The Ponzo Illusion

Probability Distributions and z Scores

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Getting Started	General Instructions \| Introduction to Your Study \| Experimental Design \| Stating a Hypothesis
Descriptive Statistics	Histograms \| Central Tendency \| Standard Deviation \| Confidence Intervals
Comparing Two Samples	Samples and Populations \| Choosing a T-Test \| Independent T-Test \| P-Values and T-Tables
Important Concepts	The Normal Distribution \| Z Scores \| Probability Distributions
Levels	You are currently on Probability Distributions at level 3. Level 1 \| Level 2 \| Level 3
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Explanation

When The Distribution is Not Normal
Z-tables are used to convert z-scores to probabilites. They assume that the data from which the z-scores are calculated is normally distributed. You can see this fact in the structure of the z-table:

A normal distribution is symmetrical and z-tables usually only present one half of the distribution, leaving you to subtract from 1 to find the proportion in the other direction. This would not work for any distribution that is not symmetrical.
The entry in the z-table for z = 0 is 0.5, reflecting that the mean of a normal distribution is at its centre.
The pattern of values in the z-table follow the bell shape of the normal distribution.

If your data is not normal, the z-table will give you misleading answers.

Exploration

The game below allows you to draw frequency histograms and compare them to a normal distribution. To make any bar higher or lower, simply click on the chart at the height you want. The total number of data points does not change, so the chart re-draws itself each time you make a change.

The game will draw a red line indicating the shape of the normal distribution closest to your distribution. When you use z-tables, you are assuming that this is the shape of the distribution of your data. The game reports the percentage of your data (as shown by the blue histogram) to the right of three different values. They are the mean (which has a z-score of 0), one standard deviation above the mean (z=1) and two standard deviations above the mean (z=2). The numbers in brackets tell you the actual values of the mean and the points one and two standard deviations from the mean.
When you assume that data is normally distributed, you assume that the percentages of data above each of these points are as follows:
50% of the data lies above the mean
16% of the data lies more than one standard deviation above the mean
2% of the data lies more than two standard deviations above the mean
Try the following and see how different the normal curve is from the actual histogram:
Make the distribution flat;
Make the distribution skewed left or right;
Make the distribution multi-modal;
Try to make the percentage of data above the mean as high as you can.
( You need to enable Java to see this applet. )

Application

Now we can look at your data and see how well it matches the calculations made from a z-table. The frequency histogram for estimation error for the present sample is shown below.

The mean of estimation error for the present sample is 6.1 and the standard deviation is 1.86. There are 20 values in this data.

20 values are below the mean. There is a calculator below to help you answer these questions and a help topic on calculating percentages above.
What percentage, to the nearest whole number (0 to 100), of your data lie below the mean?
Z-tables say that half (50%) of the data in a normal distribution lie below the mean. How many values would you expect to find below the mean in your data if it were normally distributed?
Enter your answer as a whole number:
19 values in your data lie more than one standard deviation from the mean. What is that as a percentage (to the nearest whole number)?
Z-tables say that 34% of the data in a normal distribution lie more than one standard deviation from the mean. How many values would you expect to find in that range in your data if it were normally distributed?
Enter your answer as a whole number:
Calculator
( You need to enable Java to see this applet. )
Help Think for a moment about whether the distribution of your data is sufficiently normal for the probabilities from a z-table to be of use. If not, think about whether this is because the population from which your sample is drawn is not normally distributed or because your sample is too small to capture the distribution of the population.

Probability Distributions