## Probability Distributions and z Scores

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### Explanation

From Z-Scores to Probabilities
When we studied z-scores, we saw that you could represent the distance between a value and the mean of a sample in terms of how many standard deviations apart they are. This is called a z-score and it allows us to say how many standard deviations above or below the mean any value lies. Z-scores can also be used to see how far apart any two values are, for example there might be 1 standard deviation between my height and yours.

It would be even more useful if you could know what proportion of a sample or a population lay between the mean and a given value. You might also want to know what proportion of a sample or population lies above or below a given value. Converting from z-scores to proportions when the underlying distribution is normal is simple and this page shows how it is done.

If you know the proportion of a population that lies in a certain range, then you know the probability of a randomly selected value falling into that range. For example, if I knew that 10% of all men were taller than 2 meters, I would know that if I picked a man at random, the probability of him being taller than 2 meters would be 0.1. If this is new to you, there is a help section on probability below. Many statistical techniques require their results to be reported in terms of probabilities so the ability to convert from z-scores to probabilities is very useful and widely used.

Converting From Z-Scores to Probabilities
Converting from z-scores to probabilities is done using z-tables, which we will show you at level two of this topic. Z-tables tell you the proportion of data that is below a given z-score in a normally distributed population.

Here is an example. Let's say that the mean of a sample of people's height is 180cm, with a standard deviation of 30cm and your own height is 190cm. The z score for 190cm (your height) is 0.33 (see the section on z-scores for more on this) so you are a third of a standard deviation above the mean. Looking up 0.33 in a z-table reveals that 63% of people in a normal distribution fall below (are shorter than) you. Clearly, this only works if the data is normally distributed.

### Exploration

Converting from z-scores to probabilities using z-tables relies on the data being normally distributed. This is because the z-tables are simply a long list of the proportion of a normal distribution that appears below each z-score.

The game below illustrates this point. It shows a normal distribution on a chart where the horizontal axis shows the number of standard deviations away from the mean. In other words, z-scores are shown along the x-axis. The mean of the distribution is at 0 on the x-axis, which is where the curve is at its highest.

There are two boxes into which you can enter a z-score. When you click 'Go' the chart will show the percentage of the data that falls between the chosen z-scores.

Z tables tell you the proportion of a normal distribution below a given point (denoted by a z-score), so to find the proportion above a point, you simply subtract the value in the z table from 1. To find the area between two points, you must find the area below the highest and then subtract the area below the lowest.

What is the total area covered by 2 standard deviations from the mean in both directions?
Is the area between -1 and 0 the same as the area between 0 and 1?
Is the area between 0 and 1 the same as the area between 1 and 2?

( You need to enable Java to see this applet. )

Here are a few other challenges for you:

• Find a range of z scores that covers 99% of the distribution
• Find a range of z scores that covers 50% of the distribution
• Find a range of z scores that covers 95% of the distribution

### Application

Here are a few examples from your own data to test your understanding.

Looking at items correctly recalled in the 100 msecs sample, a value of 15.817 taken from this data has a z score of 1, which has a corresponding value of 0.841 in a z score table.
Is 15.817 above or below the sample mean?
How many standard deviations away from the mean is the value 15.817?
What proportion of the data in your sample falls below 15.817?
What proportion of the data in your sample falls above 15.817?
Finally, a value of 8.15 in this data produces a z score of -3. Looking this up in a z table tells you that 0.999 of the data lies above this point.
How often would you expect to see a value lower than this?
Calculator
( You need to enable Java to see this applet. )
Help

 Probability Distributions