Formula for S shaped curve given specific values

[Sharidb]

I need to derive a formula that will give me an S-shaped curve, such as this one:

The x-values would go from 0 to 250 and the y-values from 0 to 50.

After searching I found this to be a sigmoid function or logistic function, but to be honest, my level of knowledge isn't high enough to understand the example explanations and derive my own formula.

Could someone point me in the right direction?

 

[hlsmith]

What is your dataset and your question. Why do you need a sigmoidal shape and no a traditional linear line?

This is easily achievable in logistic regression, but do you need to run a regression model. Logistic usually plots the probability by covariate value, in predicting a binary variable. Though you appear to have to bounded continuous variables?

You should provide much more information so we can help you and ensure this is the right approach for you!

 

[Sharidb]

Thanks for your response.
I am running an experiment where people are asked to forecast sales numbers, given a certain promotional expenditure by the company.
I have tested this first with a linear relationship between the promo expenditure and the rise in sales: y = 1/5x

in a following version I will test if people are able to do this when the relationship between the expenditure and rise in sales follows an S-shape, rather than a linear shape. Similar to the data with the linear equation, promotional expenditures will vary from zero to 250 units. The sales uplift for 250 units should correspond to 50 uplift, similar to the linear equation.

What I need is a formula so that I can calculate the uplift values for 10, 20, 30, .. until 250. When I plot these values it should return the S shape.

I hope this is more clear? is such a formula easy to derive?

 

[hlsmith]

Seems achievable. Some type of calculus derivative or based formula. Which s not an area I am familiar with well enough to help.