Title: The normal curve and the probability of getting accurate estimates from a sample
Course: UGQuants (SSPS) > Section: Probability

The normal curve and the probability of getting accurate estimates from a sample

When data are the result of many different random effects (such as successive tosses of a coin, throws of dice or the values of a large number of different variables), the distribution of values that results from these effects takes the shape of a normal curve:

If you find it helpful to think of an empirical example, think of the X-axis as the proportion of heads obtained in the results of countless very long series of coin tosses, and the Y-axis as the number of series with such a result. The curve, therefore, represents the distribution of the values of X.

The normal curve has the following properties

Bell shaped | The mean, mode and median of X all coincide at the 'peak' or highest point of the curve: i.e. at the highest value of Y. Most values cluster around the mean, and as we travel further from it, in either direction, the number of observations (or cases) decreases.
Symmetrical | The curve to the left side of the mean is a mirror image of the right side, and there are an equal number of observations above and below the mean.
Continuous | The variable can take any value above or below the mean, and there are an infinite number of values between any two points on the X-axis (which is why you need to imagine an infinitely long series of coin tosses.)
Asymptotic | Which means that the line never actually touches the X-axis: there is always the possibility that a yet more 'extreme' value can occur beyond the 'tails' of the curve (e.g. one further head in a yet longer series of tosses).
The mean and standard deviation | Any curve can be defined perfectly by knowing only the mean and standard deviation of the observations. A higher mean relative to the standard deviation will produce a taller higher and ‘thinner’ curve (the values will cluster more tightly around the mean) while a higher standard deviation will produce a flatter, more spread out curve.
'Theoretical' | In the sense that a perfect normal curve is a logical idea rather than an actually occurring empirical reality. It is a kind of ruler against which we can measure the empirical results that we obtain.

Normal curves with very different shapes are thus possible, depending on the value for the mean and the standard deviation. Because the normal curve has so many useful applications, statisticians use a standard normal curve with a mean of zero and a standard deviation of 1, and then standardise other normal distributions to this one in order to make calculations. This might appear a cumbersome procedure, but since the formula for standardisation is so simple, in practice it is very useful. Standard normal statistical tables, such as those reproduced in the appendix to Fielding and Gilbert (or available freely on many websites), report the area under the curve corresponding to different distances from the mean on the X axis. We can use these tables to calculate exactly what the probabilities are of any one case in a set of normally distributed cases taking a value that is within a given distance from the mean of all cases. This has many useful applications as we shall see.