# How Ratio Bins Are Calculated From Age And Difficulty

### From Panamath

## Contents |

## How Ratio Bins Are Calculated From Age And Difficulty

If the user has not entered customized settings under Advanced Options on the main panel then the program will generate appropriate settings for numerosity ratio and display time. It does this by taking the users age and selected difficulty, adjusting the age based on the difficulty, then finding appropriate values for numerosity ratio and display time using this adjusted age. The following sections show exactly how adjusted age, numerosity ratio, and display time are calculated.

## Adjusted Age

- Very Easy Difficulty
- If user's age is between 0 and 20, then the formula .5 * age is used. If the user is between 20 and 40, then 3/20 * age + 7 is used. Otherwise, an adjusted age of 13 is used. In all subsequent algorithms, this adjusted age is used instead of the real age. This casuses the numerosity ratio and display time to scale with age and difficulty.

- Easy Difficulty
- If user's age is between 0 and 20, then the formula 3/5 * age is used. If the user is between 20 and 40, then 3/20 * age + 9 is used. Otherwise, an adjusted age of 15 is used.

- Medium Difficulty
- On medium difficulty, the user's real age is used in all calculations. This means that a 20 year old will have an adjusted age of 20, a 40 year old will have an adjusted age of 40, etc.

- Hard Difficulty
- On hard difficulty, the adjusted age is given by the formula 4.116713430316e-01 + 1.111762903248e+00 * actualAge + 9.684975424473e-03 * actualAge^2. Note that at this difficulty, this formula is always used, unlike for very easy and easy difficulty, in which after 40 the same value is used.

- Very Hard Difficulty
- On very had difficulty, the adjusted age is given by the formula 4.755221349509e-01 + 1.477883254935e+00 * actualAge+ 1.382024390740e-02 * actualAge^2. This formula is used for all input ages.

## Numerosity Ratio

Once the adjusted age has been calculated by the above formulas, it is run through a function to determine appropriate numerosity ratios based on the adjusted age and the difficulty. When calculating the numerosity ratios, the weber fraction is used. The [Fraction] is the amount of change in a stimulus relative to the background "noise" in the stimulus. The important thing about the weber fraction is that it is roughly constant for a certain stimulus. In vision lab experiments, the weber fraction for panamath experiments is approximately weberFraction = adjustedAge ^ -.55. This value is used to calculate scaled values for numerosity ratio according to the following rules:

- If the adjusted age is less than 40 or is on very hard or hard difficulty, then the following four values are used as the numerosity ratios.
- (.75*w) + 1
- (1.25*w) + 1
- (2*w) + 1
- 2*(w + 1)

- If the adjusted age is greater than 40 and the user is on very easy, easy, or medium difficulty, then the following values are used for numerosity ratio, based on difficulty.
- Very Easy Difficulty
- 1.5
- 1.8
- 2.8
- 4.0

- Easy Difficulty
- 1.4
- 1.5
- 2.5
- 3.5

- Medium Difficulty
- 1.2
- 1.3
- 1.8
- 3.0

- Very Easy Difficulty

## Display Time

Display time is the simplest to calculate. It is based only on adjusted age, and is calculated according to the following rules:

- If the user's adjusted age is between 0 and 20, it is calculated according to the formula 3.23713 * e^(-0.0892973 * adjustedAge) + 0.0573327.
- If adjusted age is between 20 and 40, a display time of .6 seconds is used
- If adjusted age is between 40 and 70, display time is calculated according to the formula 0.0467 * adjustedAge - 1.2667
- If adjusted age is greater than 70, then display time is set to 2 seconds