## Probability Distributions and z Scores

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### Explanation

Converting z Scores to Probabilities With Z Tables
Once you have calculated a z score, you can answer questions concerning the probability of new measurements taking certain values. For example, if you planned to sell shoes in a shop and you knew that shoe sizes were normally distributed, you could work out how many customers for each shoe size you would be likely to see.

There is a formula for converting from z scores to probabilities, but it is rarely used by hand. Most people use either computer software or z score tables to calculate probabilities from z scores.

Here are the first few lines of a z score table.
 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 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.591 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

The bold figures down the left hand side represent the first decimal place in the value of z that you wish to look up and the values across the top represent the second decimal place of the value. This allows you to look up values up to two decimal places without needing a very long table. In the extract above, for example, the z score 0.17 is highlighted - the corresponding value in the table is 0.5675 (shown in bold).

What do the values that you look up represent? In the table above, the values represent the proportion of a standard normal distribution that falls to the left of (that is, below) the given z score. So in our highlighted example, we see that 0.5675 (about 57%) of a normal distribution lies below the point that is 0.17 standard deviations above the mean.

Some tables show negative z scores too. Some (like ours above) do not. In such cases, you must look up the positive value (so if you have a z score of -2, look up 2) and subtract the value you get from 1. Returning to our example again, if we wanted to look up -0.17, we would find the value for 0.17 (0.5675) and subtract that from 1, giving 0.4325. That tells us that 43% of the values in a normal distribution lie below the point 0.17 standard deviations below the mean.

• To find the proportion above a given z score, simply look up the z score in the table and subtract the value you find from 1. This works because all the probabilities must add up to one, so the proportion below a point plus the proportion above a point must sum to one.
• To find the proportion between two z scores, look up both z scores and subtract the value from the table for the lowest from that for the highest. For example, looking up z = 1.96 in the table gives 0.975. The value for z = -1.96 is 0.025. Subtract 0.025 from 0.975 and you get 0.95. So 95% of a standard normal distribution lies between -1.96 and +1.96 of the mean.

### Exploration

This section will let you see how some calculations are made using z-tables and z-scores.

 Z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 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 0.567 0.571 0.575 0.2 0.579 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 0.629 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 0.691 0.695 0.698 0.702 0.705 0.709 0.712 0.716 0.719 0.722 0.6 0.726 0.729 0.732 0.736 0.739 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 0.831 0.834 0.836 0.839 1 0.841 0.844 0.846 0.848 0.851 0.853 0.855 0.858 0.86 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 0.901 1.3 0.903 0.905 0.907 0.908 0.91 0.911 0.913 0.915 0.916 0.918 1.4 0.919 0.921 0.922 0.924 0.925 0.926 0.928 0.929 0.931 0.932 1.5 0.933 0.934 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 0.949 0.951 0.952 0.953 0.954 0.954 1.7 0.955 0.956 0.957 0.958 0.959 0.96 0.961 0.962 0.962 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 0.977 0.978 0.978 0.979 0.979 0.98 0.98 0.981 0.981 0.982
Enter a z-score between -2 and 2 in the box below and click one of the buttons. The prgram will calculate the proportion of data above or below your chosen z-score, and explain how the calculation is done.

Enter an example z-score here
Proportion above the chosen z-score
Proportion below the chosen z-score
Proportion between the chosen z-score and the mean

Go back to the level one game to see how these values cover a normal distribution plotted on a histogram.

### Application

Below we present some z-scores calculated from your data, looking at items correctly recalled in the 100 msecs sample.

Use the z-table below to answer the following questions. Enter values to 3 decimal places.

Remember to subtract the value you find from one when the z-score is negative.

 What proportion of your data would you expect to find below 13,which has a z-score of -0.47? What proportion of your data would you expect to find above 15,which has a z-score of 0.57? What proportion of your data would you expect to find between 14,which has a z-score of 0.05 and 16, which has a z-score of 1.1?

The z table is shown below.

 Z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 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 0.567 0.571 0.575 0.2 0.579 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 0.629 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 0.691 0.695 0.698 0.702 0.705 0.709 0.712 0.716 0.719 0.722 0.6 0.726 0.729 0.732 0.736 0.739 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 0.831 0.834 0.836 0.839 1 0.841 0.844 0.846 0.848 0.851 0.853 0.855 0.858 0.86 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 0.901 1.3 0.903 0.905 0.907 0.908 0.91 0.911 0.913 0.915 0.916 0.918 1.4 0.919 0.921 0.922 0.924 0.925 0.926 0.928 0.929 0.931 0.932 1.5 0.933 0.934 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 0.949 0.951 0.952 0.953 0.954 0.954 1.7 0.955 0.956 0.957 0.958 0.959 0.96 0.961 0.962 0.962 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 0.977 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 0.987 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 0.992 0.993 0.993 0.993 0.993 0.993 0.994 2.5 0.994 0.994 0.994 0.994 0.994 0.995 0.995 0.995 0.995 0.995 2.6 0.995 0.995 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 2.8 0.997 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 2.9 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.999 0.999 0.999 3 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 3.1 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 3.2 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 3.3 1 1 1 1 1 1 1 1 1 1 3.4 1 1 1 1 1 1 1 1 1 1 3.5 1 1 1 1 1 1 1 1 1 1 3.6 1 1 1 1 1 1 1 1 1 1 3.7 1 1 1 1 1 1 1 1 1 1 3.8 1 1 1 1 1 1 1 1 1 1 3.9 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1
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 Probability Distributions