Standard Normal Distribution and Z-Table

Foundations of Statistics

The Universal Reference Distribution

The standard normal transforms any normal distribution into a common scale, enabling probability calculations across diverse contexts. Mastering Z-tables and Z-scores is essential for hypothesis testing, confidence intervals, and comparing across different datasets.

Medical Diagnostics — Z-scores help identify abnormal lab results by comparing to reference populations
Manufacturing — Standardized measurements enable quality control across different production lines
Education — Standardized tests use Z-scores to compare performance across different test versions

The Z-distribution is the Rosetta Stone of statistical inference.

Core Concepts

The standard normal distribution $Z \sim N(0, 1)$ is the reference distribution for all normal probability calculations. Its ubiquity stems from the Central Limit Theorem: regardless of the underlying distribution, standardized sums converge to $N(0,1)$ .

DfStandard Normal

The standard normal random variable $Z$ has pdf:

\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}, \quad z \in \mathbb{R}

Any normal $X \sim N(\mu, \sigma^2)$ can be standardized via $Z = (X - \mu)/\sigma \sim N(0,1)$ .

Standard Normal PDF

\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}

Here,

$\phi(z)$ =Standard normal PDF
$z$ =Standard score (number of SDs from mean)

Derivation: Why $\int_{-\infty}^{\infty} e^{-z^2/2} \, dz = \sqrt{2\pi}$

ThNormalization Constant of the Gaussian

We must show $\int_{-\infty}^{\infty} e^{-z^2/2} \, dz = \sqrt{2\pi}$ .

Proof (polar coordinates trick): Let $I = \int_{-\infty}^{\infty} e^{-z^2/2} \, dz$ . Then:

I^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2)/2} \, dx \, dy

Switch to polar coordinates $(r, \theta)$ : $x^2 + y^2 = r^2$ , $dx\,dy = r\,dr\,d\theta$ :

I^2 = \int_0^{2\pi} \int_0^{\infty} e^{-r^2/2} \, r \, dr \, d\theta = 2\pi \int_0^{\infty} r \, e^{-r^2/2} \, dr

Substitute $u = r^2/2$ , $du = r\,dr$ :

I^2 = 2\pi \int_0^{\infty} e^{-u} \, du = 2\pi \cdot 1 = 2\pi \implies I = \sqrt{2\pi}

Key Properties

ThMoments of the Standard Normal

For $Z \sim N(0,1)$ :

E[Z] = 0, \quad \text{Var}(Z) = 1

All odd central moments vanish by symmetry: $E[Z^{2k+1}] = 0$ .

The even moments are: $E[Z^{2k}] = \frac{(2k)!}{2^k \cdot k!} = (2k-1)!!$

In particular: $E[Z^2] = 1$ , $E[Z^4] = 3$ , $E[Z^6] = 15$ .

Why the 4th Moment Matters

The excess kurtosis of the standard normal is $\kappa = E[Z^4] - 3 = 0$ . This is why the normal is called "mesokurtic." Distributions with $\kappa > 0$ are leptokurtic (heavy tails); $\kappa < 0$ are platykurtic (light tails).

MGF and CGF of the Standard Normal

M_Z(t) = e^{t^2/2}, \quad K_Z(t) = \frac{t^2}{2}

Here,

$M_Z(t)$ =Moment generating function
$K_Z(t)$ =Cumulant generating function

Symmetry and the CDF

The standard normal satisfies $\Phi(-z) = 1 - \Phi(z)$ , so only positive $z$ -values need be tabulated. Also, $\Phi(z)$ has no closed form — it requires numerical integration or lookup tables.

The Z-Table

Using the Z-Table

The Z-table gives $\Phi(z) = P(Z \leq z)$ . Key identities:

$P(Z > z) = 1 - \Phi(z)$
$P(a < Z < b) = \Phi(b) - \Phi(a)$
$P(|Z| < z) = 2\Phi(z) - 1$

Common Critical Values

z_{0.05} = 1.645, \quad z_{0.025} = 1.96, \quad z_{0.01} = 2.326, \quad z_{0.005} = 2.576

Here,

$z_{\alpha}$ =Critical value such that P(Z > z_α) = α
$1.96$ =For 95% confidence (two-sided)
$2.576$ =For 99% confidence (two-sided)

Worked Example: Z-Score Calculation

Example: Height Distribution

Suppose adult male heights are $X \sim N(\mu = 70, \sigma^2 = 9)$ inches (so $\sigma = 3$ ). Find $P(X > 76)$ .

Step 1: Standardize: $Z = \frac{X - 70}{3}$

Step 2: $P(X > 76) = P\left(Z > \frac{76 - 70}{3}\right) = P(Z > 2) = 1 - \Phi(2)$

Step 3: From the Z-table: $\Phi(2) = 0.9772$

Step 4: $P(X > 76) = 1 - 0.9772 = 0.0228$

So approximately $2.28\%$ of men are taller than 76 inches.

Connection to the Central Limit Theorem

ThCLT and the Standard Normal

Let $X_1, X_2, \ldots$ be i.i.d. with mean $\mu$ and variance $\sigma^2 > 0$ . Then:

\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0, 1) \quad \text{as } n \to \infty

This is why the standard normal appears everywhere: it is the limiting distribution of standardized sample means, regardless of the population distribution.

Berry-Esseen Bound (CLT Rate)

The Berry-Esseen theorem quantifies the rate of CLT convergence:

\sup_z \left| P\left(\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \leq z\right) - \Phi(z) \right| \leq \frac{C \cdot \rho}{\sigma^3 \sqrt{n}}

where $\rho = E[|X - \mu|^3]$ and $C$ is a universal constant (best known: $C < 0.4748$ ). The convergence rate is $O(1/\sqrt{n})$ .

Relationship to Other Distributions

If $Z_1, Z_2, \ldots, Z_k$ are i.i.d. $N(0,1)$ , then $Z_1^2 + \cdots + Z_k^2 \sim \chi^2_k$ (chi-squared with $k$ df).
The ratio of two independent standard normals follows a Cauchy distribution: $Z_1/Z_2 \sim \text{Cauchy}(0,1)$ .
$|Z|$ follows a half-normal distribution.
$Z^2 \sim \chi^2_1 = \text{Gamma}(1/2, 1/2)$ .

Specific Applications

Hypothesis testing — The Z-test uses $Z = (\bar{X} - \mu_0)/(\sigma/\sqrt{n})$ to test $H_0: \mu = \mu_0$ when $\sigma$ is known.
Confidence intervals — A $100(1-\alpha)\%$ CI for $\mu$ (known $\sigma$ ): $\bar{X} \pm z_{\alpha/2} \cdot \sigma/\sqrt{n}$ .
Standardization of data — Converting any normal to standard normal for probability calculations.
Quality control — Six Sigma methodology uses the standard normal to compute defect rates.

Key Takeaways

Summary: Standard Normal and Z-Table

Standard normal: $Z \sim N(0, 1)$ , obtained via $Z = (X-\mu)/\sigma$
PDF: $\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}$ ; no closed-form CDF
Symmetry: $\Phi(-z) = 1 - \Phi(z)$ ; all odd moments are zero
Even moments: $E[Z^{2k}] = (2k-1)!!$ ; kurtosis $= 0$ (mesokurtic)
MGF: $M_Z(t) = e^{t^2/2}$
Common critical values: $z_{0.025} = 1.96$ (95%), $z_{0.005} = 2.576$ (99%)
Foundation for CLT, z-tests, and confidence intervals when $\sigma$ is known

Standard Normal Distribution and Z-Table

Standard Normal Distribution and Z-Table

The Universal Reference Distribution

Core Concepts

DfStandard Normal

Standard Normal PDF

Derivation: Why $\int_{-\infty}^{\infty} e^{-z^2/2} \, dz = \sqrt{2\pi}$

ThNormalization Constant of the Gaussian

Key Properties

ThMoments of the Standard Normal

MGF and CGF of the Standard Normal

The Z-Table

Common Critical Values

Worked Example: Z-Score Calculation

Connection to the Central Limit Theorem

ThCLT and the Standard Normal

Relationship to Other Distributions

Specific Applications

Key Takeaways

Summary: Standard Normal and Z-Table

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Standard Normal Distribution and Z-Table

Standard Normal Distribution and Z-Table

The Universal Reference Distribution

Core Concepts

DfStandard Normal

Standard Normal PDF

Derivation: Why ∫−∞∞e−z2/2 dz=2π\int_{-\infty}^{\infty} e^{-z^2/2} \, dz = \sqrt{2\pi}∫−∞∞​e−z2/2dz=2π​

ThNormalization Constant of the Gaussian

Key Properties

ThMoments of the Standard Normal

MGF and CGF of the Standard Normal

The Z-Table

Common Critical Values

Worked Example: Z-Score Calculation

Connection to the Central Limit Theorem

ThCLT and the Standard Normal

Relationship to Other Distributions

Specific Applications

Key Takeaways

Summary: Standard Normal and Z-Table

Premium Content

Need Expert Statistics Help?

Derivation: Why $\int_{-\infty}^{\infty} e^{-z^2/2} \, dz = \sqrt{2\pi}$