Is the average of acceleration magnitude valid?

I am doing two tasks of flipping card and lifting a load. I am collecting data for this two tasks for a week. I want to find the mean acceleration for each day. As it is 3-axis I thought it is good to take magnitude.

I have a 3-axis accelerometer sensor. I have the 100 readings collected. I found the acceleration magnitude for each record by using formula: $$\sqrt{a_x^2 + a_y^2 + a_z^2}$$

I would like to find the total average acceleration of my readings. Should I find the average of the magnitude of acceleration?

Can I find it by dividing the sum of values by 100 or (sum of magnitude values)/time?

Is it valid ?

Any suggestions or help is greatly appreciated.

• It depends on what problem you're trying to solve. Is it possible for you to give a bit more information about what kind of thing you're using the average value for? – Suzu Hirose Oct 24 '16 at 1:20
• Suzu Hirose, Thank you for replying. I am doing two tasks of flipping card and lifting a load. I am collecting data for this two tasks for a week. I want to find the mean acceleration for each day. As it is 3-axis I thought it is good to take magnitude. – Katherine Oct 24 '16 at 1:30
• @kathy - Is the crux of your question whether you should divide by the number of samples or the total time passed? – Myridium Oct 24 '16 at 2:27
• Yes Myridium ! And also is it a correct way of doing ? Is the resultant obtained gives the correct mean acceleration ? – Katherine Oct 24 '16 at 2:32
• As it was already said, the answer depends on what do you want to do with these values. Your description of the experiment is very vague. Repeating what you already said does not add any new information. You did not even say acceleration of what are you measuring. And how and why. – nasu Oct 24 '16 at 2:58

There is an important consideration here. When are the readings from your accelerometer collected? Are the readings correlated in any way to periods of large acceleration or small acceleration? If so, your readings will be skewed. I'm going to suppose that your accelerometer gives readings every $0.1s$ for instance, and that this time interval does not correlate with the flipping of cards or lifting of loads (i.e. you aren't doing something strenuous every $0.3s$ exactly). In other words, that the readings are essentially at random times.

You can divide the sum of the magnitudes by the number of readings. This will give you an estimate of the typical or mean magnitude of acceleration experienced throughout the period of measurement. This quantity is useful if, for instance, acceleration is putting stress on the equipment. This measure of mean magnitude gives you an idea of the average amount of acceleration/force that is being experienced without caring about its direction.

However, if you want to know the magnitude of the mean acceleration, then this is a different question! It's first necessary to calculate the mean acceleration in the form $(a_x,a_y,a_z)$ (again, divide by number of samples) and then take the magnitude. The magnitude of the mean probably isn't as useful, because the mean acceleration is a measure of what constant, directed acceleration $(a_x,a_y,a_z)$ experienced over the entire duration of measurement would result in a final velocity equal to that predicted by the changing measurements.

Whether you use the mean magnitude or magnitude of the mean depends on the information you want to know.

• Thank you for providing valuable information. I will be having 2 sessions of data each day and each session (task) duration is between (11sec t0 30sec). – Katherine Oct 24 '16 at 3:10
• You have given two ways of finding mean. One is mean magnitude of acceleration and other being magnitude of mean acceleration. In my instance I assume to take the magnitude of mean acceleration. When should one consider calculating mean magnitude of acceleration ? Again Thank you for your input. – Katherine Oct 24 '16 at 3:14
• No problem @kathy. Which one you should use depends on what you want to know. If acceleration is putting stress on some instruments for example, and you want to know the average amount of stress they experience, then you should calculate the mean of the magnitude. The magnitude of the mean is less useful I think, but it really depends on what kind of information you want to know about. If you elaborate on what you're using the measurement for, I may be able to ascertain which is more appropriate. Also see my updated answer. – Myridium Oct 24 '16 at 4:27
• Myridium ! I am using this measurement to track the performance of an injured person doing tasks over a week or a month period. Definitely there will be stress the person experiences in the initial stages of recovery. I would like to know the improvement in acceleration in doing tasks over a period of time. I hope you understood my scenery. – Katherine Oct 24 '16 at 15:24
• @kathy - I'm in no position to evaluate how best to track physical rehabilitation, but I think what you are looking for is most definitely the mean of the magnitude. Presumably, you do not care so much about the direction of acceleration-- you just want to know how fast the person is capable of accelerating their limbs and body. – Myridium Oct 25 '16 at 9:36

Average acceleration seems like it would only be useful for finding overall change in velocity. If you take the magnitude, then you can get the total change in speed. If you want much of anything else, you should integrate in time.