Normal Distribution and Z Scores | Probability Distributions

The normal distribution is a continuous probability distribution and is also called the Gaussian distribution because its density curve is symmetrical and bell-shaped. It is one of the most important distributions in statistics, and many measurements, sample statistics, and statistical methods use it directly or indirectly.

Normal distribution shape#

A normal distribution is determined by mean μ and variance σ². The mean determines the center of the curve, and the standard deviation σ determines the spread. The larger σ is, the wider and lower the curve, and the smaller σ is, the narrower and higher the curve.

Symmetrical around the average μ
The mean, median, and mode are in the same center
Both ends approach the x axis but do not intersect
The total area under the density curve is 1
The interval probability is the area under the curve for that interval

Normal distribution notation

X\sim N(\mu,\sigma^2)

probability density function

f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\!\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]

symbol	meaning
X	Continuous random variable with normal distribution
x	one possible value of X
μ	population mean. determine the center of the distribution
σ	Population standard deviation. determine the width of the distribution
σ²	population variance
f(x)	Probability density at x. Not a probability of 1 point
π、exp	pi and exponential function

Standard normal distribution and Z-score#

A normal distribution with different mean and standard deviation has the problem that it is difficult to share a single probability table. By subtracting the mean from X and dividing by the standard deviation, you can convert it to a standard normal variable Z. This is called standardization.

standardization

Z=\frac{X-\mu}{\sigma}

standard normal distribution

Z\sim N(0,1)

The Z-score represents how many standard deviations the observed value is from the mean. If z=1.5, it is 1.5 standard deviations above the mean, and if z=−2, it is 2 standard deviations below. Standardization only changes the scale, not the relative position of the values or the shape of the distribution.

symbol	meaning
Z、z	Standard normal variables and their realizations
X	Random variable before standardization
μ	population mean of X
σ	population standard deviation of X
N(0,1)	Standard normal distribution with mean 0 and variance 1

Find probability from density curve#

Since the normal distribution is continuous, the probability of one point is 0. What is actually calculated is the area under the curve of the interval. P(a≤X≤b) represents the probability that X falls between a and b.

Probability of 1 point

P(X=x)=0

interval probability

P(a\le X\le b)=\int_a^b f(x)\,dx

total area

\int_{-\infty}^{\infty}f(x)\,dx=1

Cumulative distribution function and Z table#

The cumulative distribution function of the standard normal distribution is usually written as Φ(z), and it represents the left-sided probability that Z is less than or equal to z. Traditional Z tables often show this cumulative area, but you need to check which table shows the area on the left, right, or from the average to z.

left side cumulative probability

\Phi(z)=P(Z\le z)

symmetrical relationship

\Phi(z)=1-\Phi(-z)

right tail probability

P(Z>z)=1-\Phi(z)

z value	Left cumulative probability Φ(z)	interpretation
−∞	0	No area on the left
0	0.5000	Center of symmetry. half left and right
1.645	Approximately 0.9500	Approximately 5% right hem
1.960	Approximately 0.9750	The right hem is approximately 2.5%. -1.96 to 1.96 is about 95%
+∞	1	including total area

Why is the normal distribution important?#

Some biometric measurements can approach a normal distribution under certain populations and conditions. However, you should not assume normality just because the data is continuous. Judgments will be made based on research background, graphs, and diagnostic methods.

Many sample statistics approximate a normal distribution under appropriate conditions and with a large enough sample size.
Binomial distribution etc. can be approximated by normal distribution if the conditions are met.
Normal theory or its approximations are often used in confidence intervals, hypothesis testing, and regression analysis.

How are the normal, χ², t, and F distributions connected?#

χ², t, and F are not independent curves. Starting from independent standard normal variables, they can be constructed through squaring, addition, division, and ratios. This relationship is the common basis for understanding variance estimation, t-tests, chi-square tests, and ANOVA.

Sum of squares: χ² distribution

U=\sum_{i=1}^{\nu}Z_i^2\sim\chi^2_{\nu}

Standard normal divided by estimated standard error: t distribution

T=\frac{Z}{\sqrt{U/\nu}}\sim t_{\nu}

Ratio of two standardized variations: F distribution

F=\frac{U_1/\nu_1}{U_2/\nu_2}\sim F_{\nu_1,\nu_2}

symbol	meaning
Z、Zᵢ	standard normal variables that are independent of each other
U、U₁、U₂	Random variable following a χ² distribution
ν、ν₁、ν₂	Degree of freedom. Greek letter ν is pronounced nu
T	Random variable that follows the t distribution
F	Random variable following F distribution