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# Small Sample Statistics

## Small Sample Statistics

(OP)
Greetings,

4.10, 3.98, 4.86, 4.39, 4.57, 4.44

I am being asked "what is the error." Thery want to know what the +/- is on my numbers. I have averaged them (4.39) and said +/- 2SD. I am tracking changes - so I will make these same measurements under a different condition and want to know if they are statistically different. But at the new condition I will only have two values.

How can I answer this? And does it make a difference if I discard the first two results (procedural error).

Thanks.

### RE: Small Sample Statistics

What "error" are you trying to define:  measurement error, material variability, process variability, ??  If you have 2 sets of data and want to determine whether the data sets are from the same population or are statistically different, then you can use a t-test.  However, with this few of data points in each set (4 and 2), there will have to be a very large difference in the data set means for the difference to be significant.  Futher, the estimate of the population standard deviation with only 4 or 6 measurements is is not going to be very good, so to quote an "error" as +/-nSD is not likely to be useful.

### RE: Small Sample Statistics

Calculate an average of all measurements (=4,39). Then calculate (xi-xavg) for each measurement, sum the differences(-0,29;-0,41;0,47;0;0,18;0,05) and divide by the number of masurements(=6) to get 0,233.This is called (according to terminology in my books)"the average absolute error of measurement".Divide it by the average(4,39)and multiply by 100 and this is relative percent error of measurement.
There is another way, namelly  to estimate the inerval  where the average will be found with a certain probability:calculate the average(=4,39) and stdev(=0,3187),
number of degrees of freedom N-1 (=5) and from tables find out the value of students t for double sided alfa=0,01(=3,36)and calculate the term t*stdev/sqrt(N-1)(=3,36*0,3187/2,36)=0,478.
So your average value will be 98% likely  in the range 4,39-0,478...4,39+0,478.
You can find the students t in excel:
choose confidence limits CL[%]
calculate alfa=(100-CL)/100
call TINV(alfa, N-1)
m777182

### RE: Small Sample Statistics

(OP)
Thanks m777182. That looks good.

For SWC.... here's the story. I have a single value at many (~50) conditions. I've been asked to repeat one condition ~10 times to "determine the scatter." Soon I will have 12 values at condition X. The confidence limits and student't t-test is where I was leaning.

It is not practical to test each condition 10 times. But in this case it is reasonable to assume the multiple tests at condition X will be representative of all tests(literature confirms this).

### RE: Small Sample Statistics

The "scatter" you determine at condition X can be assumed to represent the scatter at the other 49 condtions, PROVIDED that the test measurement error is not a function of the conditions and the variability of the process is not a function of the conditions.  Both are assumptions that you have to validate based on your knowledge of the measurement method and the process that you are measuring.

### RE: Small Sample Statistics

I think that your problem calls for an approach that is known as Experimental Design. It will lead you to the best solution with the least expencive(=time consuming) sequence of steps and with the hihgest statistical confidence.
m777182

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