Normal distribution: One, two and three standard deviations
One important property of normal distributions is that if we consider intervals centered on the mean and a certain extent proportional to the standard deviation, the probability of these intervals is constant regardless of the mean and standard deviation of the normal distribution considered.
For example, if we consider an interval centered on the mean with amplitude one standard deviation up and down, the probability is 68.2%.
If the amplitude is two standard deviations up and down, the probability is 95.5%.
If the amplitude is three standard deviations up and down, the probability is 99.7%.
Modifying the parameters of the normal distribution we can see that the probabilities of the intervals remain unchanged.
The mean is represented by a triangle that can be seen as an equilibrium point. By dragging it we can modify the mean. We can get the same effect by moving the point at the top of the curve.
By dragging the other point of the curve we can modify the standard deviation.
The gray dots control the vertical and horizontal scales of the graphic. By pressing the right button and dragging we can move left and right.
REFERENCES
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Normal distribution
The Normal distribution was studied by Gauss. This is a continuous random variable (the variable can take any real value). The density function is shaped like a bell.
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Calculating probabilities in Normal distributions
It may be interesting to familiarize ourselves with the probabilities correspondig to different intervals in normal distributions.
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Student's t-distributions
Student's t-distributions were studied by William Gosset(1876-1937) when working with small samples.
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Binomial distribution
When modeling a situation where there are n independent trials with a constant probability p of success in each test we use a binomial distribution.
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Normal approximation to Binomial distribution
In some cases, a Binomial distribution can be approximated by a Normal distribution with the same mean and variance.
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Poisson distribution
Poisson distribution is discrete (like the binomial) because the values that can take the random variable are natural numbers, although in the Poisson distribution all the possible cases are theoretically infinite.
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