I have population data for the last 50 years (total population for a country), that I need to make linear. In its original it has a positive relationship (looking at a scatter plot) but it is not linear. There are three smaller periods with large increases in population, which makes the scatter plot look almost like a jigsaw. I have tried transforming the data with logaritms, square roots etc but I do not succeed in making the scatter plot like more or less linear. Any advice?

Try doing 1/x or maybe x^3

Populations normally grow exponentially, so transforming your data with logarithms should work. What base did you use?

You might also try the natural logarithm.

You should remove these outliers or extreme values.

When you have a dataset with a positive relationship that is not linear, transforming the data using mathematical functions like logarithms and square roots is a good approach. However, if these transformations haven't been successful in achieving linearity, there are a few other techniques you can try:

1. Polynomial Regression: Instead of fitting a straight line, you can fit a polynomial curve to the data. Polynomial regression allows for more flexibility in capturing non-linear relationships. You can try fitting a polynomial of higher degree (e.g., quadratic, cubic) to see if it better captures the jigsaw-like pattern in your data.

2. Piecewise Linear Regression: Since you mentioned that there are three smaller periods with large increases in population, you can try dividing your dataset into these periods and fitting separate linear regression models to each segment. This way, you can model the non-linear relationships within each period separately.

3. Smoothing Techniques: You can use smoothing techniques like moving averages or locally weighted scatterplot smoothing (LOWESS) to smooth out the jigsaw-like pattern and identify trends. These methods create a smoothed line that approximates the underlying pattern in the data, making it easier to visualize any linear trends.

4. Time Series Analysis: If your data represents population changes over time, you can consider applying time series analysis techniques such as autoregressive integrated moving average (ARIMA) models or exponential smoothing. Time series analysis is specifically designed to handle data that exhibits patterns or trends over time.

It may be helpful to experiment with different combinations of these techniques to find the best approach for your specific dataset. Additionally, visualizing the transformed data and evaluating the residual plots can give you insights into the effectiveness of each transformation method.