The issue is accuracy. If i tell you that this bridge will support 100,000 kilos you might feel safe. If I then state this is an estimate with a standard deviation of 95,000 kilos you might not feel so safe. In any case, I'll let you drive across first.

So what level of accuracy do you need? The related question is what level of treatment differences should be significant? If your existing materials will build a bridge that holds 100,000 kilos and your treatments will build a bridge that holds 100,000.01 kilos should we care enough to do the thousands of replicates it might take to detect the difference?

What is the variability in your data?

Often one worries about the underlying distribution being normal (Gaussian). Depending on how the data are analyzed this might be the data or it might be the residuals. However with so few replicates you cannot reliably assess the distribution of your data in either case.

If the underlying distribution is skewed or asymmetric, then three replicates will likely underestimate the true mean and underestimate the true standard deviation. Of course if the underlying distribution is very skewed and asymmetric then one might question if the mean and standard deviation statistics have any validity to begin with.

Dear Sadeq

for every laboratory testing, there is a degree of significance, as you are testing asphalt concrete specimens, 0.95 significance is sufficient for your purposes, the standard deviation of test results from the mean value should be within -+ 5%.

Dear Mohammed

The sample size for any study depends on the standard deviation of the variable ( from previous studies ) and the margin of error you decided . My advice to use G*Power .

G*Power software is effective tool to calculate sample size for many ranges of experiments. Also, you can determine effect size and power of the test, G*Power is free to download and easy to use after reading the manual, the download link:

http://download.cnet.com/G-Power/3000-2054_4-10647044.html

The link to download G*Power manual:

http://www.gpower.hhu.de/fileadmin/redaktion/Fakultaeten/Mathematisch-Naturwissenschaftliche_Fakultaet/Psychologie/AAP/gpower/GPowerManual.pdf

Good Luck

Hi Mohammed, I have found your questions and the answers very interesting. I think you are talking about lab prepared samples. In that case, I can tell you from my experience, 3 samples should be fine to achieve statistically significant test results (eg, 5% confidence level, as mentioned by Dr. Saad), if you doing E* and flow number test. Because there is relatively less variability for lab samples compared to field samples. In binder testing, I have found even lesser variability among the replicates. But for field cores or probably for fatigue testing, the variability is high and you may think of larger number of replicates. Nevertheless, the higher the number of replicates, more confident you will be about your results and conclusions. 

Go to a random number generator that generates numbers from a Gaussian distribution. Select a mean and a standard deviation of your choice. Get three numbers that you know are from a Normal distribution. That is the result of your experiment. Now do this twice more. For each set of three numbers calculate a mean and standard deviation. The real result of your experiment will be one of these sets of three. You will not be able to tell which one, and you will not get to see the others. Do you care which one you would get? If all outcomes are equally good, then 3 samples is great and costs much less. However, if you actually care about the outcome then I suspect that you will not be happy with any of the outcomes.

If I take 1,000,000 sets of three samples, take the average of all sets, and then take the average of those averages, my expected outcome will be the mean and standard deviation of the underlying distribution. However, you do not get to do that experiment. You get one value out of the 1,000,000.

Three replicates is a terrible choice under most circumstances. If variability is so low that it makes no practical difference to my outcome then 3 replicates are fine. If the difference between treatments is so large that statistical testing is stupid. There are also cases where there are no choices. You have to build and test 100,000 bridges each year. Under these conditions initial tests may have only one replicate. This happens in insecticide companies, where an initial screening test may be done with a single replicate because they must try so many different products and the failure rate is very high. There are consequences to using too few replicates.

Something else to consider. I have run a chemical assay to detect DNA sequences. If you add all the same chemicals at the same proportions you (nearly) always get exactly the same answer. If you run tens or hundreds of thousands of samples there comes a point where fatigue sets in. An air bubble gets into the pipette. the chemicals degrade slightly. A host of other minor problems show up. The solution was to run paired samples. If both of the pair give the same result move on. Even though I have a nearly invariant process, I now have a check for human error and bad luck. If it is important, I can rerun the 1 in 2,000 samples that show a problem. It is a quality control issue.

For conventional testing one can find answer in standards e.g. EN 12697-24 (fatigue),

EN 12697-26 (modulus) or in other (e.g. AASHTO) Specifications.

In general the number of specimens at a given confidence level (for road materials it is usually 95%) depends on the required  repeatability and reproducibility conditions.

The precision of the test here among others depends very much on the test method specimen preparation especially  the temperature and frequency etc.

On my experience at 4-point bending fatigue 6 specimens were needed at a given strain level but sometimes additional specimens had to be prepared to reach the required precision. So for 3 strain levels one must prepare at least 3x6=18 specimens. At the same time I have prepared 18 specimens plus 2-4 specimens for reserve. Than for a given strain level specimens were selected randomly from 20 or 22.

Testing of asphalt mixes is expensive and time consuming. At a given test method for obtaining higher precision one can increase the number of specimens but there is a rational limit after which:

a)  increasing the number of specimens makes the whole test expensive;

b) the precision of the test reaches a "practical" limit. For asphalt mixes this limit is determined by many conditions (test method, preparation method, etc...). After this limit the precision practically cannot be improved.

Dear Mohammed Sadeq, 

to put it in a nutshell, a Three samples should be fine. 

you have to consider many factors that control this point such as the degree of significance and standard deviation. 

Good luck , 

Dear Sadeq,

The sample size will base on confidence limit (% error you will allow in estimate) and normality in data. Any statistics book you can look at for the formula. 

Check ASTM E122, it has a way to estimate the standard deviation, and from there the number of specimens needed to reach a given level of confidence.

For statistical analysis. Three replicates will be enough.

good question

Maybe a better question to start with is to understand variability in your samples.

1) Source. If the materials that you use to make asphalt are mined from the ground, then there will be variability in their composition. That variability will be small if you measure in aggregate (this ton of material is indistinguishable from that ton). However, the differences will be more apparent if you measure at smaller units: this kilogram versus another kilogram.

2) Homogeneity: The asphalt is mixed in a large machine. While the mixing looks vigorous and complete, that is not the case if you measure small samples carefully. This I know from pesticide application. A tank of pesticide may be 500 gallons. When measured microliter by microliter it is far from uniform despite vigorous agitation.

3) People: How uniform is the process? A skiploader bucket full of gravel weighs 400000.0000 grams and every load is exactly 400000.0000 grams? I simply do not believe this.

You should care because product failure does not usually happen in aggregate. Failure starts at the weakest point and propagates.

However, you must balance these issues against various costs in time and funding.

Maybe you could try, just once, to run two experiments each the same and each with a sample size of three. However run them a few months apart. Do you get the same answer? If yes, then maybe a sample size of three is fine in your case.

Agreed with Ebert

Dear Mohammed

My be my answer is too late.

Anyway, you have to keep in you mind that the variation or differences than mean must be at least less than 25%

Need to find a good estimate for the standard deviation . Also a reasonable significance level maybe 5%. Assume a normal distribution and find the sample size for a two tailed distribution or confidence levels. Conduct a pilot study and estimate error or standard deviation is one sixth the max minus min.