# How ion Calculates Rate of Improvement

Updated

## Models

There are generally two different methods for calculating Rate of Improvement from an academic sense.

• The simple model. `y=(x2 - x1) / w`
• Where y = ROI, x2  = last data point, x1 = first data point, w = number of weeks between x2 and x1
• The problem with this method is it ignores fluctuations within the period between x2 and x1
• In the example below, the ROI calculated using this method would be 1.52. However this does not represent the actual growth of the student - as they didn't see a surge in test scores until the last three data points.
• The Linear Regression model. `y = mx + b`
• y = mx + b is not the formula of a linear regression - but we use the linear regression to get the components of the slope of a line formula - which gives us our ROI.
• Where y = ROI, m = slope, x = Date, b = intercept
• Using a linear regression for calculating ROI is much more reliable as it takes into account the fluctuations in scores - and levels outliers.
• In the example above, the intervention ROI is 1.36 - which is much more telling of the progress the student made.