Uniform and normal frequency distributions

There are two typical frequency distributions that we may also commonly encounter in practice. These are the uniform frequency distribution and the normal frequency distribution.

Uniform frequency distribution

A uniform distribution means that all values occur equally often in our population. For example, we can count how many months the years 2000, 2001, ..., 2009 had:

If we roll a classical idealized die, we have the probability $\frac16$ of rolling a 1, the probability $\frac16$ of rolling a 2, and so on. Thus we speak of a uniform probability distribution, because each outcome has the same probability.

A non-uniform distribution would be if we had a clinking die on which the probability $\frac13$ was always a six. The other sides cannot also have probability $\frac13$. Similarly, we could count the number of days in each month - not all months have the same number of days, so it would not be a uniform frequency distribution.

Normal probability distribution

A normal distribution is sometimes also called a Gaussian distribution. It is only used for continuous random variables. A standardised normal distribution has the following form:

This curve is described by a so-called Gaussian function, which has the form:

$$ \Large f(x) = \frac{1}{\sigma\sqrt{2\pi}} \mathrm{e}^{-\frac{{(x-\mu)}^2}{2\sigma^2}} $$

The function is quite complicated, you don't have to remember it. But we can see that we can parameterize the normal distribution using the mean and standarddeviation/variance. We can then get different plots for different values of the mean and variance:

The normal distribution is important because it fairly faithfully simulates the various distributions we may encounter in real life. We can see that the curve always has one global maximum, which is also equal to the mean value. For example, the red curve has a maximum at X = 0, so the average value of this quantity is zero. The curve is symmetric, it is axisymmetric with a line that is perpendicular to the axis X and passes through the mean, the point X = 0. In other words - the curve looks the same to the left of zero as it does to the right of zero. The green curve has the same properties, but with zero but with the point X = −2.

The normal distribution can faithfully describe the distribution of intelligence or IQ among people. The average IQ is - by definition - 100. In doing so, there are approximately equal numbers of people who have IQs above 100 and below 100. There are approximately as many slightly below average intelligent people as there are slightly above average intelligent people, and there are as many morons as there are geniuses. Although it doesn't seem like it sometimes :-). A graph showing the approximate distribution of IQ:

For a normal distribution, standard deviation plays a big role. If we have a distribution that is normal and has a standard deviation of $\sigma$, then it must be true that 68% of the values are in the interval $\left<\mu-\sigma, \mu+\sigma\right>$. That is, 68% of the values differ from the mean by at most one standard deviation. Approximately 95% of the values must then lie within the interval $\left<\mu-2\sigma, \mu+2\sigma\right>$ and 99.7% of the values within the interval $\left<\mu-3\sigma, \mu+3\sigma\right>$. This is clearly illustrated in the following figure:

References and sources

• Some of the images are from the Wikipedia articles Normal distribution and Intelligence.
• Gaussian distribution on Wikiskripta.eu