diagramm instead? How does that change the meaning?

Yes, you can change the word order in German while keeping the basic meaning:

Die Wissenschaftlerin erklärt dem Publikum die Zahlen im Diagramm.
Im Diagramm erklärt die Wissenschaftlerin dem Publikum die Zahlen.

German has the verb in second position rule: the finite verb (erklärt) must be the 2nd element in the main clause. The first position can be taken by different elements:

Die Wissenschaftlerin – subject-first (neutral, default)
Im Diagramm – place-first (emphasis on where the numbers are)

The second version puts a bit more emphasis on the location/context (in the diagram), but the core meaning remains the same.

What is the difference between Zahlen and Nummern?

Both can be translated as numbers, but they’re used differently:

die Zahl (pl. die Zahlen)
- general concept of numbers, quantities, figures
- often used in mathematics, statistics, data, graphs
- e.g., die Zahlen im Diagramm, hohe Zahlen, Zahlen analysieren
die Nummer (pl. die Nummern)
- specific identifying number, label, or code
- e.g., phone number, bus number, room number
- Telefonnummer, Hausnummer, Bestellnummer

In a diagram with data, Zahlen (figures) is the natural word.
Nummern would sound like you are talking about ID numbers or labels, not the data values themselves.

Is Publikum treated as singular or plural in German?

Publikum is a collective noun:

It refers to many people, but
Grammatically it is neuter singular: das Publikum

So you say:

Das Publikum ist begeistert. – The audience is enthusiastic.
Pronoun: Es klatscht laut. – It claps loudly.

If you really want to stress the individuals, you’d use something like:

die Zuschauer – the spectators
die Zuhörer – the listeners

These are then plural and take sie (they) as a pronoun.

Could the sentence be in past tense, and how would that look with erklären?

Yes. For narration or reports, German often uses the perfect tense (with haben + past participle) in speech, and the simple past in writing.

Present (original):
- Die Wissenschaftlerin erklärt dem Publikum die Zahlen im Diagramm.
  – The scientist explains the numbers in the diagram to the audience.
Perfect (spoken past):
- Die Wissenschaftlerin hat dem Publikum die Zahlen im Diagramm erklärt.
  – The scientist has explained / explained the numbers in the diagram to the audience.
Simple past (written narrative):
- Die Wissenschaftlerin erklärte dem Publikum die Zahlen im Diagramm.
  – The scientist explained the numbers in the diagram to the audience.

All three forms use the same verb erklären, just in different tenses.