Often, the real use of statistics is in telling us whether groups are more different than we would expect them to be by chance. Let's imagine that you design a new garden fertilizer that is supposed to improve plant growth. You select a sample of plants. After measuring their current size, you give half of them the fertilizer while the others are left to grow naturally. You think your fertilizer works, so you expect that plants given the fertilizer will have grown more than plants that were left to grow naturally when you measure their size a few weeks later.
Now imagine that you look at the growth of the two groups of plants and you find that the fertilized plants grew an average of 7 cm while the natural plants grew an average of 5 cm. You may think that this proves that your hypothesis is true: the fertilizer works to increase plant growth. But you cannot conclude this without knowing how much plants vary in growth rate to begin with. Maybe if you just measured ...