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.

'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
'das Mittel' and 'der Durchschnitt' can mean the same thing in mathematics: the value calculated from several numbers. In this sense, 'Durchschnitt' is usually more natural in everyday language, for example with grades, prices, or temperatures. 'Mittel' sounds more technical or appears in fixed expressions such as 'arithmetisches Mittel' (arithmetic mean). The words can be confused because both refer to the average value. If you are speaking generally about numbers in everyday life, 'Durchschnitt' is usually the simpler choice.

Der Durchschnitt der letzten drei Tests liegt bei 82 Punkten.
The average of the last three tests is 82 points.
Im Mathekurs berechnen wir das Mittel aus fünf Messwerten.
In math class, we calculate the mean from five measured values.