Normal Curve

The properties of the bell curve were studied by Carl Friedrich Gauss (1777 - 1855) and his contemporaries, and Pierre-Simon LaPlace (1749 - 1827) proved the first version of the Central Limit Theorem in 1810. The modern function that describes the Gaussian distribution or the Normal Distribution is as follows:

\[ f(x) = \frac{1}{\sigma \sqrt{2\pi} } e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \]

where \(\mu\) is the mean and \(\sigma\) is the standard deviation. In this course, we use the notation \(N(\mu, \sigma)\) to refer to a normal distribution with the indicated mean and standard deviation.

Standard Normal

For the purposes of calcuation, especially when the integrals involved must be done by hand, mathematicians developed a simplification where

• \(\mu = 0\)

• \(\sigma = 1\)

One can readily verify by inspection of the above intgral that, computational, this simplication should make the integration more easily accomplished.

Standardized Scores

Statistics before computers involved tables. Any score within the experimentally gathered data could be converted to a standardized score. In this way, the estimation or calculation of percentiles was possible for any data point in the sample data. A stadardized score called a \(z\)-score was computed, and the relevant value was found in a table.

The formula for calcuating a \(z\)-score is as follows:

\[z = \frac{x - \mu}{\sigma}\]

In modern statistical practice, we often estimate \(\mu\) and \(\sigma\). Thus, we often calculate \(z\)-scores based upon the sample average and sample standard deviation.

Example 1

The ACT traditionally has scores that follow a \(N(21,5)\) distribution. If Molly scored a 29 on the ACT, in what percentile was here score?

\(z\)-score

We first calculate her standardized score:

\[z_M = \frac{29 -21}{5} =\frac{8}{5} = 1.6\]

The percentile relative to this \(z\)-score can be found using the R command pnorm():

pnorm(1.6)

0.945200708300442

Thus, Molly scored in the 94th percentile on the ACT.

Example 2

What if we knew that Dolly scored in the 87th percentile on the ACT. What was her score?

First, we have to convert the percentile into a standardized score. For this, we use the R function qnorm().

qnorm(0.8)

0.841621233572914

If \(z \approx 0.8416\), then we can calcuate using the \(z\)-score formula:

\[0.8416 = \frac{ACT_D - 21}{5}\]

When we simplify and rearrange the equation, we find that:

\[ACT_D = 5 (0.8416) + 21\]

or equivalently that:

\[ACT_D \approx 25.21\]

Thus, Dolly scored a 25 on the ACT.