A smoother line is a line that is fitted to the data that helps you explore the potential relationships between two variables without fitting a specific model, such as a regression line or a theoretical distribution. Smoother lines are most useful when the curvature of the relationship does not change sharply. Smoother lines added to graphs are calculated using the lowess smoothing method.
The lowess smoothing method is a common technique for determining a smoothing line. Lowess stands for locally-weighted scatterplot smoother. The routine selects a fraction (default f = 0.5) of all points, using the data closest in x-value on either side of the (x,y) point. For each data point, Minitab does a weighted linear regression, giving points closest to each x-value the most weight in the smoothing and limiting the effect of outliers. You can specify parameters to modify both the degree of smoothing and the effect of outliers. You can also specify the weight of the smoothing parameter. The larger the weights, the more the smoothed values follow the data; the smaller the weights, the less jagged the pattern is in the smoothed values.
You can add a lowess smoother line to scatterplots, matrix plots, histograms, and time series plots.
The lowess routine calculates a new, smoothed y-value for each x-value.
The routine selects a fraction (default f = 0.5) of all points, using the data closest in x-value on either side of the (x,y) point. The selection often results in more points selected from one side of the x-value than the other. The following example shows the fraction of data selected for a given point. The shaded area holds the 0.5 fraction closest to the solid red data point.
Minitab calculates weights using the x-distance between each point in the selected fraction and the point to be smoothed:
The following graph shows the relationship between the weights (vertical axis) and the x-values (horizontal axis) for the fraction of selected points. Points closest to each x-value have the greatest weight in the smoothing.
Minitab performs a weighted linear regression on all points in the selected fraction of the data, using the weights from step 2 to produce an initial smoothed value.
Finally, Minitab limits the influence of outliers on the results by using further iterations (default n = 2) of step 3 (called "robust steps"), with new weights calculated as follows: