Lesson Z-SCORES AND THE Z-TABLE CALCULATOR
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This lesson reviews the concept of Z-Scores. It includes the use of the Z-Table Calculator. The Z-Table Calculator is one of the tools and tutorials provided on the <a href = "http://davidmlane.com/hyperstat/" target = "_blank">http://davidmlane.com/hyperstat</a> website. The Z-Table Calculator allows you to find the percentage of the total normal distribution that a particular data point value will be above, below, between, or outside of. It also allows you to find the data point values that are a specified percentage above, below, between, or outside of those data point values. ACCESSING THE Z-TABLE CALCULATOR The Z-Table calculator can be accessed by clicking on the hyperlink titled <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> If you are using Firefox and are having trouble accessing this tool, or connecting to any website that loads applets to your computer, then the problem may be caused by having the wrong version of firefox installed on your computer and / or having the wrong version of java installed on your computer. To obtain the latest version of java, you would need to go to <a href = "http://www.java.com/en/" target = "_blank">www.java.com</a> and download the latest java version on your computer. That website will allow you to test your computer beforehand to determine if you have the latest version of Java installed. To obtain the latest version of Firefox, you would need to go to <a href = "http://www.mozilla.com/en-US/" target = "_blank">www.mozilla.com</a> and download the latest Firefox on your computer. You should have no trouble using Internet Explorer, but if you are experiencing problems, check with Microsoft at <a href = "http://www.microsoft.com/en/us/default.aspx" target = "_blank">www.microsoft.com</a> MEAN AND STANDARD DEVIATION The Mean is the average of all your data divided by the number of data items. The Standard Deviation is a measure of the dispersion of your data about the mean. Both these are inputs used by the Z-Table Calculator. NORMAL DISTRIBUTION The Z-Table calculator assumes your data is normally distributed. Normally distributed data has the same mean, median, and mode. the Median is the data point value where 50% of your data points are below and 50% of your data points are above. The Mode is the data point value that the majority of your data points are clustered around. In a normal distribution, 50% of the data points are below the mean and 50% of the data points are above the mean. In a normal distribution, the Mean is the same as the Median. In a normal distribution, the most frequently occurring data point Value is the Mean. In a normal distribution, the majority of the data points are clustered around the mean. GROUPED DATA POINTS Sometimes data is combined into groups to minimize the number of data point values in the distribution. A group would consist of a range of data values. group 1 might be data values 0 to 100 group 2 might be data values 101 to 200 etc. Grouping allows large numbers of data values to be compressed into smaller groups of data values. A specified value of the group may be chosen to represent the group. If the midpoint of the range is used, then group 1 above would be identified by the data value of 50; group 2 by the data value of 150, etc. The calculation of the mean, median, and mode of the distribution would take into consideration the number of data points within each group. USING THE Z-TABLE CALCULATOR Click on the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to access the calculator. It will open in a separate window so you can keep it open and readily accessible when you need it. You will see two graphs. The top graph allows you to enter the mean and the standard deviation and then to select the criteria you are interest in. For example, enter a mean of "100" and a standard deviation of "15". Select "between" and enter 85 and 115. It will tell you that approximately 68.3% of the population will have scores between 85 and 115. Select "between" and enter 70 and 130. It will tell you that approximately 95.5% of the population will have scores between 70 and 130. Select "between" and enter 55 and 145. It will tell you that approximately 99.7% of the population will have scores between 55 and 145. The results are in ratio form. To translate to percentage form, simply multiply the results by 100% As an example, .997 ratio is equal to 99.7%. Z-SCORE The Z-Score is a normalized distribution. The Mean is always 0. the Standard Deviation is always 1. Our example above assumed a mean of 100 and a standard deviation of 15. We took 3 measurements. measurement 1 was a score of 85 to 115. measurement 2 was a score of 70 to 130. measurement 3 was a score of 55 to 145. Since the mean was 100 and the standard deviation was 15, measurement 1 referenced 1 standard deviation about the mean. 85 was 1 standard deviation below the mean and 115 was one standard deviation above the mean. Go to the table again (it should still be there in a different window unless you closed it. If you closed it, just click on it again and it will open). Enter a mean of 0 and a standard deviation of 1. Select "between" and enter -1 and 1. It will tell you that approximately 68.3 percent of the population will have scores that are between 1 standard deviation below the mean and 1 standard deviation above the mean. This is exactly the same percentages it told you when you entered a mean of 100 and a standard deviation of 15 and then selected "between" and then entered 85 to 115. If you enter a mean of 0 and a standard deviation of 1 and select "between" and enter -2 to 2, you will see that the percentage is 95.5. This is the same as when you entered a mean of 100 and a standard deviation of 15 and selected "between" and entered 70 to 130. If you enter a mean of 0 and a standard deviation of 1 and select "between" and enter -3 to 3, you will see that the percentage is 99.7. This is the same as when you entered a mean of 100 and a standard deviation of 15 and selected "between" and entered 55 to 145. Every distribution with a mean and a standard deviation can be converted to a Z-score with a mean of 0 and a standard deviation of 1. By converting a data point value to a Z-score, you can determine how many standard deviations from the mean that data point value is. CONVERTING FROM MEAN AND STANDARD DEVIATION TO Z-SCORE. Given the mean and the standard deviation, you would convert to Z-score as follows: Z-Score = [data point value minus the mean] divided by the standard deviation. Let x = the measurement. Let m = the mean. Let s = the standard deviation. Let z = the z-score. Formula would be: z = (x-m)/s In our example, 100 was the mean and 15 was the standard deviation. Given a measurement of 115, then the z-score would be equal to (115-100)/15 = 15/15 = 1 Given a measurement of 85, then the z-score would be equal to (85-100)/15 = -15/15 = -1 IMPACT OF DIFFERENT STANDARD DEVIATIONS ON THE Z-SCORE The z-score will always have 99.7 percent of the population within 3 standard deviations about the mean. The actual data point values involved, however, become very different depending on the standard deviation. Assume the following: mean of 100 and standard deviation of 5 mean of 100 and standard deviation of 15 mean of 100 and standard deviation of 30 With a standard deviation of 5, 99.7 percent of the population will score between 85 and 115. With a standard deviation of 15, 99.7 percent of the population will score between 55 and 145. With a standard deviation of 30, 99.7 percent of the population will score between 10 and 190. With all of these measurements, the Z-score would be -3 to +3. That's because the z-score is normalized. All means become 0 and all standard deviations become 1. If you had a score of 115 on a test and you wanted to know how you did relative to the population, you would need to know the mean and standard deviation of the test results to determine that. Alternatively, you would need to know the Z-score, but in order to know the z-score you would need to know the mean and the standard deviation of the test results anyway. Assume you achieved a score of 130. Assume the mean of the test results was 100. Assume the standard deviation of the test results was 5. The Z Table Calculator would tell you that your score of 130 was greater than 100% of the population. Now assume the standard deviation of the test results was 15. The Z-Table Calculator would tell you that your score of 130 was greater than 97.7% of the population. Now assume the standard deviation of the test results was 30. The Z-Table Calculator would tell you that your score of 130 was greater than 84% of the population. It's the same score and the same mean, but the standard deviation is different yielding different percentages. Z-TABLE CALCULATOR AND SPECIFIED PERCENTAGES If you know the mean and the standard deviation and you want to know what data measurements can be expected to fall within specified percentages of the population, then you would use the bottom graph of the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> For example, if the mean is 100 and the standard deviation is 15 and you want to know what score you would have to have in order that 75 percent of the population would get scores below that, you would do the following: Enter a mean of 100 and a standard deviation of 15 in the bottom graph. Enter a shaded area of .75 in the bottom graph. Select "below" and it would tell you that a score of 110.1173 would be required in order to have 75% of the scores of the population be below that. If you wanted to know what scores would be required in order to have 95.45% of them be within the specified limits, then you would select "between" and you would enter .9545 in the Shaded Area and it would tell you that in order for you to have 95.45% of your scores fall between the specified values, those values would have to be from 70 to 130. The two graphs work interchangeable. You can enter a mean of 0 and a standard deviation of 1 in the top graph and then select between and it will tell you the percentage of the total distribution that lies between those values. You can then go to the bottom graph and enter a mean of 100 and a standard deviation of 15 and enter a shaded area of whatever you just found out from the top graph and then select between and it will tell you the values required in order that the specified percentage of the population fall between those values. Example: Go to the top graph and enter a mean of 0 and a standard deviation of 1 and then select "between" and enter -1.5 to 2 It will tell you that the percentage of the total distribution between those values is .910443 Now go to the bottom graph and enter a mean of 100 and a standard deviation of 20 and enter a shaded area of .910443 and select "between". It will tell you that in order to have 91.0443 percent of the population between selected values, those values would have to be 66.0452 and 133.9548.
