- #1
SamBam77
- 27
- 0
I am measuring the thickness of many identically prepared objects. In order to obtain the most accurate value, I measure each object multiple times, in different locations, and average these values to get the average thickness of that object, along with the standard deviation. But I am really interested in the thickness of the average object prepared with this technique. So I produce many objects in the same way and, using the same instrument, measure each object’s average thickness.
At this point, I have a set of averages:
Object_1: µ_1 ± σ_1
Object_2: µ_2 ± σ_2
…
Object_i: µ_i ± σ_i
Where µ_i and σ_i are the average thickness and standard deviation of the ith object.
How would I go about computing the overall average thickness, and its standard deviation, from these data?
To find the overall average, I know it should be:
µ_overall = (N_1 * µ_1 + N_2 * µ_2 + … + N_i * µ_i) / (N_1 + N_2 + … + N_i)
and if each sample’s average is computed using the same number of measurements, this simplifies to just the average of the averages,
µ_overall = (µ_1 + µ_2 + … + µ_i) / i
But what about the standard deviations? I am not sure.
Does it have to do something with sum of the squares of the individual standard deviations (variances)?
And a related concept question that I am not clear on in my mind at this point:
Let’s say that the measurements performed on each object, individually, have a very narrow standard deviation, but the average that results from each object’s measurement vary greatly between objects.
The resulting overall standard deviation must be large, right?
At this point, I have a set of averages:
Object_1: µ_1 ± σ_1
Object_2: µ_2 ± σ_2
…
Object_i: µ_i ± σ_i
Where µ_i and σ_i are the average thickness and standard deviation of the ith object.
How would I go about computing the overall average thickness, and its standard deviation, from these data?
To find the overall average, I know it should be:
µ_overall = (N_1 * µ_1 + N_2 * µ_2 + … + N_i * µ_i) / (N_1 + N_2 + … + N_i)
and if each sample’s average is computed using the same number of measurements, this simplifies to just the average of the averages,
µ_overall = (µ_1 + µ_2 + … + µ_i) / i
But what about the standard deviations? I am not sure.
Does it have to do something with sum of the squares of the individual standard deviations (variances)?
And a related concept question that I am not clear on in my mind at this point:
Let’s say that the measurements performed on each object, individually, have a very narrow standard deviation, but the average that results from each object’s measurement vary greatly between objects.
The resulting overall standard deviation must be large, right?