Advanced Statistical Analysis of Sampled Data
The Measurement Data Entry Worksheet contains a variety of options for analyzing data samples to estimate measurement uncertainty due to repeatability error.  These include analyzing measured values, measurement data recorded in sampled cells, or measurement data recorded as sampled mean values and standard deviations. Mixed samples, consisting of sample mean values with specified sample sizes together with estimated population standard deviations and degrees of freedom, can also be analyzed.   
 
Data Entry Options
Up to 2,000 sampled values can be entered or imported into the Measurement Results table.  The Measurements Results data entry table changes depending upon which sampling option has been selected.
 
The Sampled Values option is selected when entering individual repeat measurements.  The data can be entered as measured values or deviations from the nominal or specified value of the parameter. 
 
The Sampled Cells option is selected when entering a sampled value that has been observed one or more times.  In this case, the value can be entered as a measured value or a deviation from nominal.  The corresponding number of times that the value was observed is entered under the Number Sampled column.
 
The Sampled Mean Values option is selected when entering mean values obtained from sets of repeat measurements.  The mean values or deviations from the parameter nominal value are entered along with the standard deviation and sample size for each set of repeat measurements.  
 
The Mixed option is selected when statistics from a prior sample analysis will be used to characterize the repeatability uncertainty of a measurement process.  The recently collected measured data are entered on the Measurement Results table and a Mean Value is computed and displayed in the Computed Statistics portion of the Worksheet.  The standard uncertainty from the prior sample analysis is entered into the Distribution Uncertainty Estimate field and the associated sample size is entered into the Applicable Sample Size field.  With the Mixed option, we use these variables to represent the repeatability of the measurement process and to estimated confidence limits for the computed mean value with a specified confidence level.
 
The Data Import Profile Screen, which is accessed by selecting the Import Data option on the Tools menu, can be used to import data directly into the Measurement Results table from the following external application files: Microsoft Access, dBASE III or IV, Microsoft Excel 3.0 through 2000, Lotus 1-2-3 WK1 through WK4, HTML and delimited text files.   
 
Computed Statistics
The computed statistics also vary slightly depending upon which Sampling Option has been chosen.  The computed statistics for all four sampling options include the Sample Size, Sample Mean, Mean Deviation, and Standard Uncertainty.  The user can input a Confidence Level (%) or Coverage Factor to compute the Confidence Limits.  The display precision (e.g., 1 decimal place, 2 decimal places, ...) for the computed statistics can also be set by the user. 
 
The Coefficient of Skewness and Coefficient of Kurtosis are also computed for the Sampled Values and Sampled Cells options.  Skewness is a measure of the asymmetry of the sample distribution.  Kurtosis is a measure of its "peakedness."  The Within Sample Sigma and Between Sample Sigma statistics are also computed for the Sampled Mean Values option.  The Between Sample Sigma is the standard deviation of the mean values and the Within Sample Sigma is an indicator of the variation within samples.  
 
Additional statistical analysis tools include Normality testing and Outlier Removal testing, which are accessed via the Tools menu at the top of the Worksheet.  Normality testing is used to determine if the sampled data can be assumed to be normally distributed.  Outlier identification is done using a combination of statistical testing and the application of Chauvenet’s criterion. 
 
The Use Uncertainty in the Mean option box is checked if the uncertainty is to be reported for the mean value (or overall mean for the sampled mean values option).  If this option box is left unchecked, then the standard uncertainty applies to individual measured values  The standard uncertainty for the mean value is computed somewhat differently depending upon the Sampling Option chosen.

In some cases, only a single  measurement or deviation may be available for entry into the Measurement Results table.  If so, then a standard uncertainty will not be computed.  However, if a standard uncertainty has been obtained elsewhere that characterizes the repeatability uncertainty in a measurement process, it would be beneficial to be able to enter it into the Computed Statistic section of the Worksheet, along with the appropriate sample size.  But, if data are entered into this section of the Worksheet, UncertaintyAnalyzer responds by deleting the measurement sample information in the Measurement Results table.  To prevent this from happening, the Override Sample Size and Uncertainty option box must be checked.

Sample Histogram
A Sample Histogram plot, located in the upper right portion of the Worksheet, displays the relative frequency of the sampled data values.  The histogram Auto-cell feature automatically determines the optimal number of histogram cells to display for sampled data.  Alternatively, the user can enter a preferred number of histogram cells for plotting.

Correction Factors
If a bias (i.e., systematic offset) from the parameter's nominal value has been input via the Parameter Bias Uncertainty Worksheet or Parameter Template, then the user can select whether or not to include the Parameter Bias Correction factor to the computed mean value.

A fixed offset in the subject (or measuring) parameter value or reading can also result from environmental factors (e.g., temperature, humidity, air pressure, ...).  The user can select whether or not to adjust the computed mean value by an Environmental Correction.  The environmental correction is estimated using the Environmental Factors Uncertainty Worksheet, which is accessed via the Environmental Factors button on the Main Screen Error Source List.