In our previous article, we have seen that what is Normal Distribution and it’s computation by using the Empirical Rule. Today in this article we will study how to calculate the probabilities of the normally distributed data sets. The key features of this article are

  • Normal Distribution
  • Gauss Distribution
  • Bell Curve
  • Standardization
  • Z-Score
  • Standard Score

There are various different types of species distributions and which in turn have various applications also. But one of the distribution is widely used in the statistical studies which is known as the Normal Distribution or also can be termed as the Gauss Distribution or the Bell Curve. This is the continuous distribution with the algebraic equation of the probability density as


µ is the mean

σ is the standard deviation

Generally, the normal distribution is as below, the meaning of below distribution is always in symmetrical form and the graph is known as the bell curve

As this is the continuous distribution the integral calculus can directly calculate the probabilities associated with it. The tables of the probabilities are always available as the normal distribution can be applied in many situations. This distribution can be easily observed by the mean and the standard deviation of the interest. Maybe you will not be able to understand the notations, but the probability can be written as

This equation computes the area below the curve from the negative infinity to x = c, as shown in the shaded region.

Further we can also compute the probability P(a < X ≤ b). in such types of cases the shaded region should be limited. The probability for the distribution is related to the area below the curve for the particular range of the values. The area below the normal curve will be unity.

This discussion is mostly applicable to the populations rather than the normal samples. We cannot get the continuous normal distribution from the samples.

Z-Scores and Normal Distribution

if the condition is modeled by the normal distribution and mean µ and the standard deviation σ, then we can calculate the probabilities by standardizing the normal distribution. Hence, the exponential function of the probability can be (x – µ) / σ. Let z be the equation, which is Z-Score or Standard Score. Now by using the integral calculus

In the above equation if /σ, ten the standard normal distribution will be

The graph will be in the form of

The standard deviation of the standard normal curve is unity and the mean is z = 0. The peak of the curve will be 0.399 approximately.

This all can be seen as the complicated mathematics and the pointless changes in the variables. But this process has a particular purpose, by which we can standardize the normal curve with the mean and the standard deviation to the standardized normal curve. We can calculate the probabilities of any normal distribution for any standard distribution. Those who do not understand the integral calculus even they can also calculate the probabilities of the normally distributed data sets.

Using Standard Normal Distribution Tables

The table of the standard normal distribution includes the probabilities for the range of values –∞ to x (or z) i.e. P(X ≤ x). this will be the same probability as

And graphically this can be represented as,

Conclusion: in this article, we have seen how to determine the normal distribution and its fundamental characteristics. We have also seen how to standardize the random variable using the Z-Score and the calculation of the probabilities for the normal distribution.