# Normal Distribution: Definition, Properties and Examples

The normal distribution is a continuous, symmetric, bell-shaped probability distribution defined by two parameters: the mean $\mu$ and the standard deviation $\sigma$. It is also called the Gaussian distribution, after Carl Friedrich Gauss, and it models many real-world measurements such as height, blood pressure and cholesterol level [1][2]. This article covers the definition, the formula, the 68-95-99.7 rule, a full worked example, and the mistakes that trip people up.

## Quick Answer

- A normal distribution is a continuous probability distribution with a symmetric, unimodal, bell-shaped curve [3].
- It has two parameters: the mean $\mu$ sets the center, and the standard deviation $\sigma$ sets the spread. A larger $\sigma$ makes the curve wider and flatter [1].
- The total area under the curve equals 1, or 100%, and probabilities are areas under the curve [1][4].
- About 68.27% of values fall within $\mu \pm 1\sigma$, 95.45% within $\mu \pm 2\sigma$, and 99.73% within $\mu \pm 3\sigma$ [3].
- Any normal distribution converts to the standard normal $Z \sim N(0,1)$ using $z = (x - \mu)/\sigma$, so one table or function handles every case [1][4].

## What the Normal Distribution Means

In plain terms, the normal distribution describes data that cluster around an average value and thin out symmetrically as you move away from it in either direction. Most observations sit near the middle, and extreme values are rare on both sides.

The precise statistical definition: a continuous random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ when its probability density function is

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$

The shorthand is $X \sim N(\mu, \sigma^2)$ [3]. When $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution, written $Z \sim N(0, 1)$, and a particular value of $Z$ is a z-score [1]. A z-score counts how many standard deviations a value sits from the mean.

## How It Works

The formula has three moving parts.

| Symbol | Meaning | Effect on the curve |
|---|---|---|
| $\mu$ | Population mean | Moves the center left or right |
| $\sigma$ | Population standard deviation | Controls width. Larger $\sigma$ means wider and flatter [1] |
| $x$ | Any value of the variable | The point where you evaluate the density |

The density itself is not a probability. Probabilities come from areas under the curve, found by integrating the density between two limits [4][3]. That integral has no simple closed-form solution, so it is computed numerically [5]. In practice you standardize first:

$$z = \frac{x - \mu}{\sigma}$$

Then you look up the area to the left of $z$ in a standard normal table, or let software do it [2][6]. The table at the NIST handbook, for example, gives the area from 0 to $z$. To find $P[x \le 1.53]$ you take the table value 0.43699 and add 0.5 for the probability below zero, giving 0.93699 [6]. For a value to the right of a z-score, you compute $1 - P(Z \le z)$. Between two z-scores, you subtract the smaller area from the larger [2].

Because the probability for one standard deviation is identical on every normal curve, you only ever need the standard normal [4]. If a population has mean 10 and standard deviation 5, the probability from 10 to 15 equals the probability from 0 to 1 on the standard normal [4].

## Worked Example

The dataset is a class survey of heights in centimeters for 50 students. Here are the values.

| 158 | 162 | 165 | 166 | 167 | 168 | 169 | 170 | 170 | 171 |
| 171 | 172 | 172 | 173 | 173 | 174 | 174 | 175 | 175 | 175 |
| 176 | 176 | 176 | 177 | 177 | 177 | 178 | 178 | 178 | 179 |
| 179 | 179 | 180 | 180 | 180 | 181 | 181 | 182 | 182 | 183 |
| 183 | 184 | 185 | 185 | 186 | 187 | 188 | 189 | 190 | 192 |

**Step 1. Sample size.** $n = 50$.

**Step 2. Sample mean.** The heights sum to 8848, so $\bar{x} = 8848/50 = 176.9600$ cm.

**Step 3. Sample standard deviation.** Using the $n-1$ denominator, $s = 7.2758$ cm.

**Step 4. Z-score for 180 cm.** $z = (180 - 176.9600)/7.2758 = 0.4178$.

**Step 5. Probability above 180 cm.** $P(X > 180) = 1 - \text{norm.cdf}(180, 176.9600, 7.2758) = 0.3380$. Standardizing gives the same answer: $1 - \text{norm.cdf}(0.4178) = 0.3380$.

In Excel the equivalent formula is `=1-NORM.DIST(180,176.9600,7.2758,TRUE)`, which returns 0.3380. In R it is `1 - pnorm(180, mean=176.9600, sd=7.2758)`, also 0.3380.

```python
import numpy as np
from scipy import stats
heights = np.array([...])  # 50 student heights
mean = heights.mean()
sd = heights.std(ddof=1)
z = (180 - mean) / sd          # z = 0.4178
p = 1 - stats.norm.cdf(180, mean, sd)  # p = 0.3380
```

Values computed by the code (it has no print statements):

```
mean = 176.9600 cm
sd   = 7.2758 cm
z    = 0.4178
P(X > 180) = 0.3380
```

So the fitted normal model predicts that about 33.8% of students are taller than 180 cm. In the actual sample, 15 of the 50 students (30%) are. You can reproduce this with the [Normal Distribution Calculator](/tools/normal-distribution-calculator).

## How to Interpret It

The z-score of 0.4178 says that 180 cm sits about 0.42 standard deviations above the mean. That is a common value, not an unusual one, which is why the tail probability is a large 0.3380.

The 68-95-99.7 rule gives you a fast sanity check. With $\mu = 176.96$ and $\sigma = 7.28$, roughly 68% of heights fall between about 169.7 cm and 184.2 cm, and roughly 95% between about 162.4 cm and 191.5 cm [3]. A height of 180 cm sits comfortably inside the first band, so a probability near one third above it makes sense.

When you report a normal probability, always state the mean and standard deviation you used. The same value of $x$ gives different probabilities under different parameters, because every change in $\mu$ or $\sigma$ changes the shape of the curve [1].

## When to Use It (and when not to)

Use the normal distribution when your variable is continuous and its histogram is symmetric, unimodal and bell-shaped, and when you can justify the mean and standard deviation as its parameters [2][3]. It is the default model for measurement error, quality control charts, and sampling distributions of means. It also underpins many significance tests through its critical values [6].

Do not use it when the data are counts of rare events, strictly positive with a long right tail, or bounded on one side with a hard limit. Not every bell-shaped curve is a normal curve, because a true normal curve has a specific fixed relationship between its height and its width [2]. If your histogram is skewed, consider alternatives such as the [exponential distribution](/blog/data-analysis/exponential-distribution), the [Weibull distribution](/blog/data-analysis/weibull-distribution-formula-examples), or the [generalized extreme value distribution](/blog/data-analysis/generalized-extreme-value-distribution) for maxima and minima.

## Normal Distribution vs Standard Normal Distribution

The standard normal is not a different distribution. It is the normal distribution with $\mu = 0$ and $\sigma = 1$, used as a common reference so that one table or function serves every problem [1][6].

| Feature | Normal $N(\mu, \sigma^2)$ | Standard normal $N(0, 1)$ |
|---|---|---|
| Mean | Any real number $\mu$ | 0 |
| Standard deviation | Any positive $\sigma$ | 1 |
| Notation for a value | $x$ | $z$ |
| Typical use | Modeling raw data | Looking up probabilities |
| Conversion | Standardize with $z = (x-\mu)/\sigma$ | Already standardized |

If you are new to the family of distributions, the overview in [Probability Distributions: Definition, Types and Examples](/blog/data-analysis/probability-distributions-explained) puts the normal curve in context alongside discrete models like the [Poisson distribution](/blog/research-skills/poisson-distribution-formula-and-examples).

## Common Mistakes

- **Treating the density value as a probability.** The height of the curve at $x$ is not $P(X = x)$. For a continuous variable that probability is zero. Fix: always compute an area over an interval [4].
- **Forgetting to standardize.** Tables are built for $N(0,1)$. Fix: convert with $z = (x-\mu)/\sigma$ before looking anything up [6].
- **Mixing up the direction of the tail.** $P(Z \le z)$ and $P(Z \ge z)$ are different numbers. Fix: for the right tail use $1 - P(Z \le z)$ [2].
- **Using the sample standard deviation as if it were the population value.** In the worked example $s = 7.2758$ estimates $\sigma$. Fix: report which one you used and remember that estimates carry uncertainty [5].
- **Assuming normality because a histogram looks roughly bell-shaped.** Fix: check symmetry and the height-to-width relationship, and use a formal test or plot when the decision matters [2].
- **Applying the normal model to bounded or skewed data.** Heights, weights and waiting times often have hard limits or long tails. Fix: inspect the distribution first, and consider a skewed alternative such as a [right skewed distribution](/blog/research-skills/right-skewed-distribution-meaning-and-examples).

## Limitations

The normal distribution assigns nonzero probability to every real number, including impossible ones. A normal model for adult height puts a small but positive probability on negative heights. For values far into the tails, those tiny probabilities are meaningless in practice, so tail estimates from a normal fit should be treated with caution.

The model is also fully determined by two parameters, which is a strength and a weakness. Real data with outliers, skew or multiple peaks will not fit well, and forcing a normal curve onto them produces misleading probabilities. The cumulative distribution function has no simple closed formula and must be computed numerically, so exact hand calculation is not possible beyond table lookups [5]. Finally, the sample mean and sample standard deviation are estimates, so any probability you compute from them inherits that uncertainty [5].

## Frequently Asked Questions

### What is the difference between the normal distribution and the Gaussian distribution?

They are two names for the same thing. The distribution is called normal in most statistics texts and Gaussian after Carl Friedrich Gauss, who studied it [1][3]. You will see both terms used interchangeably, along with "bell curve."

### What are the two parameters of a normal distribution?

The mean $\mu$, which sets the center, and the standard deviation $\sigma$, which sets the spread [3]. Changing either one changes the shape of the curve, so the same value of $x$ can be common under one set of parameters and rare under another [1].

### What is the 68-95-99.7 rule?

It is the share of a normal population that falls within one, two and three standard deviations of the mean: 68.27%, 95.45% and 99.73% respectively [3]. It gives you a quick way to judge whether a value is unusual without computing anything.

### How do I find a probability for a normal distribution by hand?

Standardize your value with $z = (x-\mu)/\sigma$, then read the area from a standard normal table [6]. For $P[x \le 1.53]$, the table gives 0.43699 for the area from 0 to 1.53, and adding 0.5 for the area below zero gives 0.93699 [6]. For the right tail, subtract from 1.

### Can I use the normal distribution for any bell-shaped data?

No. A normal curve has a specific relationship between its height and width, so a roughly bell-shaped histogram is not automatically normal [2]. Check for symmetry and unimodality, look at a normal probability plot, and consider a different distribution if the data are skewed or bounded.

## References

1. [6.4: Normal Distribution - Statistics LibreTexts](https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Mostly_Harmless_Statistics_(Webb)/06%3A_Continuous_Probability_Distributions/6.04%3A_Normal_Distribution)
2. [6.1: Introduction to the Normal Distribution - Statistics LibreTexts](https://stats.libretexts.org/Courses/Citrus_College/Statistics_C1000%3A_Introduction_to_Statistics/06%3A_Continuous_Probability_Distribution/6.01%3A_Introduction_to_the_Normal_Distribution)
3. [6.5.1. What do we mean by "Normal" data?](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc51.htm)
4. [6.2 Using the Normal Distribution - Introductory Business Statistics 2e | OpenStax](https://openstax.org/books/introductory-business-statistics-2e/pages/6-2-using-the-normal-distribution)
5. [1.3.6.6.1. Normal Distribution](https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm)
6. [1.3.6.7.1. Cumulative Distribution Function of the Standard Normal Distribution](https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm)

## Related Articles

- [Probability Distributions: Definition, Types and Examples](/blog/data-analysis/probability-distributions-explained)
- [Uniform Distribution: Definition, Formula and Examples](/blog/data-analysis/uniform-distribution)
- [Frequency Distribution: Definition, Table and Examples](/blog/data-analysis/frequency-distribution)
- [Exponential Distribution: Definition, Formula and Examples](/blog/data-analysis/exponential-distribution)
- [Bernoulli Distribution: Definition, Formula and Examples](/blog/data-analysis/bernoulli-distribution)
- [Fundamental Statistics: Core Concepts Explained](/blog/research-skills/fundamental-statistics-core-concepts-explained)
- [Poisson Distribution: Formula and Examples](/blog/research-skills/poisson-distribution-formula-and-examples)
- [Bell Curve: Normal Distribution Explained](/blog/research-skills/bell-curve-normal-distribution-explained)