It sounds like Statistics::LSNoHistory might be what you're looking for. It will calculate a Least-Squares linear regression for a given set of input data and return the slope (m) and intercept (k) for a line of the form y = m*x + k. It will also calculate Pearson's r correlation coefficient, which is simply a measure of how well the line fits your data. If by "error" you mean variance, you can calculate that easily after the equation of the line is known (simply sum the squares of the differences of the y-values for each data point and the value predicted by the equation, then divide by the number of data points use the variance_x and variance_y methods).

Update: here is some code that uses your example data:

while( my ( $distrib_type, $href ) = each %{ $distributions } )
{
    my $regobj = Statistics::LSNoHistory->new(
                     points => [ %{ $href->{distribution} } ] );

    print "\nData for $distrib_type:\n";
    printf("Slope: %.2f\n", $regobj->slope);
    printf("Intercept %.2f\n", $regobj->intercept);
    printf("Correlation Coefficient: %.2f\n", $regobj->pearson_r);
    printf("Variance (y): %.2f\n", $regobj->variance_y);
}
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Output:

Data for fast_increase:
Slope: 0.72
Intercept 1.05
Correlation Coefficient: 0.89
Variance (y): 22.78

Data for slow_increase:
Slope: 0.36
Intercept 0.53
Correlation Coefficient: 0.89
Variance (y): 5.69

Data for random:
Slope: 0.12
Intercept 2.77
Correlation Coefficient: 0.28
Variance (y): 6.11
[download]