Summary of Standard Normal Distribution Z-Table
Standard Normal Distribution Z-Table Explained for Students
Introduction
The standard normal distribution is a foundational concept in statistics. It describes how values of a standardized variable $Z$ (with mean $0$ and standard deviation $1$) are distributed. This study material explains how to read and use the standard normal (Z) table, interpret cumulative probabilities, and apply them in practical problems.
What the Z-table shows
The Z-table gives cumulative probabilities for a standard normal random variable $Z$. Each table entry tells you the probability that $Z$ is less than or equal to a given value: $P(Z \le z)$.
Definition: The standard normal distribution is the normal distribution with mean $0$ and standard deviation $1$. The cumulative distribution function is $F(z)=P(Z\le z)$.
Two common table formats
- Positive z values table: lists $P(Z\le z)$ for $z\ge 0$. Rows give the first decimal of $z$, columns give the second decimal.
- Negative z values table: lists $P(Z\le z)$ for $z\le 0$. Because of symmetry, values for negative $z$ can also be found using the positive table.
Definition: Symmetry property of the standard normal: $P(Z\le -z)=1-P(Z\le z)$ for $z\ge 0$.
How to read the table
- Split the z-value into a row and column component. For example $z=1.23$ has row $1.2$ and column $0.03$.
- Find the row labeled $1.2$ and the column labeled $0.03$; the table entry is $P(Z\le 1.23)$.
- For negative $z$, use the negative table directly or apply symmetry: $P(Z\le -1.23)=1-P(Z\le 1.23)$.
Example 1: Reading a positive z
To find $P(Z\le 0.85)$:
- Row: $0.8$, Column: $0.05$.
- Table value: $0.8023$.
- So $P(Z\le 0.85)=0.8023$.
Example 2: Using symmetry for a negative z
To find $P(Z\le -1.40)$:
- Option A: Use negative values table entry directly: $0.0808$.
- Option B: Use symmetry: $P(Z\le -1.40)=1-P(Z\le 1.40)=1-0.9192=0.0808$.
Common probability queries and how to compute them
- Probability greater than a value: $P(Z>z)=1-P(Z\le z)$.
- Probability between two values: $P(a<Z\le b)=P(Z\le b)-P(Z\le a)$.
- Two-tailed probabilities: For symmetric bounds $\pm c$, $P(|Z|>c)=2\left(1-P(Z\le c)\right)$.
Example 3: Between values
Find $P(-0.5<Z\le 1.2)$:
- $P(Z\le 1.2)=0.8849$ (from table).
- $P(Z\le -0.5)=0.3085$ (from table).
- So $P(-0.5<Z\le 1.2)=0.8849-0.3085=0.5764$.
Practical applications
- Confidence intervals: Critical Z-values like $z_{0.975}=1.96$ are used to build confidence intervals for means when population variance is known.
- Hypothesis testing: Compare test statistics standardized as $Z$ against table probabilities to compute p-values.
- Quality control: Determine the probability of extreme deviations in manufacturing measurements.
Quick reference table: common Z and probabilities
| Z | $P(Z\le z)$ | Use case |
|---|---|---|
| $0$ | $0.5000$ | Median of distribution |
| $1.00$ | $0.8413$ | One-sided significance ~0.1587 |
| $1.64$ | $0.9495$ | Approx 90% one-sided critical value |
| $1.96$ | $0.9750$ | 95% two-sided critical value (each tail 0.025) |
| $2.58$ | $0.9951$ | 99% two-sided critical value |
Tips and pitfalls
- Always confirm whether table gives $P(Z\le z)$ or tail probabilities; most standard tables give cumulative $P(Z\le z)$.
- For high precision, use statistical software rather than manual table lookup.
- Pay attention to rounding: read the table entry to the same decimal precision as the table provides.
💡 Did you know?Fun fact: The area under the standard normal curve integrates to 1, and the shape is completely determined by just two parameters: mean $0$ and standard deviation $1$, which makes it a standard reference for comparing different normal distributions.
Summary
- The Z-table lists cumulative probabilities $P(Z\le z)$ for the standard normal $Z$.
- Read a z-value by splitting into row (first decimal) and column (second decimal).
- Use symmetry $P(Z\le -z)=1-P(Z\le z)$ to find negative values.
- Convert between cumulative, tail, and between probabiliti
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Standard Normal Table
Klíčová slova: Standard Normal Distribution Table
Klíčové pojmy: Z-table gives cumulative probabilities $P(Z\le z)$ for standard normal, Read z by row (first decimal) and column (second decimal), For negative z, $P(Z\le -z)=1-P(Z\le z)$, $P(Z>z)=1-P(Z\le z)$, $P(a<Z\le b)=P(Z\le b)-P(Z\le a)$, Two-tailed prob: $P(|Z|>c)=2\left(1-P(Z\le c)\right)$, Common critical values: $z_{0.975}=1.96$, $z_{0.995}=2.58$, Use software for high precision instead of manual lookup, Always check if table lists cumulative or tail probabilities, Round to table precision when reading values
## Introduction
The **standard normal distribution** is a foundational concept in statistics. It describes how values of a standardized variable $Z$ (with mean $0$ and standard deviation $1$) are distributed. This study material explains how to read and use the standard normal (Z) table, interpret cumulative probabilities, and apply them in practical problems.
## What the Z-table shows
The Z-table gives **cumulative probabilities** for a standard normal random variable $Z$. Each table entry tells you the probability that $Z$ is less than or equal to a given value: $P(Z \le z)$.
> Definition: The standard normal distribution is the normal distribution with mean $0$ and standard deviation $1$. The cumulative distribution function is $F(z)=P(Z\le z)$.
### Two common table formats
- **Positive z values table**: lists $P(Z\le z)$ for $z\ge 0$. Rows give the first decimal of $z$, columns give the second decimal.
- **Negative z values table**: lists $P(Z\le z)$ for $z\le 0$. Because of symmetry, values for negative $z$ can also be found using the positive table.
> Definition: Symmetry property of the standard normal: $P(Z\le -z)=1-P(Z\le z)$ for $z\ge 0$.
## How to read the table
1. Split the z-value into a row and column component. For example $z=1.23$ has row $1.2$ and column $0.03$.
2. Find the row labeled $1.2$ and the column labeled $0.03$; the table entry is $P(Z\le 1.23)$.
3. For negative $z$, use the negative table directly or apply symmetry: $P(Z\le -1.23)=1-P(Z\le 1.23)$.
### Example 1: Reading a positive z
To find $P(Z\le 0.85)$:
- Row: $0.8$, Column: $0.05$.
- Table value: $0.8023$.
- So $P(Z\le 0.85)=0.8023$.
### Example 2: Using symmetry for a negative z
To find $P(Z\le -1.40)$:
- Option A: Use negative values table entry directly: $0.0808$.
- Option B: Use symmetry: $P(Z\le -1.40)=1-P(Z\le 1.40)=1-0.9192=0.0808$.
## Common probability queries and how to compute them
- Probability greater than a value: $P(Z>z)=1-P(Z\le z)$.
- Probability between two values: $P(a<Z\le b)=P(Z\le b)-P(Z\le a)$.
- Two-tailed probabilities: For symmetric bounds $\pm c$, $P(|Z|>c)=2\left(1-P(Z\le c)\right)$.
### Example 3: Between values
Find $P(-0.5<Z\le 1.2)$:
- $P(Z\le 1.2)=0.8849$ (from table).
- $P(Z\le -0.5)=0.3085$ (from table).
- So $P(-0.5<Z\le 1.2)=0.8849-0.3085=0.5764$.
## Practical applications
- **Confidence intervals**: Critical Z-values like $z_{0.975}=1.96$ are used to build confidence intervals for means when population variance is known.
- **Hypothesis testing**: Compare test statistics standardized as $Z$ against table probabilities to compute p-values.
- **Quality control**: Determine the probability of extreme deviations in manufacturing measurements.
## Quick reference table: common Z and probabilities
| Z | $P(Z\le z)$ | Use case |
| --- | ---: | --- |
| $0$ | $0.5000$ | Median of distribution |
| $1.00$ | $0.8413$ | One-sided significance ~0.1587 |
| $1.64$ | $0.9495$ | Approx 90% one-sided critical value |
| $1.96$ | $0.9750$ | 95% two-sided critical value (each tail 0.025) |
| $2.58$ | $0.9951$ | 99% two-sided critical value |
## Tips and pitfalls
- Always confirm whether table gives $P(Z\le z)$ or tail probabilities; most standard tables give cumulative $P(Z\le z)$.
- For high precision, use statistical software rather than manual table lookup.
- Pay attention to rounding: read the table entry to the same decimal precision as the table provides.
Fun fact: The area under the standard normal curve integrates to 1, and the shape is completely determined by just two parameters: mean $0$ and standard deviation $1$, which makes it a standard reference for comparing different normal distributions.
## Summary
- The Z-table lists cumulative probabilities $P(Z\le z)$ for the standard normal $Z$.
- Read a z-value by splitting into row (first decimal) and column (second decimal).
- Use symmetry $P(Z\le -z)=1-P(Z\le z)$ to find negative values.
- Convert between cumulative, tail, and between probabiliti