Measurements such as human height or repeated laboratory results often cluster around a central value. In some populations and under suitable conditions, a normal distribution can approximate that pattern. Check the model against the observed data before using it.
In 1733, Abraham de Moivre derived a normal approximation to the binomial distribution for large . Pierre-Simon Laplace later extended this result, which is now associated with the de Moivre-Laplace theorem and the development of the central limit theorem.
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The normal distribution is one possible model. Use it when observations are roughly symmetric around a mean and the model fits the data well enough for the question being studied.
The normal density has a symmetric bell shape. The visualization below shows the standard normal curve with μ=0 and σ=1:
The curve has four defining characteristics:
Shape and Symmetry
The curve is bell-shaped and symmetrical about the vertical line passing through the mean (μ). This means the left and right parts of the curve are perfect mirrors of each other.
Central Point
The normal density is symmetric and has one peak. Therefore, its mean, median, and mode all coincide at μ.
Inflection Points
The curve has inflection points at x=μ±σ, which means the curve changes from concave to convex (or vice versa) at a distance of one standard deviation from the mean.
Horizontal Asymptote
The density remains positive for every finite x and approaches 0 as x tends to either infinity.
The formula below gives the probability density at each value of x.
If X is normally distributed with mean μ, variance σ2, and σ>0, its density function is:
fμ,σ(x)=σ2π1e−21(σx−μ)2
This formula applies to every x satisfying −∞<x<∞.
The constants and parameters are:
π≈3.1416 is the circle constant
e≈2.7183 is Euler's number
μ is the mean of the distribution
σ>0 is the standard deviation
The parameter μ shifts the center of the curve, while σ controls its spread. Find probabilities from areas under the density curve. The height f(x) at one point is a density value.
In practice, we often use the standard normal distribution with mean μ=0 and standard deviation σ=1. To convert a regular normal distribution to standard normal, we use the transformation:
Z=σx−μ
The variable Z is the standard score, or z-score. It measures how many standard deviations x lies above or below the mean. Standardization maps every normal distribution to the same Z scale, so one standard normal table works for all of them.
Why use z-score?
With this transformation, we can compare data from different distributions. For example, you can compare math scores with physics scores, even though their means and standard deviations are different.
Suppose X is normally distributed with μ=50 and σ=10. Find the probability that X lies between 45 and 62.
Step1: Transform to z-score
z1=1045−50=10−5=−0.5
z2=1062−50=1012=1.2
Step2: Use the standard normal distribution table
The transformed interval is P(−0.5<Z<1.2). Subtract the cumulative probability at the lower boundary from the cumulative probability at the upper boundary:
P(a<Z<b)=P(Z<b)−P(Z<a)
From the standard normal distribution table:
P(Z<−0.5)=0.3085
P(Z<1.2)=0.8849
Step3: Calculate the final probability
P(45<X<62)=P(−0.5<Z<1.2)=0.8849−0.3085=0.5764
So, the probability that X lies between 45 and 62 is 0.5764 or approximately 57.64%.
