
der Durchschnitt
average
The arithmetic mean, calculated by dividing the sum of several values by their number.
Der Durchschnitt der Prüfungsergebnisse lag bei einer Note von 2,7.
The average of the exam results was a grade of 2.7.
average / cross-section
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average
The arithmetic mean, calculated by dividing the sum of several values by their number.
Der Durchschnitt der Prüfungsergebnisse lag bei einer Note von 2,7.
The average of the exam results was a grade of 2.7.

average
The normal, usual, or middle level; the standard with which something is compared.
Seine sportlichen Leistungen liegen deutlich über dem Durchschnitt seiner Altersklasse.
His athletic performance is clearly above the average for his age group.

on average
Used as a phrase to express what is typically the case or true on average.
Im Durchschnitt trinkt jeder Deutsche über 100 Liter Kaffee pro Jahr.
On average, every German drinks over 100 liters of coffee per year.

cross-section
A cut surface that is created when you cut through a body or object.
Der Durchschnitt des Baumstammes zeigte deutlich die Jahresringe.
The cross-section of the tree trunk clearly showed the annual rings.
'Durchschnitt' is made up of 'durch' (through) and 'Schnitt' (cut). The literal meaning is a 'cut through' something, as with a tree trunk. This idea is transferred to describe the average value (a 'cut through' the data) or the norm.
Auf dem Durchschnitt des Apfels konnte man die Kerne sehen.
On the cross-section of the apple, you could see the seeds.
The phrase 'im Durchschnitt' (on average) functions like an adverb and often appears at the beginning of a sentence or in the middle of a sentence (position 3 or after the verb). It introduces a generalizing statement about an average value.
Im Durchschnitt schlafen die Menschen in Deutschland sieben Stunden pro Nacht.
On average, people in Germany sleep seven hours per night.
To compare something with the norm, 'Durchschnitt' (average) is often used with the prepositions 'über' (above) or 'unter' (below) + dative. Typical formulations are 'über dem Durchschnitt liegen' (to be above average) or 'unter dem Durchschnitt sein' (to be below average).
Ihre Intelligenz liegt weit über dem Durchschnitt.
Her intelligence is far above average.
When referring to the mathematical mean, the typical verbs are 'berechnen' (to calculate), 'ermitteln' (to determine), or 'bilden' (to form). The result then 'liegt bei' (is at) a certain value.
Können Sie bitte den Durchschnitt dieser drei Zahlen berechnen?
Could you please calculate the average of these three numbers?
remedy / means
When talking about numbers, “Mittel” and “Durchschnitt” can refer to the same calculated value. “Mittel” sounds more mathematical, while “Durchschnitt” is also common in everyday language. In a mathematical calculation, “Mittel” is especially suitable; otherwise, “Durchschnitt” is often more natural.

Der Durchschnitt der fünf Messwerte beträgt 18 Grad.
The average of the five measurements is 18 degrees.
Das arithmetische Mittel der fünf Messwerte beträgt 18 Grad.
The arithmetic mean of the five measurements is 18 degrees.