There is a bewildering array of catalogued properties of stars. We know their masses, radii, luminosities, surface temperatures, colors, distances, spectra, ... (see the tables in the Appendices of the test). Given this huge amount of information, the question is how can we make sense of the data? An obvious first step is to graph the data. For example, one could plot the radius versus the surface temperatures of stars and see what one finds. The reason one goes through this exercise is because such plots can indicate whether the two properties are correlated, that is, if there is a physical connection between them. (They may not be independent quantities in physical and mathematical senses).
In the early 1900's, Ejnar Herstzprung and Henry Norris Russell independently made the discovery that the luminosity of a star is related to its surface temperature. (They actually used a quantity known as the Magnitude of the star and the Spectral Class of the star.) This result is amazing. A schematic Hertzsprung-Russell diagram is shown below:
Hertzsprung and Russell both used the spectral class (which is related to the temperature) in their plots. They ordered the stars as O, B, A, F, G, K, and M. Since the O stars are the hottest, this means that in the HR diagram, the temperature axis is odd in that the temperature decreases as one moves to the right.
When luminosity versus temprature plots are mode, stars do not fall randomly on the graph; rather they are confined to specific regions. This tells you that there is some physical relationship between the the luminosity and temperature of a star. To try to make this point more clear, let's look at people. People have many defining characteristics, not all of which are related. So, we can plot properties of people and see what the plots look like. For example,
The plot of weight versus IQ is a scatter diagram. That is, a person who weighs 150 lbs can either have a high or low IQ, while a person who weighs 300 lbs can either have a high or low IQ. The weight of a person is not a good predictor of IQ, that is, there isn't a physical relationship between weight and IQ.
The plot of weight versus height for a sample of people, however, shows a well-defined (albeit noisy) relationship in the sense that the taller you are, the more you are likely to weigh. This correlation suggests something about the structure of a person in that the weight and height are connected.
From this simple exercise we learned something important about people. We did not learn how people are put together, but we did learn that there is a correlation between height and weight. It is up to the theorists to explain why height and weight are related in the manner that they are.
Similarly, it is clear that the HR diagram is telling us about how stars are put together (unfortunately, it is again up to theorists to tell us exactly what the HR diagram is telling us).
What are some other things we can deduce from the HR diagram?
What might this be telling us about the lengths of the various stages in a star's evolution?
Further, even along the Main Sequence, stars are not distributed smoothly. There are many more low luminosity stars than there are high luminosity stars. A plot of this distribution is referred to as the Luminosity Function.
The plot is for the Solar Neighborhood, but it is representative of most stars in the Galaxy.
When we consider Main Sequence stars, is there any hint about whether an individual star evolves along the Main Sequence or whether a star once on the Main Sequence does not change its position? Theoretically, we believe that this is true and when we look at the Mass-Luminosity relationship this seems to be true.