Equations

Calibration

Weighted Regression

When data with more than one repetition is used to create a calibration curve, weighted regression can be selected. The count error for data collected in this manner is represented as the standard deviation of the counts. Usually the count error for higher concentrations is larger than that for lower concentrations. When using 1/SD^2 as the weight for the calibration curve table, lower concentrations are given more weight because it is more desirable that the curve pass through points having lower error than points having higher error.

The weight of each point is calculated as follows.

 

where

S_i : standard deviation of each point

n : number of point

When using 1/x, 1/y, or 1/x^2 as the weight, s_i^-2 in the above equation is replaced with 1/x, 1/y, or 1/x^2, respectively, to calculate the weight.

When using 1/x or 1/x^2 as the weight, the weight cannot be at a level where x may have a value of 0.

When using 1/y as the weight, the weight cannot be at a level where y may have a value of 0.

Only linear regressions can be weighted.

Internal Standard

When an internal standard is selected, the count of each data point in the calibration curve is divided by the ratio of the count per concentration of the internal standard of the same level.

When yy_s /(y_i/x_i)(x_i=0, y=y_s)

 

where

x_i : concentration of internal standard

y_i : count of internal standard

y_s : count of sample data

This value is used as the measured value of y in the following sections.

Correlation Coefficient

This value is calculated using the following formula.

 

 

where

: average of x_i: average of y_ix_i : measured value of x

y_i : measured value of y

This is available only for linear regressions.

Coefficients of Calibration Curves

Coefficients are calculated as follows.

where

n : number of points

x_i : numbered value of x

y_i : measured value of y

w_i: weight of each point (w_i=1.0 in the case of unweighted regressions)

1. y = ax

 

2. y = ax + b

 

 

3. y = ax² + bx

 

 

4. y = ax² + bx + c

 

 

 

where

 

 

 

 

 

5. log (y) = a (log x) + b

 

 

6. y = ax + b + bkg (Standard Addition)

 

 

where

y_bkg: count of the background

7. y = ax + [blank]

    

   where

   Blk: number of counts in the calibration blank

% Relative Error (%RE)

% Relative Error (%RE) is calculated as,

 

Where,  is true value for the calibration standard, and  is measured concentration of the calibration standard.

If is 0 (i.e. blank), this value cannot be calculated. In that case, the value will be empty.

% Relative Error (%RE) is displayed in the Calibration Curve table for Data Analysis software.

Relative Standard Error (%RSE)

Relative standard error% represents a metric for calibration curve fitting. The value is calculated as,

 

Where  is the true value for the calibration level ,  is the measured concentration of the calibration level ,  is the number of terms in the fitting equation (Average of Response Factor=1, Linear=2, Quadratic=3), and  is the number of available calibration points. If  is 0 (i.e. blank), the level is not used. This means the level is skipped from the above equation. For example, even if there are 4 levels for calibration curve and if one level is used for blank (the expected concentration is 0), the other 3 levels will be used for the calculation.

% Relative Standard Error (%RSE) is displayed in the Calibration Curve table and Batch table for Data Analysis software.