Very great comment done by Grumpy_Mike: it brings myself to question again: "what can an accelerometer measure?" The acceleration - but what is it? (and which?)
First statement: acceleration is NOT "the speed" - it is the change of speed (difference).
But when I jump - it changes the speed (it slows me down, stops, and I speed up again when I am falling down to earth again. What would it measure during these phases?
I would "assume": nothing, all the time zero, never mind if I still "fly" upwards, stop or fall backwards to ground.
BTW: I think, an accelerometer cannot measure the earth gravitational acceleration (g): when no change in speed happens, e.g. stationary on ground - zero.
Even when "flying freely" in the space (I jump and feets not on ground anymore) - also nothing (even the speed changes).
I would assume: when you jump, you lift your body and try to jump into the air - as soon as your feet has left the ground (and no force anymore) - it measures zero acceleration.
Never mind if you slow down during jumping/"flying" (due to earth gravity), you stop at top point and you reverse direction (falling down now to earth) - it will always measure zero during all phases, never mind if you change velocity (in which direction you "fall").
I imagine the accelerometer as this: it has a tiny mass inside: it can realize if the shell, the housing, starts moving but the internal mass does not yet (due to inertia). When you jump and you have the ground (no forces anymore) - both are affected by the same: both speed up, slow down, stop at the same time, or fall back with the same "speed".
The sensor will not see anymore "different forces", both, the housing (reference) and the sensor (the tiny internal mass) are moving now with the same speed (even the velocity is changed).
Imagine it as: it is actually an "inertia measurement device": is the reference (housing/shell) moving with other speed as the sensor mass (the tiny "pendulum" inside)? If both are affected by the same "effect" (slowing down, speeding up), in the same way: no indication that there is an acceleration (the difference is zero).
So, jumping up, falling down, when in "free motion" should give you acceleration of zero, even you slow down on the way up or you speed up again on the way down.
Or again: a speed change is not really any indication for an accelerometer: Just if the speed change is different to the "expected speed change" due to the acceleration affecting both "parts" in sensor (intertia) - it will realize if speeding up faster as "expected" or "slowing down" more as "expected". If all changes speed in the same way = a constant acceleration, potentially measured as zero in an "inertia sensor".
As long you are in free fall, even speeding up with the gravity acceleration g - there is no change: the acceleration factor has not changed.
The same for jumping up into "free air": the first acceleration (pushing back from earth surface) is gone, now "free movement". Even it slows you down on the jump: there is not any force, not any difference for object and sensor (both are affected in the same way, no difference): the acceleration (earth gravity) slows you down, you start to fall back to earth, you stop, you turn the direction, you fall down, even with increasing speed - the acceleration remains the same. All measured remains the same (zero).
Potentially, your accelerometer will measure zero as value as soon you have "left the earth ground" until you land again on earth.
But tricky:
You cannot assume, a measured value zero means "free floating in the air", a jump has been done. The same value is measured if you are stationary (zero), sitting on the ground:
So, in math terms: you need a second derivative: you have to track if the acceleration has changed (acceleration is the first derivative of speed - a speed change, a 2nd derivative the change of acceleration - needed here).
Now you track if also the acceleration has changed (the second derivative): if so, you know there was a force to speed up (or down). If this becomes zero: no force and now "freely moving" or "stationary" . During "free movement" - no indications (all zero), even sitting on the ground.
You would know it was a jump, if you have measured a "positive" acceleration ( a change: the 2nd derivative). If it has stopped (zero) - the body/object is "freely moving" or back on rest. No statement about the speed, e.g. if stopping or reversing direction.
You can just "realize" the end of the free fall, person has landed again on the ground - if you measure now a "negative" acceleration: when "flying stops", the sensor does not move anymore, but the internal mass keeps going to fly still in same direction. This abrupt stop creates a difference and you see now a "negative" acceleration.
So, in order to know if a person has jumps: the difference at start on acceleration value, a long time zero (freely moving) and a negative acceleration when it lands.
So, when the person starts its jump, as long as it has not left the earth ground - there should be a positive acceleration. If strong enough to leave the ground - you have to do the math (maybe he was just stretching his body).
When zero again - no idea if he is in "free fall" or has stopped the body stretching, still sitting on ground.
So, my conclusion: even you can measure the acceleration (from sensor any value) - without the knowledge about the mass of the moving body/object - you do not have a clue if it was a jump or not, strong enough to "leave the ground".
BTW: the change of acceleration gives you small hint: if slow - the mass might be large. But you do now know how strong the force to cause acceleration. It results in: if you do not know the mass of the accelerated object - you cannot make any statement about "what happens".
You have to consider mass (weight, on earth) and the change of acceleration (2nd derivative of speed), in order to get a clue "how strong the force applied" was and if body could have really left the earth ground for a "free flight".
Just measuring any values on sensor does not seem to make any sense to me: you have to do the math in order to "predict" what the body is doing now ("flying freely through the air" or just stretching/moving his body and still not strong enough to "fly").
Good luck.