Standard Normal Distribution: What It Is and How to Use Z-Tables

Key takeaways

• N(0,1) tells you that you're looking at a standard normal distribution, where the mean is zero and the standard deviation is one.

• You can find your cumulative probability, the likelihood that a value falls at or below a specific z-score, by looking up Φ(z).

• About 68 percent of values in a standard normal distribution fall within one standard deviation of the mean.

You can use the 68-95-99.7 rule to estimate probabilities quickly before turning to a z-table for an exact value. Learn how the standard normal distribution works, how to calculate and interpret z-scores, and how to read a z-table in the following article. If you're ready to start building data analysis skills, consider enrolling in the IBM Data Analyst Professional Certificate to work with Python, IBM Cognos Analytics, and Tableau to analyze and visualize data.

What is the standard normal distribution?

A standard normal distribution is a normal distribution with a mean (𝜇) of zero and a standard deviation (𝜎) of one. Like every normal distribution, it forms a symmetric, bell-shaped curve, but its center and spread are always fixed. That means every standard normal distribution you encounter looks identical.

Instead of describing a specific real-world data set, you can use the standard normal distribution as a reference model. When you convert values from any normal distribution into z-scores, you can compare measurements that originally used different units or scales. This gives you a common framework for calculating probabilities, interpreting results, or analyzing data consistently across different problems.

You'll run into the standard normal distribution throughout statistics, data science, machine learning, and other quantitative fields. It's the foundation you'll build on for z-scores, confidence intervals, hypothesis testing, and probability calculations [1].

Learn more: 7 Statistical Analysis Methods Beginners Should Know

What is the standard deviation of a normal distribution?

The standard deviation of a normal distribution tells you how far data values typically fall from the mean. You can think of it as the setting that controls the spread of the bell-shaped curve, while the mean controls where that curve sits.

When you're looking at a larger standard deviation, expect a wider, flatter curve because the values spread over a broader range. A smaller standard deviation gives you a narrower, taller curve since the values cluster more closely around the mean.

In a standard normal distribution, the standard deviation is always one. That fixed value is what lets you read a z-score as a distance in standard deviation units from the mean. For example, a z-score of two puts you two standard deviations above the mean, while a z-score of negative one represents a value one standard deviation below the mean [2].

The 68-95-99.7 rule, also known as the empirical rule, tells you roughly what share of values fall within a given range of the mean. In a standard normal distribution, about 68 percent of values fall between a z-score of negative one and one, 95 percent fall between negative two and two, and 99.7 percent fall between negative three and three [3].

Is the standard normal distribution mean 1?

No. In a standard normal distribution, the mean is always zero. The standard deviation is one. This specific combination of the mean and standard deviation distinguishes the standard normal distribution from the broader category of normal distributions.

What is the standard normal distribution formula?

The general formula for a normal distribution's probability density function uses μ for the mean and σ for the standard deviation. Substitute a mean of zero and a standard deviation of one, the values fixed in a standard normal distribution, and the formula simplifies to:

Since μ equals zero, you no longer subtract it from x. Since σ equals one, you no longer divide by it. What remains is a version of the formula built entirely around z, the variable you'll use throughout the standard normal distribution.

You'll also encounter a specific shorthand for this distribution: N(0,1). The N indicates a normal distribution, and the two numbers in parentheses give you the mean and standard deviation, in that order. Once you recognize this notation, you can read Z ∼ N(0,1) and identify a standard normal random variable without the full formula written out.

What is a z-score?'

A z-score tells you how many standard deviations separate a data point from the mean. When you see a positive z-score, you're looking at a value above the mean. A negative z-score puts you below it. A z-score of zero means you're sitting right at the mean.

You get a z-score by standardizing a raw value, which shifts it onto a common scale where the mean is zero, and the standard deviation is one.

Standardizing lets you compare scores that started on completely different scales. Say you took two exams with different scoring systems. Convert both scores to z-scores, and you can line them up side by side on the same scale. You'll find this useful anytime you're comparing results from different measurements. Since every standardization score shares the same scale, you'll also lean on z-scores when you calculate probabilities using the standard normal distribution [4].

What is a probability density function?

A probability density function (PDF) shows you how the values of a continuous random variable are distributed. Rather than assigning a probability to each value, it shows you where values cluster along the number line and where they thin out.

When you're working with continuous data, the probability of observing one exact value is effectively zero. You'll calculate probability over intervals instead. To find the probability that a value falls within a given range, look at the area under the PDF curve between the two endpoints. Where the curve rises taller, you're looking at ranges where values occur more often; where it dips lower, those ranges are less likely.

For a normal distribution, the PDF gives you the familiar bell-shaped curve. You'll see it peak near the mean, since values close to the average occur more frequently than values farther away. As you move away from the center in either direction, the curve drops off, showing you how much less likely those more extreme values are [5].

What is the standard normal CDF?

The standard normal cumulative distribution function (CDF) tells you the probability that a value falls at or below a specific z-score.

The PDF shows you how likely a single point is. The CDF tracks the cumulative probability along the curve from the left tail to the value of interest. As the z-score increases, the cumulative probability increases with it, since you're including more of the area under the curve. Move far enough to the right, and the probability approaches one, because the full curve accounts for all of it.

You can calculate this by using a calculator, statistical analysis software, or the z-table to find the probability tied to any z-score [6].

How to use a standard normal distribution table (z-table)

You can use a z-table to look up the cumulative probability for any z-score. The result tells you the likelihood that a value in the standard normal distribution is less than or equal to that z-score.

To read one:

1. Find your z-score: Round to two decimal places if needed.

2. Locate the row: Use the ones digit, tenths place, and sign of your z-score. A z-score of 1.23 falls in the row labeled 1.2.

3. Locate the column: Use the hundredths place. For 1.23, use the 0.03 column.

4. Read the probability: Where your row and column meet is the cumulative probability through that z-score. For 1.23, that value is 0.8907, meaning about 89.07 percent of values fall at or below that point.

Z-tables typically report the area to the left of a z-score. If you need the probability above a z-score, or between two z-scores, you can find it using the cumulative probabilities you already pulled from the table.

What does a z-score tell you?

Say you scored 82 on an exam with a mean of 75 and a standard deviation of five. Your z-score comes out to 1.4, meaning you landed 1.4 standard deviations above the average. Look that up in a z-table, and you'll find you scored higher than about 92 percent of everyone who took the exam. That's what a z-score gives you: your exact standing, pulled from nothing but your score, the mean, and the standard deviation.

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