In the context of this exercise, what can be determined on the relationship between the mean, standard deviation, and the shape of a normal distribution?
Also, what does it tell us about the percentage of values that are within one standard deviation of the mean?
We can infer that these values affect the location and shape of the normal distribution in the graph.
The mean determines where the distribution will be centered around, so changing this value effectively ‘shifts’ the distribution on the graph. The shape will not be changed if we change this value on the example graph.
The standard deviation has a stronger effect on the shape. Smaller standard deviations will cause the distribution shape to be thinner, as the data is not as spread out. Larger standard deviations will cause the distribution shape to be flatter and wider, since the data is more spread out.
If we observe the graph on the right hand side of the exercise, changing either the mean or standard deviation does not have much of an effect on the percentage of values within one standard deviation of the mean. For a normal distribution, around 68% of the values will always fall within +/- 1 standard deviation of the mean.
that nearly all values are taken to lie within three standard deviations of the mean, and thus it is empirically useful to treat 99.7% probability as near certainty. The usefulness of this heuristic especially depends on the question under consideration.