production date 2/5/00

z Scores

Table of Contents Objectives
What are Z scores? Learn about z scores.
A Real Example Calculate z score with real values.
Finding Exact z Score Values Using an Applet, Table, or Statlets.
Computer Project 13 Using Statlets to find normal curve tail areas.
Computer Project 14 Using Statlets to find normal curve critical values.
Finding Raw Scores Learn to convert a z score to a raw score.
Transformation Rules Learn how to calculate means and standard deviations if data are simply transformed.
Additional Information Discover other distributions
Questions/Test Take the End of Chapter Test
Report Send a Chapter Report to your Instructor


What are Z scores?

Z scores are very important position measures. Z scores use standard deviations that you learned to calculate in Chapter 5 in their formula. You will need to pay particular attention to the calculation and understanding of z scores.

Z scores tell how many standard deviations away from the mean a score resides. Z scores can be positive or negative. A positive z score indicates that the value is above the mean. A negative z score indicates that the value is below the mean. The two equations on the left are used to calculate z scores. The first equation is for populations. In the first equation the Greek letters mu and sigma stand for the population mean and standard deviation. Note that beside the score represented by the letter x, only population parameters are on the right side of the equation. The second equation is used for sample data. Note that the equivalent sample statistics are used in place of the population parameters found in the first equation.

The figure on the left illustrates a normal distribution . According to Chebychev's rule, in any distribution the proportion of scores between the mean and k standard deviations contains at least 1-1/k2 scores. Doing the math, Chebychev's rule would give at least 75% of the scores between the mean and 2 standard deviations (±2 z), and 89% of the scores would reside between the mean and ±3 z. However. knowing that this is a normal distribution, gives us further information. In a normal distribution approximately 68% of the scores reside between the mean and ±1 z, approximately 95% of the scores reside between the mean and ±2 z and approximately 99% of the scores reside between the mean and ±3 z. This is known as the Empirical Rule.

In a normal distribution z scores can range outside of ±3. However, in this applet, the z score values are restricted and can only range from -3 to +3. When the applet is first drawn, the normal distribution is drawn in blue. The mean of the distribution is represented by a black line in the center. The area between -1 z and +1 z is filled in red. Below the distribution are several lines of text. The first line indicates that the positive z value is equal to 1. At this time, the red area extends one standard deviation, and therefore one z score above the mean. The next line indicates that the percentage of scores between the mean and the positive z value is 34.13 percent. The third line indicates that the percentage of scores in the normal distribution that are above the positive z value is 15.87. These values added together equal 50 percent. Half the scores (50%) are above the mean in a normal distribution. Lines four-six give the same information for the negative z value. The seventh line indicates that between these two z values 68.26 % of the scores can be found. According to the Empirical rule, in a normal distribution approximately 68% of all the scores will be between ±1 z. Click and drag on the bottom of the normal distribution to change the z values to ±2. Do the percentage between these two values correspond to the Empirical Rule? You may drag and positive z value between 0 and +3. Negative z values can be set between 0 and -3.

A Real Example    

[Image]Scores on intelligence tests (IQs) are normally distributed in children. IQs from the Wechsler intelligence tests are known to have means of 100 and standard deviations of 15. In almost all the states in the United States (Pennsylvania and Nebraska are exceptions) children can be labeled as mentally retarded if their IQ falls to 70 points or below. What is the maximum z score one could obtain on an intelligence test and still be considered to be mentally retarded? The z score calculation on the right provides the answer of -2.

If gifted children need IQs of 130 and above, what is the minimum z score on an intelligence test that a gifted child can obtain? Answer +2.

Using the normal curve Empirical Rule, what percentage of the children in the United States are thought to be neither retarded nor gifted? What percent have IQs between ±2 z?

What percentage of the children would be thought to be retarded?

What percentage of the children would be expected to be gifted in the typical school system?

For the answers to these questions view this answer page.

Finding Exact z score values    

When answering the above questions, you were using the Empirical Rule. If the distribution was not normal, you could use Chebychev's rule. The Empirical Rule's percentages were approximate. To find exact percentages between means and certain z scores, you will either need to be able to use z score tables found in almost every statistics text, use the z score lookup applet, or use the Plot/Probability Distributions procedure in Statlets.

Using A Table

This link opens a typical z score table in a new browser window. Open the z score table link, and look at the tabled values. The z score table contains 4 columns of z values and associated with each of the z values are 4 columns of p values. Looking at the first row you see a z value of 0 and an associated p value of .5. In a normal distribution the mean z is 0 and half (.5) of the scores are below the mean. This is what these values are telling you. In the next two columns you see a z value of 1 and an associated p value of 0.841. These two entries indicate that below a z value of 1 are 84.1% of the scores. The rest of the table is used in exactly the same manner. There are only positive z values listed, and the associated p values indicate the proportion of scores below the specified z value. Scrolling down close to the bottom of the table are two entries indicating that a z score of 1.96 has 97.5% of the scores below it.

Using the Z Score Applet

The z score lookup applet on the left calculates proportions for z scores to the third decimal point of accuracy. To use it enter a z score value (typically a number between ±4 in the box. Then, click the Z proportions button. Enter a value of 1 and click the Z Proportions button. The applet indicates that the proportion of scores between the mean and the z value are 0.34099 (rounded 0.341). When we evaluated a z= 1 in the table section above, we were told that the proportion of scores below a z of 1 was 0.841. The difference in these two procedures is that using the table, the proportion of .5 below the mean in included, while in the applet it is not. In the applet, negative values can be entered, in the table, they are not directly available.
[Image]Using the applet, the first line of text below the calculate button displays the proportion of scores between the mean (0), and the provided z value. This line is colored red, and corresponds to the red area in the figure on the right. The second (blue) line displays the proportion of scores beyond the provided z value. This blue line corresponds to the blue area in the figure on the right.

Answer the following questions:

Use both the table, and the applet to answer the following:
  1. What is the percentage of scores between 0.5 Z and -0.5Z?
  2. What is the percentage of scores between -1.8 Z and +1.8Z?
  3. What is the percentage of scores above a Z of 1.21?
  4. What is the percentage of scores below a Z of 0.87?


For the answers to these questions, view this answer page.

Using Statlets

The Statlet's procedures for evaluating normal distributions, and z scores are found using the Plot/Probability Distributions menu items. If you have not previously read the material from the user manual for this section, or if you would like to review these procedures, please do so.

In the Real Example section above, it was noted that IQs are normally distributed, and have means of 100 and standard deviations of 15. If you apply for graduate school, you may be asked to take the Graduate Record Examination (GEE). This test has a mean of 500 and a standard deviation of 100 and is normally distributed. After selecting a Normal distribution using the Input tab, the means and standard deviations of both these distributions can be plotted using the PDF tab as illustrated in the figure below. Usually you will only want a plot of a single distribution, but using the Options button, you can enter up to five different distributions to be plotted.


Clicking the CDF tab creates cumulative density functions for both the distributions. This CDF plot is shown directly below. Notice how both normal distributions show the classic S-shaped CDF plot that is characteristic of a normal distribution. Because both plots are drawn using the same horizontal axis, and because the first graph has a much smaller standard deviation, it is a bit pushed together along the horizontal axis. It's CDF plot is still S-shaped. It may be a bit difficult to see because of the common axis.


While the Survivor, Log Survivor and Hazard outputs are not typically discussed in introductory material, the Tail Areas and Critical Values tabs are quite important. The figure shown directly below shows both the options, and the output from the Tail Areas tab for our two distributions. Remember that distribution 1 has a mean = 100 and a standard deviation = 15, while distribution 2 has a mean = 500 and a standard deviation = 100. Notice in the options input box, that the means (z= 0) and values for z=1 for both distributions have been entered. In the Lower Tail Area (<) section. For the value of 100 (mean on the first distribution, and a z = -4 in the second distribution, the value of .5 is given for distribution 1. This makes sense, 100 is at the mean of the first distribution, and 50% of the scores in that distribution lie below the mean. For the second distribution a value of 3.2E-500 is displayed. This is a number in scientific notation. The E-500 part indicates that the decimal point as shown should be moved back 500 places. So the value should be a decimal point, followed by 499 zeros and finally 32. This value is very close to zero. Again this makes sense, for the second distribution 100 converts to a z score equal to -4. Almost all the scores in a normal distribution are between plus and minus 3, so almost zero would be below a z score of -4.0.

The other values should be just as easy to interpret. In the second row, the value of 500 is the mean of the second distribution and 50% of the scores are indicated to be below it. However, 500 in the first distribution is a z value of 26.67 and virtually all the values would be below such a large z score. The output correctly indicates that a proportion of 1.00 (all of them) are below.


Finally, the Critical Values tab, displays the equivalent raw scores in distributions when provided with proportion above information in the options box. The figure directly below displays both the output, and the options used for the two distributions. Looking at the first tail area provided (.841) remember that this is approximately the proportion below a z value of 1.0. Notice that for both distributions the critical values provided are very near z values that would be 1.0. The second tail area provided (.5) is the proportion below a z equal to 0, or the mean. In both distributions, the mean is the critical value provided. The tail area of 0.025 should give low values in the distribution (only 2.5% of the scores would be lower) and indeed the reported values are well below the respective means. The last two tail areas provided (0.975 and 0.99) should give high values in the distribution. Again, the displayed values are quite high in their respective distributions.


Computer Project 13   

Many tests, including some quite important personality tests, are scored on what is called a t scale. T scores have means of 50 and standard deviations of 10. The Minnesota Multiphasic Personality Inventory (MMPI) is one such personality test. It has many subtests that are normally distributed with means of 50 and standard deviations equal to 10. Using Statlets Plot/Probability Distributions procedure, enter the characteristics of the MMPI, and then using the Tail Areas tab, calculate the percentage of scores below a raw score of 62.6. If your instructor requests, submit this project report.

Computer Project 14   

Computer project 14 can be done using the information from project 13. This time, however, use the Critical Values tab to answer the following question.
If a large sample of subjects took the MMPI, and you want to do extensive follow-up interviewing with all the subjects that scored in the top 10% on the Schizophrenia Scale, above what score would you start selecting your follow-up subjects? If your instructor requests submit this fourteenth project report.

Finding Raw Scores   

If you know a person's Z score and the mean and standard deviation, you can easily find their raw score in the distribution.

Graduate Record Examination (GRE) scores have means equal to 500 and standard deviations of 100. If a person receives a Z score on the GRE of 1.45, what would their raw score be?
Hint: use the Z score formula:
The answer can be found on this page.

Transformation Rules   

Often new data sets are formed by transforming old data sets. A constant may be added to, subtracted from, multiplied or divided into an old data set to form the new data set. If the mean and standard deviation for the old data set were previously calculated, these rules will help in the calculation of the mean and standard deviation for the new data.

Adding a constant

  1. The new mean is equal to the old mean + the constant.
  2. The new standard deviation is identical to the old standard deviation.

Subtracting a constant

  1. The new mean is equal to the old mean - the constant.
  2. The new standard deviation is identical to the old standard deviation

Multiplying by a constant

  1. The new mean is equal to the old mean times the constant.
  2. The new standard deviation is equal to the old standard deviation times the absolute value of the constant.

Dividing by a constant

  1. The new mean is equal to the old mean divided by the constant.
  2. The new standard deviation is equal to the old standard deviation divided by the absolute value of the constant.

Additional Information    

Besides the normal distribution, Statlets provides percentage and probability information for 23 other distributions. In many introductory statistics textbooks, the binomial distribution is discussed before the normal distribution. Use the Plot/Probability Distributions procedure, but instead of using the normal distribution, choose the binomial distribution. Produce the PDF and CDF plots. How different are the shapes of these plots from those done using the normal distribution?

Questions/Test   

This link allows you to take a computer scored end-of-chapter test. If your instructor requests to see the results of this examination, you can either copy and e-mail or print the feedback you will receive immediately after taking the test.

Report   

Please send a report indicating your understanding of this chapter to your instructor. You will need to know both your and your instructor's e-mail addresses.