# The Standard Normal Distribution Table

## Calculating the Probability of Values to the Left of a Z-Score on a Bell Curve

Normal distributions arise throughout the subject of statistics, and one way to perform calculations with this type of distribution is to use a table of values known as the standard normal distribution table in order to quickly calculate the probability a value occurring below the bell curve of any given data set whose z-scores fall within the range of this table.

The table found below is a compilation of areas from the standard normal distribution, more commonly known as a bell curve, which provides the area of the region located under the bell curve and to the left of a given z-score to represent probabilities of occurrence in a given population.

Anytime that a normal distribution is being used, a table such as this one can be consulted to perform important calculations. In order to properly use this for calculations, though, one must begin with the value of your z-score rounded to the nearest hundredth then find the appropriate entry in the table by reading down the first column for the ones and tenths places of your number and along the top row for the hundredths place.

### Standard Normal Distribution Table

The following table gives the proportion of the standard normal distribution to the left of a z-score. Remember that data values on the left represent the nearest tenth and those on the top represent values to the nearest hundredth.

 z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5 0.504 0.508 0.512 0.516 0.52 0.524 0.528 0.532 0.536 0.1 0.54 0.544 0.548 0.552 0.556 0.56 0.564 0.568 0.571 0.575 0.2 0.58 0.583 0.587 0.591 0.595 0.599 0.603 0.606 0.61 0.614 0.3 0.618 0.622 0.626 0.63 0.633 0.637 0.641 0.644 0.648 0.652 0.4 0.655 0.659 0.663 0.666 0.67 0.674 0.677 0.681 0.684 0.688 0.5 0.692 0.695 0.699 0.702 0.705 0.709 0.712 0.716 0.719 0.722 0.6 0.726 0.729 0.732 0.736 0.74 0.742 0.745 0.749 0.752 0.755 0.7 0.758 0.761 0.764 0.767 0.77 0.773 0.776 0.779 0.782 0.785 0.8 0.788 0.791 0.794 0.797 0.8 0.802 0.805 0.808 0.811 0.813 0.9 0.816 0.819 0.821 0.824 0.826 0.829 0.832 0.834 0.837 0.839 1.0 0.841 0.844 0.846 0.849 0.851 0.853 0.855 0.858 0.85 0.862 1.1 0.864 0.867 0.869 0.871 0.873 0.875 0.877 0.879 0.881 0.883 1.2 0.885 0.887 0.889 0.891 0.893 0.894 0.896 0.898 0.9 0.902 1.3 0.903 0.905 0.907 0.908 0.91 0.912 0.913 0.915 0.916 0.918 1.4 0.919 0.921 0.922 0.924 0.925 0.927 0.928 0.929 0.931 0.932 1.5 0.933 0.935 0.936 0.937 0.938 0.939 0.941 0.942 0.943 0.944 1.6 0.945 0.946 0.947 0.948 0.95 0.951 0.952 0.953 0.954 0.955 1.7 0.955 0.956 0.957 0.958 0.959 0.96 0.961 0.962 0.963 0.963 1.8 0.964 0.965 0.966 0.966 0.967 0.968 0.969 0.969 0.97 0.971 1.9 0.971 0.972 0.973 0.973 0.974 0.974 0.975 0.976 0.976 0.977 2.0 0.977 0.978 0.978 0.979 0.979 0.98 0.98 0.981 0.981 0.982 2.1 0.982 0.983 0.983 0.983 0.984 0.984 0.985 0.985 0.985 0.986 2.2 0.986 0.986 0.987 0.987 0.988 0.988 0.988 0.988 0.989 0.989 2.3 0.989 0.99 0.99 0.99 0.99 0.991 0.991 0.991 0.991 0.992 2.4 0.992 0.992 0.992 0.993 0.993 0.993 0.993 0.993 0.993 0.994 2.5 0.994 0.994 0.994 0.994 0.995 0.995 0.995 0.995 0.995 0.995 2.6 0.995 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 2.7 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997

### An Example for Using the Table to Calculate Normal Distribution

In order to properly use the above table, it's important to understand how it functions. Take for example a z-score of 1.67. One would split this number into 1.6  and .07, which provides a number to the nearest tenth (1.6) and one to the nearest hundredth (.07).

A statistician would then locate 1.6 on the left column then locate .07 on the top row. These two values meet at one point on the table and yield the result of .953, which can then be interpreted as a percentage which defines the area under the bell curve that is to the left of z=1.67.

In this instance, the normal distribution is 95.3% because 95.3% of the area below the bell curve is to the left of the z-score of 1.67.

### Negative z-Scores and Proportions

The table may also be used to find the areas to the left of a negative z-score. To do this, drop the negative sign and look for the appropriate entry in the table. After locating the area, subtract .5 to adjust for the fact that z is a negative value.  This works because this table is symmetric about the y-axis.

Another use of this table is to start with a proportion and the find a z-score.  For example, we could ask for a randomly distributed variable, what z-score denotes the point of the top 10% of the distribution?

Look in the table and find the value that is closest to 90%, or 0.9. This occurs in the row that has 1.2 and the column of 0.08.  This means that for z = 1.28 or more, we have the top 10% of the distribution and the other 90% of the distribution are below 1.28.

Sometimes in this situation, we may need to change the z score into a random variable with a normal distribution. For this, we would use the formula for z-scores.