The normal distribution is a unimodal symmetric continuous distribution defined by two parameters: the expected value or arithmetic mean (μ) and the variance (σ2). Since the normal distribution is symmetric, the values of the arithmetic mean, mode, and median are the same.
The normal distribution is also known as the Gaussian distribution after the German mathematician and physicist Carl Friedrich Gauss. He famously predicted the location of the dwarf planet (asteroid) Ceres in 1801 after it had been lost from view behind the Sun, based on the assumption of a normal distribution of errors in the astronomical measurements of its orbit. Later, the concept was adopted in the social sciences.
It is a very common law that natural and social phenomena converge to mean values, and that extreme values are rare. We can see this in the examples of people's height and weight, measurement errors, marathon times, reading speed, job satisfaction, exam scores, etc. That is why the normal distribution is very important in statistics.
Another reason why the normal distribution is so important is that many statistical tests can be performed if the random variable that records the results of the observed phenomenon is normally distributed. Statistical tests that assume a normal distribution are sufficiently "powerful" even if the variables are only approximately normally distributed, or when the assumption of normality is severely violated.
The normal distribution can also be used as an approximation in the case of some variables that are inherently discontinuous, such as exam scores, intelligence, etc.
The normal distribution curve is bell-shaped and symmetrical about the arithmetic mean. The value of the arithmetic mean determines the center of the distribution, and the variance determines the width of the distribution, when displayed graphically. The curve is defined for all elements of the set of real numbers.
The probability density function of the normal distribution curve is:
The total area under the normal distribution curve is always 1, and the maximum of the curve is located at the point:
Most of the values of a normally distributed random variable X, about 99.7%, fall within a range of 3 standard deviations from the expected value. About 95.44% of the values lie within two standard deviations of the expected value, i.e., within the interval [μ - 2σ, μ + 2σ]. This is sometimes called the 2 sigma rule.
Changing the distribution parameters μ and σ2 does not change the bell shape of the normal curve, only its location on the abscissa and its scale. The total area under the curve remains the same regardless of the parameters, as does the 2-sigma rule.
In the pictures we see two random variables that are normally distributed, but the distribution has different parameters. In the image on the left, we see a normal distribution with a higher variance, i.e. a higher variability of results around the mean, and therefore it is wider than the other one, where the variability of results is lower.
empirical rule
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