Normal Distribution
This page features the essential Standard Normal Distribution table, universally recognized as the Z-table. This table is arguably one of the most important and frequently used tools in applied statistics. It specifically relates to the Standard Normal Distribution, which is a particular instance of the normal (or Gaussian) distribution. This specific distribution is defined by having a mean ($\mu$) of exactly $\mathbf{0}$ and a standard deviation ($\sigma$) of exactly $\mathbf{1}$.
The primary function of the Z-table is to allow you to determine the area under the curve of the standard normal distribution up to a certain point. This area represents the cumulative probability associated with a given Z-score. A Z-score (or standard score) is a measure that indicates how many standard deviations a particular data point is away from the mean of the distribution. A positive Z-score means the data point is above the mean, while a negative Z-score means it is below the mean.
$$ Z = \frac{x - \mu}{\sigma} $$ For the standard normal distribution itself, as $\mu=0$ and $\sigma=1$, a value 'x' is simply its own Z-score. However, the Z-table is invaluable for *any* normal distribution, because you can convert any observation 'x' from a normal distribution with mean $\mu$ and standard deviation $\sigma$ into a Z-score using the formula shown above. Once converted, you can use the Z-table to find probabilities related to that original observation.
Z-tables are typically presented in various formats, but a common layout shows the cumulative probability $P(Z \le z)$, which is the probability that a randomly selected value from the standard normal distribution is less than or equal to a given Z-score, 'z'. Due to the perfect symmetry of the normal distribution around its mean (0 for the standard normal), tables often only provide probabilities for positive Z-scores. Probabilities for negative Z-scores can be derived using the relationship $P(Z \le -z) = 1 - P(Z \le z)$.
Using the table involves locating the row that corresponds to the integer part and the first decimal place of your Z-score, and then finding the column corresponding to the second decimal place. The value at the intersection of that row and column is the cumulative probability $P(Z \le z)$. For example, to find $P(Z \le 1.96)$, you would look for the row labeled '1.9' and the column labeled '.06'.
The Z-table is a fundamental tool in statistical inference and analysis, used extensively in:
- Hypothesis Testing: Comparing sample means or proportions to expected values and determining statistical significance (calculating p-values).
- Constructing Confidence Intervals: Defining ranges within which a population parameter is likely to fall, based on sample data.
- Calculating Percentiles: Finding the value below which a given percentage of observations fall.
- Finding Probabilities: Determining the likelihood of a value falling within a certain range or above/below a specific point for any normally distributed variable after standardization.
By providing the relationship between Z-scores and cumulative probabilities, this resource enables the application of the powerful properties of the normal distribution to a vast array of real-world data and problems.
| $z = \frac{(x - μ)}{σ}$ | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.0 | .0000 | .0040 | .0080 | .0120 | .0159 | .0199 | .0239 | .0279 | .0319 | .0359 |
| 0.1 | .0398 | .0438 | .0478 | .0517 | .0557 | .0596 | .0636 | .0675 | .0714 | .0753 |
| 0.2 | .0793 | .0832 | .0871 | .0910 | .0948 | .0987 | .1026 | .1064 | .1103 | .1141 |
| 0.3 | .1179 | .1217 | .1255 | .1293 | .1331 | .1368 | .1406 | .1443 | .1480 | .1517 |
| 0.4 | .1554 | .1591 | .1628 | .1664 | .1700 | .1736 | .1772 | .1808 | .1844 | .1879 |
| 0.5 | .1915 | .1950 | .1985 | .2019 | .2054 | .2088 | .2123 | .2157 | .2190 | .2224 |
| 0.6 | .2257 | .2291 | .2324 | .2357 | .2389 | .2422 | .2454 | .2486 | .2518 | .2549 |
| 0.7 | .2580 | .2611 | .2642 | .2673 | .2704 | .2734 | .2764 | .2794 | .2823 | .2852 |
| 0.8 | .2881 | .2910 | .2939 | .2967 | .2995 | .3023 | .3051 | .3078 | .3106 | .3133 |
| 0.9 | .3159 | .3186 | .3212 | .3238 | .3264 | .3289 | .3315 | .3340 | .3365 | .3389 |
| 1.0 | .3413 | .3438 | .3461 | .3485 | .3508 | .3531 | .3554 | .3577 | .3599 | .3621 |
| 1.1 | .3643 | .3665 | .3686 | .3708 | .3729 | .3749 | .3770 | .3790 | .3810 | .3830 |
| 1.2 | .3849 | .3869 | .3888 | .3907 | .3925 | .3944 | .3962 | .3980 | .3997 | .4015 |
| 1.3 | .4032 | .4049 | .4066 | .4082 | .4099 | .4115 | .4131 | .4147 | .4162 | .4177 |
| 1.4 | .4192 | .4207 | .4222 | .4236 | .4251 | .4265 | .4279 | .4292 | .4306 | .4319 |
| 1.5 | .4332 | .4345 | .4357 | .4370 | .4382 | .4394 | .4406 | .4418 | .4430 | .4441 |
| 1.6 | .4452 | .4463 | .4474 | .4485 | .4495 | .4505 | .4515 | .4525 | .4535 | .4545 |
| 1.7 | .4554 | .4564 | .4573 | .4582 | .4591 | .4599 | .4608 | .4616 | .4625 | .4633 |
| 1.8 | .4641 | .4649 | .4656 | .4664 | .4671 | .4678 | .4686 | .4693 | .4699 | .4706 |
| 1.9 | .4713 | .4719 | .4726 | .4732 | .4738 | .4744 | .4750 | .4756 | .4762 | .4767 |
| 2.0 | .4772 | .4778 | .4783 | .4788 | .4793 | .4798 | .4803 | .4808 | .4812 | .4817 |
| 2.1 | .4821 | .4826 | .4830 | .4834 | .4838 | .4842 | .4846 | .4850 | .4854 | .4857 |
| 2.2 | .4861 | .4865 | .4868 | .4871 | .4875 | .4878 | .4881 | .4884 | .4887 | .4890 |
| 2.3 | .4893 | .4896 | .4898 | .4901 | .4904 | .4906 | .4909 | .4911 | .4913 | .4916 |
| 2.4 | .4918 | .4920 | .4922 | .4925 | .4927 | .4929 | .4931 | .4932 | .4934 | .4936 |
| 2.5 | .4938 | .4940 | .4941 | .4943 | .4945 | .4946 | .4948 | .4949 | .4951 | .4952 |
| 2.6 | .4953 | .4955 | .4956 | .4957 | .4959 | .4960 | .4961 | .4962 | .4963 | .4964 |
| 2.7 | .4965 | .4966 | .4967 | .4968 | .4969 | .4970 | .4971 | .4972 | .4973 | .4974 |
| 2.8 | .4974 | .4975 | .4976 | .4977 | .4977 | .4978 | .4979 | .4980 | .4980 | .4981 |
| 2.9 | .4981 | .4982 | .4983 | .4983 | .4984 | .4984 | .4985 | .4985 | .4986 | .4986 |
| 3.0 | .49865 | .4987 | .4987 | .4988 | .4988 | .4989 | .4989 | .4989 | .4990 | .4990 |
| 3.1 | .49903 | .4991 | .4991 | .4991 | .4992 | .4992 | .4992 | .4992 | .4993 | .4993 |
| 3.2 | .49931 | .4993 | .4994 | .4994 | .4994 | .4994 | .4994 | .4995 | .4995 | .4995 |
| 3.3 | .49952 | .4995 | .4995 | .4996 | .4996 | .4996 | .4996 | .4996 | .4996 | .4997 |
| 3.4 | .49966 | .4997 | .4997 | .4997 | .4997 | .4997 | .4997 | .4997 | .4997 | .4998 |
| 3.5 | .49977 |