The standard normal distribution
The standard normal distribution, also called the zdistribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.
Any normal distribution can be standardized by converting its values into zscores. Zscores tell you how many standard deviations from the mean each value lies.
Converting a normal distribution into a zdistribution allows you to calculate the probability of certain values occurring and to compare different data sets.
Normal distribution vs the standard normal distribution
All normal distributions, like the standard normal distribution, are unimodal and symmetrically distributed with a bellshaped curve. However, a normal distribution can take on any value as its mean and standard deviation. In the standard normal distribution, the mean and standard deviation are always fixed.
Every normal distribution is a version of the standard normal distribution that’s been stretched or squeezed and moved horizontally right or left.
The mean determines where the curve is centered. Increasing the mean moves the curve right, while decreasing it moves the curve left.
The standard deviation stretches or squeezes the curve. A small standard deviation results in a narrow curve, while a large standard deviation leads to a wide curve.
Curve  Position or shape (relative to standard normal distribution) 

A (M = 0, SD = 1)  Standard normal distribution 
B (M = 0, SD = 0.5)  Squeezed, because SD < 1 
C (M = 0, SD = 2)  Stretched, because SD > 1 
D (M = 1, SD = 1)  Shifted right, because M > 0 
E (M = –1, SD = 1)  Shifted left, because M < 0 
Standardizing a normal distribution
When you standardize a normal distribution, the mean becomes 0 and the standard deviation becomes 1. This allows you to easily calculate the probability of certain values occurring in your distribution, or to compare data sets with different means and standard deviations.
While data points are referred to as x in a normal distribution, they are called z or zscores in the zdistribution. A zscore is a standard score that tells you how many standard deviations away from the mean an individual value (x) lies:
 A positive zscore means that your xvalue is greater than the mean.
 A negative zscore means that your xvalue is less than the mean.
 A zscore of zero means that your xvalue is equal to the mean.
Converting a normal distribution into the standard normal distribution allows you to:
 Compare scores on different distributions with different means and standard deviations.
 Normalize scores for statistical decisionmaking (e.g., grading on a curve).
 Find the probability of observations in a distribution falling above or below a given value.
 Find the probability that a sample mean significantly differs from a known population mean.
How to calculate a zscore
To standardize a value from a normal distribution, convert the individual value into a zscore:
 Subtract the mean from your individual value.
 Divide the difference by the standard deviation.
Zscore formula  Explanation 


To standardize your data, you first find the zscore for 1380. The zscore tells you how many standard deviations away 1380 is from the mean.
Step 1: Subtract the mean from the x value.  x = 1380 M = 1150 x – M = 1380 – 1150 = 230 

Step 2: Divide the difference by the standard deviation.  SD = 150 z = 230 ÷ 150 = 1.53 
The zscore for a value of 1380 is 1.53. That means 1380 is 1.53 standard deviations from the mean of your distribution.
Next, we can find the probability of this score using a ztable.
Use the standard normal distribution to find probability
The standard normal distribution is a probability distribution, so the area under the curve between two points tells you the probability of variables taking on a range of values. The total area under the curve is 1 or 100%.
Every zscore has an associated pvalue that tells you the probability of all values below or above that zscore occuring. This is the area under the curve left or right of that zscore.
Ztests and pvalues
The zscore is the test statistic used in a ztest. The ztest is used to compare the means of two groups, or to compare the mean of a group to a set value. Its null hypothesis typically assumes no difference between groups.
The area under the curve to the right of a zscore is the pvalue, and it’s the likelihood of your observation occurring if the null hypothesis is true.
Usually, a pvalue of 0.05 or less means that your results are unlikely to have arisen by chance; it indicates a statistically significant effect.
By converting a value in a normal distribution into a zscore, you can easily find the pvalue for a ztest.
How to use a ztable
Once you have a zscore, you can look up the corresponding probability in a ztable.
In a ztable, the area under the curve is reported for every zvalue between 4 and 4 at intervals of 0.01.
There are a few different formats for the ztable. Here, we use a portion of the cumulative table. This table tells you the total area under the curve up to a given zscore—this area is equal to the probability of values below that zscore occurring.
The first column of a ztable contains the zscore up to the first decimal place. The top row of the table gives the second decimal place.
To find the corresponding area under the curve (probability) for a zscore:
 Go down to the row with the first two digits of your zscore.
 Go across to the column with the same third digit as your zscore.
 Find the value at the intersection of the row and column from the previous steps.
Stepbystep example of using the zdistribution
Let’s walk through an invented research example to better understand how the standard normal distribution works.
As a sleep researcher, you’re curious about how sleep habits changed during COVID19 lockdowns. You collect sleep duration data from a sample during a full lockdown.
Before the lockdown, the population mean was 6.5 hours of sleep. The lockdown sample mean is 7.62.
To assess whether your sample mean significantly differs from the prelockdown population mean, you perform a ztest:
 First, you calculate a zscore for the sample mean value.
 Then, you find the pvalue for your zscore using a ztable.
Step 1: Calculate a zscore
To compare sleep duration during and before the lockdown, you convert your lockdown sample mean into a zscore using the prelockdown population mean and standard deviation.
Formula  Explanation  Calculation 

x = sample mean μ = population mean σ = population standard deviation 
x = 7.62 μ = 6.5 σ = 0.5 z = (7.62 – 6.5) ÷ 0.5 = 2.24 
A zscore of 2.24 means that your sample mean is 2.24 standard deviations greater than the population mean.
Step 2: Find the pvalue
To find the probability of your sample mean zscore of 2.24 or less occurring, you use the ztable to find the value at the intersection of row 2.2 and column +0.04.
The table tells you that the area under the curve up to or below your zscore is 0.9874. This means that your sample’s mean sleep duration is higher than about 98.74% of the population’s mean sleep duration prelockdown.
To find the pvalue to assess whether the sample differs from the population, you calculate the area under the curve above or to the right of your zscore. Since the total area under the curve is 1, you subtract the area under the curve below your zscore from 1.
A pvalue of less than 0.05 or 5% means that the sample significantly differs from the population.
Probability of z > 2.24 = 1 – 0.9874 = 0.0126 or 1.26%
With a pvalue of less than 0.05, you can conclude that average sleep duration in the COVID19 lockdown was significantly higher than the prelockdown average.
Frequently asked questions about the standard normal distribution
 What is a normal distribution?

In a normal distribution, data is symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center.
The measures of central tendency (mean, mode and median) are exactly the same in a normal distribution.
 What is a standard normal distribution?

The standard normal distribution, also called the zdistribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.
Any normal distribution can be converted into the standard normal distribution by turning the individual values into zscores. In a zdistribution, zscores tell you how many standard deviations away from the mean each value lies.
 What is the empirical rule?

The empirical rule, or the 689599.7 rule, tells you where most of the values lie in a normal distribution:
 Around 68% of values are within 1 standard deviation of the mean.
 Around 95% of values are within 2 standard deviations of the mean.
 Around 99.7% of values are within 3 standard deviations of the mean.
The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don’t follow this pattern.
 What is the difference between the tdistribution and the standard normal distribution?

The tdistribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the zdistribution).
In this way, the tdistribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance, you will need to include a wider range of the data.