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Normal Distribution

The normal distribution, or Gaussian distribution, is the most important continuous probability distribution, since almost all values we have in our environment are normally distributed. Body height (within a gender), the 100m times of a swimmer in different races but also something as special as the weight of several coffee packets follow the Gaussian distribution from a sufficiently large sample. 

If we perform a random experiment, such as measuring the times of a swimmer again and again, then we want to obtain a so-called density function. This tells us how often a certain event occurs. For example, we might be interested in how likely it is that the swimmer completes the 100m in a time of 1:15 min. Additionally, we might be interested in the probability that the athlete swims the 100m in under or at most 1:15min. We can answer this question with the help of the distribution function. The distribution function indicates the probability with which the result of the random experiment is less than or equal to a certain value. 


A continuous random variable X with a density function f(x) of the form

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

with the expected value µ and the variance σ² is called normally distributed (short: N(µ, σ²)). The expected value µ…

  • … is a real number, so it can also become negative.
  • … is the X-coordinate of the maximum of the density function.

The variance σ²…

  • … is the squared standard deviation σ.
  • … must always be greater than 0.
  • … determines how much the graph is stretched or compressed horizontally. Low variance means that the graph is narrow. 

Density function

In connection with the normal distribution, the density function is usually shown with its well-known bell curve. In short, this graph is used to read off the probability of this event occurring for an expected value X. The probability of this event occurring is determined by the probability of this event occurring.

The graph depicts the normal distribution of heights in centimeters measured in male subjects. The expected value µ = 180 indicates that the majority of the subjects were 180cm tall. The variance σ² in this example is 7. The probability for the expected value X = 176 is about 5%, i.e. a random male test subject is exactly 176cm tall with a probability of 5%.

Das Bild zeigt die charackteristische Glockenkurve der Normalverteilung auf kariertem Hintergrund.
Normal Distribution Density Function

Distribution function

The distribution function F(x) of the normal distribution is defined by

\(\) \[f(x) = \frac{1}{\sigma \sqrt{2 \pi}} \cdot \int_{- \infty}^{x} e^{-\frac{1}{2} \cdot \frac{(x – \mu)^2}{\sigma}} \]

Thus, the integral of the density function f(x) in the range from – to the random variable X. Accordingly, the distribution function indicates how high the probability is that the random variable takes on a value of less than or equal to X: 

\(\) \[ f(x) = Prob(X \leq x) \]

For the expected value X = 176, we obtain a probability of about 6.7% in the distribution function. A random, male person is thus shorter or exactly 176cm tall with a probability of 6.7%. 

This is what you should take with you

  • The normal distribution is the most important probability distribution.
  • Almost all random experiments occurring in nature can be represented by the Gaussian distribution.
  • The normal distribution is characterized by the density function, which gives the probability value for each event. The distribution function, on the other hand, indicates the probability that the random variable will take on a value less than x.

Other Articles on the Topic of Normal Distribution

  • You can find a concise summary of the topic here.
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