One can't emphasize enough how applications give rise to formulas and not the other way around.

Adolph Quetelet (1796-1874) was a practical man lending himself to taking actual data.

With large data collections the "Laws of Probability" should be further realized.

Quetelet was in charge of some measurements of chest-sizes of 5738 Scottish soldiers.

When does Normal Distribution apply?

There certainly is no one answer to this question. Many measurements are normally distributed and the standard deviation is considered as a help in error analysis.

Normal distribution seems to apply to most of the following measurements:

1) The heights of males in North America
2) The IQ-scores of a random individuals from a uniform population
3) The lengths of pregnancies
4) The body temperatures of humans ( or other animals)
5) The weights of newborns
6) The measures of cholesterol levels
7) The weights of manufactured products

There are certainly many more.

Test-scores at Cambridge University

1) The mean is not at the center of the test scores.
2) The distribution of the test scores are not symmetric on both sides of the mean.
3) A histogram would show that the vertical bars do not have a "bell-shape" curve.

 

US Army recruits - 1895

The data set, as presented above, is using a discrete random variable. The "Bell-curve" or normal curve uses a continuous random variable.

Even though it seems that the theory is difficult to work with at first it pays large dividends to engineers and scientists every day.

To show you how good this large collection of measurements is we look at this example:

Example Question

From the grouped data we can find the mean chest size of 39.9 inches and a standard deviation of 2.1 inches.

Using these parameters (and the Standard Normal tables)

1. Find the probability of finding a random Scottish soldier whose chest size is between 34.5 in. and 38.5 in.
2. Find probability of a random measurement being greater than 39.5 in.

 

Part A)

We find z-values for 34.5 and 38.5.

They are z = - 2.52 and z = -0.62. (Both left of the mean.)

Standard normal tables gives us a probability of

0.4910 - 0.2324 = 0.2586 = 25.9%

Part B)

We find z-values for X = 39.5. z-value is -0.14 (left of the mean).

Standard normal tables gives us a probability of
0.5000 + 0.0557 = 0.5557 = 55.6% (Theory)

A 55.6% probability the chest measurement is above 39.5 inches.
The actual numbers are that there are 3209 out of 5738 soldiers or 55.9% (Actual) measurements above 39.5 inches!!