Once you have successfully calculated a curve fit, you can view a statistics report in the Curve Fit menu’s newly enabled **Report** tab which allows you to evaluate a variety of statistics to determine how appropriate the “best” fit is.

- To print the current set of statistics, click on the
**Print**button. - To copy the statistics to the Clipboard, click on the
**Copy to Clipboard**button.

### Goodness of Fit Statistics

**r ^{2} Coef Det **

The coefficient of determination describes the variability in the response (y) due to variation in the explanatory variable (x). The r^{2} value is used to pick the best curve fit from a group of fitted functions. The closer to 1, the better the fit.

**DF Adj r ^{2}**

Because the value of r^{2} can increase simply by increasing the number of independent variables, a modified measure which adjusts for the number of terms in an equation if provided. The closer to 1, the better the fit.

**Fit Std Err**

This criterion of the goodness of fit indicates a better fit the closer it is to zero.

**F-Statistic**

Useful in evaluating the fit of a parametric equation. The higher the F-Statistic, the more significant the fit. The F-Statistic is affected by the number of coefficients; it increases if a coefficient has significance in the regression relation and decreases if it has no statistical contribution.

### Coefficient Statistics

**Std Error**

The standard error is the measure of uncertainty for each coefficient. The smaller the values, the greater the certainty of fit.

**t-Value**

A measure of which coefficients were more influential in the curve fit. The larger the t-Value of a coefficient, the greater the contribution to the fit and higher certainty with which the coefficient was determined.

**Confidence Limits**

The coefficients will fall within the interval defined by the Confidence Limits with 90, 95, or 99% certainty. The default *Level* is 95%.

**Residuals**

2D fits calculates the difference between the observed Y value and the value predicted by the fitted function. For 3D fits, this is the difference between the observed Z and the predicted Z.

**Confidence Interval**

The Confidence Interval refers to a range of possible values from repeated experiments for which there is a given probability that a response Y will occur at a given X value. You can choose the probability Level (90, 95, 99%) in the **Save** tab of the Curve Fit menu. The Y Predict value and the Confidence Interval for each point in your data set will be shown in the **Report** tab of the Curve Fit menu.

**Prediction Interval**

The Prediction Interval is similar to the Confidence Interval, except that it applies to an individual curve rather than an average curve. This means that the repeated results from a single experiment have a 95% chance of falling within the 95% Prediction Interval. Since the variability of a single curve fit is greater than that of an average curve fit, the Prediction Interval limits always fall outside the Confidence Interval limits.

### Goodness of Fit Statistics

**r ^{2} Coefficient of Determination**

The square of the sample correlation coefficient, or coefficient of determination, describes the variability in the response y_{i} due to variation in the explanatory variable x_{i}.

where:

Formal definition is the proportion of variation in the y_{i}‘s explained by the linear regression relationship between x and y. Here, SSE/SSTO represents the portion of variability in the y_{i}‘s not explained by the linear regression of y on x. In linear regression, the value of r^{2} near 1 is indicative of a strong linear relationship between the two variables under study.

**r ^{2} Adjusted for Degrees of Freedom**

Because the value of r^{2} can increase simply by increasing the number of independent variables, we introduce a modified measure which adjusts for the number of terms in an equation. The adjusted coefficient of determination adjusts r^{2} by dividing each sum of squares by its associated degrees of freedom. Like its cousin, the coefficient of determination, the closer to 1.0, the more the fitted model explains the variation in y associated with x.

**Fit Std. Err**

Known as the Fit Standard Error or the Root MSE, this criterion of the goodness of fit indicates a least-squares fit the closer it is to zero.

The Fit Standard Error is effective only when a single data set is studied.

**F-Statistic**

The F-Statistic is useful in determining the fit of a parametric equation to the data. The higher the F-Statistic, the more efficiently the equation represents the data.

F-Statistic = MSR/MSE

This criterion of fit is affected by the number of coefficients; increasing if an added coefficient has significance in the regression relation; decreasing if it has no statistical contribution.

### Coefficient Statistics

**Std Error**

The standard error conveys a measure of uncertainty to the fitted coefficients. Numerically calculated as the standard deviation:

where C_{ii} is the ith diagonal of (X’X)-1, and X is the design matrix. The smaller the value, the greater the certainty of fit.

**t-Value**

The larger the value the greater the contribution to the fit and higher certainty with which the coefficient was determined.

**Confidence Limits**

Variable at the 90%, 95%, and 99% level of confidence (default at 95%) the interval is calculated as:

where t is the t-distribution value for the specified confidence level and degree of freedom.

**Confidence Interval (at the 90%, 95%, 99% levels)**

The confidence interval refers to the range of plausible values for which there is a 0.95 probability a response y_{i} will occur at a given x_{i} value for an average curve fit.

**Prediction Interval (at the 90%, 95%, 99% levels)**

The prediction interval applies to an individual curve fit. The variability of an individual curve fit will always be larger than the variability of an average curve fit. The confidence limits will always lie inside the prediction limits.