The normal distribution is the most well-known continuous probability distribution. Any normal distribution can be converted to a standard normal distribution, i.e. standardized. This distribution is the most commonly used of all theoretical distributions in inferential statistics.

The standard normal distribution has an arithmetic mean of 0 and a standard deviation of 1. The standard normal distribution, like any theoretical distribution, allows us to know the proportion or percentage of cases below or above a certain score. In other words, the standard normal distribution allows us to find out the probability of such an outcome.

The properties of the standard normal distribution are:

  • The arithmetic mean µ=0, and the standard deviation σ=1.
  • The area under the curve is equal to 1, i.e. it covers 100% of the cases (representing the total probability across the entire population).
  • The individual results are displayed on the X axis in the form of z scores or z values ​​(standard scores) which are calculated by the formula

    Standardization of a normal random variable

Z value is a measure of the distance of an individual result from the arithmetic mean expressed in standard deviations.

Standard normal distribution of a random variable width=

The X-axis shows all the values ​​that a variable can have, expressed in the number of standard deviations from the arithmetic mean. Since this is a continuous distribution, the Y-axis shows the probability density, not absolute frequencies (a count of occurrences). Sometimes the Y-axis shows the proportion or probability of such a result, and then the value on the ordinate ranges from 0 to 1/√(2π) ≈ 0.3989. The more often a result is observed, the closer it will be to the arithmetic mean, and the further it is from the arithmetic mean, either to the positive or negative side of the distribution, the less often it will be observed.

The figure shows that 50% of the results are below and above the arithmetic mean, i.e. the probability of a result being below or above the arithmetic mean is 0.5. A result that is above the arithmetic mean will have a positive sign of the z score, and any result below the arithmetic mean will have a negative sign of the z value. The z score of a result that is equal to the arithmetic mean is equal to zero. The z score of a result that falls one standard deviation above the mean will be equal to 1, and z=-1 indicates a result that falls one standard deviation below the mean.

The percentage of results between -1z and +1z is 68.26%, or the probability that a result lies between -1z and +1z is 0.6826. The percentage of results that lie between the mean and -1z or between µ and +1z is half the area between -1z and +1z, or 34.13 (68.26/2).

95.44% of results lie between -2z and + 2z, ​​or the probability that a result lies between -2z and +2z is 0.9544. The percentage of results that lie between the mean µ and +2z is half of the area between -2z and +2z, i.e. 95.44/2=47.72. Also, between µ and -2z is 47.72 percent of the results.

The percentage of results between -3z and +3z is 99.74%, i.e. the probability that a result is between -3z and +3z is 0.9974. The percentage of results that are between the arithmetic mean µ and +3z is half of the area between -3z and +3z, i.e. 99.74/2=49.87.

The proportion of cases outside -3z and +3z is 0.26%, which means that 0.13% (0.26/2=0.13) of the results are below -3z, i.e. the probability of a result being below -3z is 0.0013, just as the probability of a result being above +3z is 0.0013.

The function of the standard normal curve is

Cumulative function of the normal distribution

Calculating the integral is complicated, so the area under the curve is read in statistical tables, or calculated using software.