Uncertainty estimation of nanoparticle size distribution from a finite number of data obtained by microscopic analysis

Various techniques are used to measure the mean particle size (PS) and the particle size distribution (PSD) and relevant uncertainties of the measured mean PS and PSD can be obtained by using statistical treatments. Among those techniques, microscopic analysis methods provide the possibility to observe individual particles, allowing detailed information about the PS. However, there is no existing method to estimate the uncertainty of the PSD from a finite number of data obtained by microscopic measurement. We found that it is possible to adopt the method of deducing the standard deviation of the sample standard deviation to estimate the PSD uncertainty from the result of a microscopic analysis when the sample data follow a normal or log-normal distribution. The recommended minimum number of data for the application of developed statistical methods was deduced by simulation with population data sets of pseudorandom numbers. It was found that the PS of gold nanoparticles (NPs) synthesized by the conventional reduction method in aqueous solution followed a log-normal distribution, indicating that this method is applicable to estimate the PSD uncertainty of synthetic Au NPs from the measured PS data obtained by microscopic methods.