Lesson MANUALLY USING A Z-TABLE
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This lesson provides some instructions on how to manually use a Z-Table. It incorporates the use of the Z-Table Calculator to confirm that you did it correctly. The Z-Table Calculator was introduced in the lesson on the Z-Table and using the Z-Table Calculator. If you don't have to look into a z-table manually, then use the z-table calculator. It makes the job much easier plus it provides you with a visual representation of the area that is selected from your criteria. Use of the z-tool calculator is covered in the lesson on the z-table and the z-table calculator. Also included in this lesson is a problem that was solved using the Z-table. Go to the bottom of this lesson to get directly to that if you wish to see a practical application. REFERENCES <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> <a href = "http://lilt.ilstu.edu/dasacke/eco148/ZTable.htm" target = "_blank">Z-Table Number 1</a> <a href = "http://www.sjsu.edu/faculty/gerstman/EpiInfo/z-table.htm" target = "_blank">Z-Table Number 2</a> <a href = "http://www.intmath.com/Counting-probability/z-table.php" target = "_blank">Z-Table Number 3</a> DEFINITION OF THE Z-SCORE The Z-Score is a normalized score from a set of distribution data that tells you how many standard deviations your score is from the mean. You need the z-score in order to use the z-table. CONVERSION OF RAW SCORE TO Z-SCORE The Z-Score is equal to [the data measurement minus the mean] divided by the standard deviation. The formula would look like this: z = (x-m)/s where: z = z-score x = data measurement m = mean s = standard deviation Example: The mean of your normal distribution is equal to 100. The standard deviation of your normal distribution is equal to 15. Your raw score is 130. Your z-score is equal to (130-100)/15 = 30/15 = 2. This means that a score of 130 is 2 standard deviations above the mean. Another example: Your mean is 100 Your standard deviation is 15 Your raw score is 70 Your z-score is equal to (70-100)/15 = -30/15 = -2 This means that a score of 70 is 2 standard deviations below the mean. Once you have the z-score, you would look into the z-table and determine the area under the normal distribution curve that is to the left of your data. You can then use this information to determine the area under the normal distribution curve that is: To the left of your data point. To the right of your data point. Between two of your data points. Outside of two of your data points. A data point is one measurement of data. If your mean is 100 and your standard deviation is 15 and one of your data measurements is 130, then 130 is one of your data points. NORMAL DISTRIBUTION CURVE USED BY THE Z-TABLE The normal distribution curve used by the Z-Table is presented in the link titled <a href = "http://www.netmba.com/statistics/distribution/normal/" target = "_blank">The Normal Distribution from NetMBA</a> In a normal distribution, the mean, the median, and the mode are the same. The mean is the average of the distribution. The median is the midpoint of the distribution. The mode is the point where most of the data in the distribution is clustered around. As can be seen from the referenced table, they all are clustered around the middle of the normal distribution curve. In a normal distribution: Approximately 68.3 percent of the data is within plus or minus 1 standard deviation from the mean. Approximately 95.5 percent of the data is within plus or minus 2 standard deviations from the mean. Approximately 99.7 percent of the data is within plus or minus 3 standard deviations from the mean. If you look at the link above from NetMBA, the standard deviations from the mean are represented by the greek letter called sigma. That letter is at the bottom of the graph on the horizontal line. It describes how many standard deviations that point is away from the mean. Sigma looks like the second symbol under the category of population parameters in the following <a href = "http://stattrek.com/Lesson1/Notation.aspx" target = "_blank">Statistics Notation Guide by Statrek</a>. The population parameters is the third major category in red counting down from the top of the middle of the page. Sigma refers to the standard deviation of a population. In this lesson we talk about population a little loosely than the formal definition of it. The formal definition talks about population as the set of data that samples are drawn from. In this lesson we talk about population as the set of data you are looking at, which could be the full set of data from which samples are drawn from (the real population), or it could be a sample of that data (the sample population). CONFIDENCE INTERVALS Confidence Intervals are lower and upper limits that are set. They are called confidence intervals because you can determine, with confidence, the proportion of the populatation that will lie within the boundaries of the limits imposed. A 95% confidence interval gives you confidence that 95% of your data will fall within those boundaries. A 99% confidence interval gives you confidence that 99% of your data will fall within those boundaries. This assumes your distribution is normal. A 99% confidence interval will be wider than a 95% confidence interval. Your data will either be inside the confidence interval limits or outside the confidence interval limits. The most commonly used confidence intervals are: 90% 95% 99% Once you decide on the confidence interval to use, you then have to decide whether your test is a two-tailed test or a one-tailed test. This will impact how the confidence interval is set. A two-tailed 95% Confidence Interval will have 2.5% of the distribution outside the limits on each end. A one-tailed 95% Confidence Interval will have 5% of the distribution outside the limits on either the high end of the distribution or the low end of the distribution, depending on which tail you are interested in. It will only apply to one end, not both. This can be seen visually in the following link titled <a href = "http://theo.x10hosting.com/examples/Statistics/ZTableConfidenceIntervals.html" target = "_blank">Confidence Intervals</a> In the following examples, it is assumed that we have already found the z-score of interest. Our data is a distribution with a mean of 100 and a standard deviation of 15 Our Scores of Interest are 70 and 130. The z-score for 70 would be equal to (70-100)/15 = -30/15 = -2 The z-score for 130 would be equal to (130-100)/15 = 30/15 = 2 FINDING A Z-SCORE IN A Z-TABLE. The procedure for finding a z-score in a z-table is the same, regardless of the table. The tables are divided into 11 columns. The first column contains the tens digit of the z-score. The second through eleventh column contains the hundreds digit of the z-score. If your z-score is 2, you would: Scroll down the left side of the table until you see a z-score of 2.0 Scroll to the right until you see a column with a heading of .00 or 0.00. The entry in that column on your row is the z-table entry you are looking for. If your z-score is 1.37, you would: Scroll down the left side of the table until you see a z-score of 1.3. Scroll to the right until you see a column with a heading of .07 or 0.07. The entry in that column on your row is the z-table entry you are looking for. If your z-score is .5, you would: Scroll down the left side of the table until you see a z-score of 0.5. Scroll to the right until you a column with a heading of .00 or 0.00. The entry in that column on your row is the z-table entry you are looking for. Z-TABLE The Z-Table is a normalized representation of a normal distribution of data. 100% of the population is represented by 1. Any proportion of the population less than that will be a fraction. .5 represents 50% of the population. .25 represents 25% of the population. .75 represents 75% of the population. When you find a z-score in a z-table, the entry for that z-score tells you the proportion of the population that is to the left (below) that z-score. Some tables will only tell you the proportion of the population to the left of that z-score to the mean only. The proportion from the mean to the left end of the distribution is missing. That represents .5 proportion of the population. If you have that kind of table, you will need to add .5 to the entry to get the proportion of the population that is to the left of that z-score. You will see how this works as we go through three example of three different types of tables. At the end, you will see a pictorial representation of the procedure used for each of the 3 z-tables shown in this lesson. Z-TABLE NUMBER 1 Our first 4 examples will be using <a href = "http://lilt.ilstu.edu/dasacke/eco148/ZTable.htm" target = "_blank">Z-TABLE NUMBER 1</a> You may link to that table now and keep the window open as we go through the examples so you don't have to link to it again. If you close it by mistake, just link to it again. This particular table contains z-scores from -3.4 to 3.4 the z-table entry give you the proportion of the population that is to the left of the z-score. Z-TABLE NUMBER 1 EXAMPLE NUMBER 1 In example 1, we want to find the proportion of the population that has a z-score less than 2. This translates to an area under the normal distribution curve that is to the left of a z-score of 2. Scroll down the left hand column until you find 2.0. Scroll to the right until you find a column titled .00 or 0.00 The z-table value for a z-score of 2.00 is equal to .9772. This means that the area under the normal distribution curve to the left of our z-score of 2.00 contains .9772 of the total distribution. .9772 is the proportion we are looking for. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "below" and enter a 2. The result is that the proportion of the population with a z-score less than 2 is equal to .977250 which confirms that we used the z-table correctly. Z-TABLE NUMBER 1 EXAMPLE NUMBER 2 In example 2, we want to find the proportion of the population that has a z-score greater than 2. This translates to an area under the normal distribution curve that is to the right of a z-score of 2. We scroll down column 1 until we find a z-score of 2.0 We scroll to the right until we find a column with a heading of .00 of 0.00 The z-table value for a z-score of 2.00 is equal to .9772. This means that the area under the normal distribution curve to the left of our z-score of 2.00 contains .9772 of the total distribution. Since we want the area to the right of the z-score, we subtract .9772 from 1 to get .0228. .0228 is the proportion we are looking for. .0228 means that 2.28% of the population have z-scores that are greater than 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "above" and enter a 2. The result is that the proportion of the population with a z-score greater than 2 is equal to .022750 which confirms that we used the z-table correctly. Z-TABLE NUMBER 1 EXAMPLE NUMBER 3 In example 3, we want to find the proportion of the population that have a z-score greater than -2 and less than 2. This translates to an area under the normal distribution curve that is to the right of a z-score of -2 and to the left of a z-score of 2. We scroll down the left hand column until we find a z-score of -2.0 We scroll to the right until we find a column with a heading of .00 or 0.00. The z-table value for a z-score of -2.00 is equal to .0228. This means that the area under the normal distribution curve to the left of our z-score of -2.00 contains .0228 of the population. .0228 means that 2.28% of the population have z-scores that are less than 2. We go back to the z-table and look for a z-score of 2. We scroll down the left column until we find a z-score of 2.0 We scroll to the right until we find a column with a heading of .00 or 0.00 The z-score of 2 yields a z-table value of .9772. This means that the area to the left of a z-score of 2 is equal to .9772 of the population. Since we want the area to the right of a z-score of 2, we subtract .9772 from 1 to get .0228 The proportion of the population to the right of a z-score of 2 is equal to .0228 We take both our values and perform an analysis to obtain what we want. We have the proportion of the population with a z-score of less than -2 equal to .0228 We have the proportion of the population with a z-score of greater than 2 equal to .0228 The proportion of the population we want is the proportion that is greater than -2 and less than 2. That would equal the proportion of the population that is NOT outside these limits. Since we have the proportion of the population that IS outside these limits, we do the following: We add .0228 and .0228 together to get .0456. We subtract that from 1 to get .9544. This means that .9544 proportion of the population has z-scores greater than -2 and less then 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "between" and enter a -2 and a 2. The result is that the proportion of the population with a z-score greater than -2 and less than 2 is equal to .9545 which confirms that we used the z-table correctly. Z-TABLE NUMBER 1 EXAMPLE NUMBER 4 In example 4, we want to find the proportion of the population that have a z-score less than -2 and greater than 2. This translates to an area under the normal distribution curve that is to the left of a z-score of -2 and to the right of a z-score of 2. This area is outside the limits imposed. We scroll down the left hand column until we find a z-score entry of -2.0 We scroll to the right until we find a column with a heading of .00 or 0.00. The z-table entry for a z-score of -2.00 is equal to .0228. This means that the area under the normal distribution curve to the left of our z-score of -2.00 contains .0228 of the population. .0228 means that 2.28% of the population have z-scores that are less than 2. We go back to the z-table and look for a z-score of 2. We scroll down the left column until we reach 2.0 and then we scroll to the right until we reach the column that has .00 or 0.00 at the top. The z-score of 2 yields a z-table value of .9772. This means that the area to the left of a z-score of 2 is equal to .9772 of the population. Since we want the area to the right of a z-score of 2, we subtract .9772 from 1 to get .0228 The proportion of the population to the right of a z-score of 2 is equal to .0228 We take both our values and perform an analysis to obtain what we want. We have the proportion of the population with a z-score of less than -2 equal to .0228 We have the proportion of the population with a z-score of greater than 2 equal to .0228 The proportion of the population we want is the proportion that is less than -2 and greater than 2. That would equal the proportion of the population that IS outside these limits. Since the entries we have are the proportion of the population that IS outside these limits, we do the following: We add .0228 and .0228 together to get .0456. That is the number we are looking for. This means that .0456 proportion of the population has z-scores less than -2 and greater then 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "outside" and enter a -2 and a 2. The result is that the proportion of the population with a z-score greater than -2 and less than 2 is equal to .0455 which confirms that we used the z-table correctly. NOTE When you are lookikng for the area under the normal distribution that is between 2 limits, get the area of the normal distribution that is outside these limits and then subtract that from 1 to get the area of the distribution curve that is within those limits. Examples 3 and 4 above used this technique. Z-TABLE NUMBER 2 Our second 4 examples will be using <a href = "http://www.sjsu.edu/faculty/gerstman/EpiInfo/z-table.htm" target = "_blank">Z-Table Number 2</a> You may link to that table now and keep the window open as we go through the examples so you don't have to link to it again. If you close it by mistake, just link to it again. This particular table contains z-scores from 0 to 3.0. It does not contain z-scores from -3.0 to 0. This means that if you want a negative z-score, you have to find it's corresponding positive z-score and convert the score yourself. The conversion process works as follows: We want a z-score of -2.5 We find a z-score of 2.5 because we can't find a z-score of -2.5 directly. We take the z-table entry you find and subtract it from 1. The result is the z-table entry we would have gotten had we been able to find a z-score entry of -2.5 directly. This will be explained at the bottom of this lesson with pictures to show you what happened. Just follow the rules for now and you will be fine. This particular z-table contains entries that represent the proportion of the population that is to the left of the entry. Z-TABLE NUMBER 2 EXAMPLE NUMBER 1 In example 1, we want to find the proportion of the population that has a z-score less than 2. This translates to an area under the normal distribution curve that is to the left of a z-score of 2. Scroll down the left hand column until you find an entry of 2.0 Scroll to the right until you find a column with a heading of .00 or 0.00 The z-table value for a z-score of 2.00 is equal to .9772. This means that the area under the normal distribution curve to the left of our z-score of 2.00 contains .9772 of the total distribution. .9772 is the proportion we are looking for. .9772 means that 97.72% of the population has z-scores that are less than 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "below" and enter a 2. The result is that the proportion of the population with a z-score less than 2 is equal to .977250 which confirms that we used the z-table correctly. Z-TABLE NUMBER 2 EXAMPLE NUMBER 2 In example 2, we want to find the proportion of the population that has a z-score greater than 2. This translates to an area under the normal distribution curve that is to the right of a z-score of 2. Scroll down the left hand column until you find a z-score of 2.0 Scroll to the right until you find a column with a heading of .00 or 0.00. The z-table value for a z-score of 2.00 is equal to .9772. This means that the area under the normal distribution curve to the left of our z-score of 2.00 contains .9772 of the total distribution. Since we want the area to the right of the z-score, we subtract .9772 from 1 to get .0228. .0228 is the proportion we are looking for. .0228 means that 2.28% of the population have z-scores that are greater than 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "above" and enter a 2. The result is that the proportion of the population with a z-score greater than 2 is equal to .022750 which confirms that we used the z-table correctly. Z-TABLE NUMBER 2 EXAMPLE NUMBER 3 In example 3, we want to find the proportion of the population that have a z-score greater than -2 and less than 2. This translates to an area under the normal distribution curve that is to the right of a z-score of -2 and to the left of a z-score of 2. This area is inside the limits imposed. We first look for a z-table entry of -2. We scroll down the left hand column until we find a z-score of -2.0. We do not find it because it is not there because this table has z-scores from 0 to 3.0 only. We look for a z-score of 2.0 rather than a z-score of -2.0 When we find it, we scroll to the right until we find a column with a heading of .00 or 0.00. The z-table value for a z-score of 2.00 is equal to .9772 Since we really wanted a z-score of -2.0, we perform the conversion indicated for this table by subtracting .9772 from 1 to get a z-score of .0228. .0228 means that 2.28% of the population have z-scores that are less than 2. Our z-score of -2 is equal to .0228. We go back to the z-table and look for a z-score of 2. We scroll down the left column until we reach 2.0 and then we scroll to the right until we reach the column that has .00 or 0.00 at the top. The z-score of 2 yields a z-table value of .9772. This means that the area to the left of a z-score of 2 is equal to .9772 of the population. Since we want the area to the right of a z-score of 2, we subtract .9772 from 1 to get .0228 The proportion of the population to the right of a z-score of 2 is equal to .0228 We take both our values and perform an analysis to obtain what we want. We have the proportion of the population with a z-score of less than -2 equal to .0228 We have the proportion of the population with a z-score of greater than 2 equal to .0228 The proportion of the population we want is the proportion that is greater than -2 and less than 2. That would equal the proportion of the population that is NOT outside these limits. Since we have the proportion of the population that IS outside these limits, we do the following: We add .0228 and .0228 together to get .0456. We subtract that from 1 to get .9544. This means that .9544 proportion of the population has z-scores greater than -2 and less then 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "between" and enter a -2 and a 2. The result is that the proportion of the population with a z-score greater than -2 and less than 2 is equal to .9545 which confirms that we used the z-table correctly. Z-TABLE NUMBER 2 EXAMPLE NUMBER 4 In example 4, we want to find the proportion of the population that have a z-score less than -2 and greater than 2. This translates to an area under the normal distribution curve that is to the left of a z-score of -2 and to the right of a z-score of 2. This area is outside the limits imposed. We scroll down the left hand column until we find an entry of -2.0. We do not find it because this table doesn't contain entries below a z-score of 0. We look for a z-score of 2 instead. We scroll down the left hand column until we find a z-score of 2.0. We scroll to the right until we find a column with a heading of .00 or 0.00. The z-table value for a z-score of 2.00 is equal to .9772. Since we are really looking for a z-score of -2, we subtract .9772 from 1 to get .0228. This means that the area under the normal distribution curve to the left of our z-score of -2.00 contains .0228 of the population. .0228 means that 2.28% of the population have z-scores that are less than -2. We go back to the z-table and look for a z-score of 2. We scroll down the left column until we reach 2.0 and then we scroll to the right until we reach the column that has .00 or 0.00 at the top. The z-score of 2 yields a z-table value of .9772. This means that the area to the left of a z-score of 2 is equal to .9772 of the population. Since we want the area to the right of a z-score of 2, we subtract .9772 from 1 to get .0228 The proportion of the population to the right of a z-score of 2 is equal to .0228 We take both our values and perform an analysis to obtain what we want. We have the proportion of the population with a z-score of less than -2 equal to .0228 We have the proportion of the population with a z-score of greater than 2 equal to .0228 The proportion of the population we want is the proportion that is less than -2 and greater than 2. That would equal the proportion of the population that IS outside these limits. Since we already have the proportion of the population that IS outsides these limits, we do the following: We add .0228 and .0228 together to get .0456. That is the number we are looking for. This means that .0456 proportion of the population has z-scores less than -2 and greater then 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "outside" and enter a -2 and a 2. The result is that the proportion of the population with a z-score greater than -2 and less than 2 is equal to .0455 which confirms that we used the z-table correctly. Z-TABLE NUMBER 3 The table we are going to use here is <a href = "http://www.intmath.com/Counting-probability/z-table.php" target = "_blank">Z-Table Number 3</a> Z-Table 3 contains z-scores above the mean only. Each z-table entry provides the proportion of the population to the left of the z-score only to the mean. That winds up leaving 1/2 the population unaccounted for. This means if you want the z-table entry to tell you the proportion of the population to the left of the z-score, you have to add .5 to whatever value the z-table gives you. This table requires two adjustments as follows: If the z-score entry you are looking for is positive, you have to find the z-table entry and add .5 to it. If the z-score entry you are looking for is negative, you have to find the corresponding positive z-score entry, add .5 to it, and then subtract that from 1. Z-TABLE NUMBER 3 EXAMPLE NUMBER 1 In example 1, we want to find the proportion of the population that has a z-score less than 2. This translates to an area under the normal distribution curve that is to the left of a z-score of 2. We scroll down the left hand column until we find a z-score entry of 2.0. We scroll to the right until we find a column with a heading of .00 or 0.00. The z-table value for a z-score of 2.00 is equal to .4772. Since this is the proportion of the population from the left of z-score entry to the mean only, we have to add .5 to it in order to get the proportion of the entire population that is to the left of the z-score. We add .5 to the z-table entry of .4772 to get .9772. This means that the area under the normal distribution curve to the left of our z-score of 2.00 contains .9772 of the total distribution. .9772 is the proportion we are looking for. .9772 means that 97.72% of the population has z-scores that are less than 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "below" and enter a 2. The result is that the proportion of the population with a z-score less than 2 is equal to .977250 which confirms that we used the z-table correctly. Z-TABLE NUMBER 3 EXAMPLE NUMBER 2 In example 2, we want to find the proportion of the population that has a z-score greater than 2. This translates to an area under the normal distribution curve that is to the right of a z-score of 2. We scroll down the left hand column until we find a z-score of 2.0. We scroll to the right until we find a column with a heading of .00 or 0.00 The z-table value for a z-score of 2.00 is equal to .4772. Since this is the proportion of the population from the left of z-score entry to the mean only, we have to add .5 to it in order to get the proportion of the entire population that is to the left of the z-score. The area to the left of the z-score of 2.00 is .4772 + .5 = .9772 This means that the area under the normal distribution curve to the left of our z-score of 2.00 contains .9772 of the total distribution. Since we want the area to the right of the z-score, we subtract .9772 from 1 to get .0228. .0228 is the proportion we are looking for. .0228 means that 2.28% of the population have z-scores that are greater than 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "above" and enter a 2. The result is that the proportion of the population with a z-score greater than 2 is equal to .022750 which confirms that we used the z-table correctly. Z-TABLE NUMBER 3 EXAMPLE NUMBER 3 In example 3, we want to find the proportion of the population that have a z-score greater than -2 and less than 2. This translates to an area under the normal distribution curve that is to the right of a z-score of -2 and to the left of a z-score of 2. This area is inside the limits imposed. We scroll down the left hand column looking for a z-score of -2.0. We don't find it because this table doesn't contain negative z-scores. We scroll down the left hand column again looking for the corresponding z-score of 2.0. When we find it, we scroll to the right looking for a column with a heading of .00 or 0.00. The z-table value for a z-score of 2.00 is equal to .4772. Since this table only includes the proportion of the population from the z-score to the mean only, we need to add .5 to it in order to get the proportion of the entire population to the left of the z-score. Our z-score becomes .4772 + .5 = .9772 Since we wanted a z-score of -2 but had to look for a z-score of 2, in order to change to a z-score of -2 we have to subtract .9772 from 1 to get .0228. This means that the area under the normal distribution curve to the left of our z-score of -2.00 contains .0228 of the population. .0228 means that 2.28% of the population have z-scores that are less than 2. We found it by getting a z-score of 2 and then subtracting it from 1 to make it a z-score of -2. We go back to the z-table and look for a z-score of 2. We scroll down the left column until we reach 2.0 and then we scroll to the right until we reach the column that has .00 at the top. The z-score of 2.00 yields a z-table value of .4772. Since this is the area to the left of the z-score only to the mean, we add .5 to it and get a z-table entry of .9772. This means that the area to the left of a z-score of 2 is equal to .9772 of the population. Since we want the area to the right of a z-score of 2, we subtract .9772 from 1 to get .0228 The proportion of the population to the right of a z-score of 2 is equal to .0228 We take both our values and perform an analysis to obtain what we want. We have the proportion of the population with a z-score of less than -2 equal to .0228 We have the proportion of the population with a z-score of greater than 2 equal to .0228 The proportion of the population we want is the proportion that is greater than -2 and less than 2. That would equal the proportion of the population that is NOT outside these limits. We add .0228 and .0228 together to get .0456. We subtract that from 1 to get .9544. This means that .9544 proportion of the population has z-scores greater than -2 and less then 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "between" and enter a -2 and a 2. The result is that the proportion of the population with a z-score greater than -2 and less than 2 is equal to .9545 which confirms that we used the z-table correctly. Z-TABLE NUMBER 3 EXAMPLE NUMBER 4 In example 4, we want to find the proportion of the population that have a z-score less than -2 and greater than 2. This translates to an area under the normal distribution curve that is to the left of a z-score of -2 and to the right of a z-score of 2. This area is outside the limits imposed. We enter z-table number 3 and look for a z-score of -2. We don't find it because this table doesn't have it. We look for a z-score of 2 instead. We scroll down column 1 until we find 2.0 Then we scroll right until we find a column with a heading of .00 The z-table value for a z-score of 2.00 is equal to .4772 Since this is the area to the left of the z-score only to the mean, we have to add .5 to it in order to get the the area to the left of the z-score all the way to the left end of the distribution curve. .4772 + .5 gives us a z-score of .9772 Since we are really looking for a z-score of -2, we subtract .9772 from 1 to get .0228. This means that the area under the normal distribution curve to the left of our z-score of -2.00 contains .0228 of the population. .0228 means that 2.28% of the population have z-scores that are less than -2. We go back to the z-table and look for a z-score of 2. We scroll down the left column until we reach 2.0 and then we scroll to the right until we reach the column that has .00 at the top. The z-score of 2 yields a z-table value of .4772. Since this is the area to the left of the z-score only to the mean, we have to add .5 to it which makes the z-score = .9772 This means that the area to the left of a z-score of 2 is equal to .9772 of the population. Since we want the area to the right of a z-score of 2, we subtract .9772 from 1 to get .0228 The proportion of the population to the right of a z-score of 2 is equal to .0228 We take both our values and perform an analysis to obtain what we want. We have the proportion of the population with a z-score of less than -2 equal to .0228. We have the proportion of the population with a z-score of greater than 2 equal to .0228. The proportion of the population we want is the proportion that is less than -2 and greater than 2. That would equal the proportion of the population that IS outside these limits. Since what we want is the proportion of the population that IS outsides these, limits, we do the following: We add .0228 and .0228 together to get .0456. That is the number we are looking for. This means that .0456 proportion of the population has z-scores less than -2 and greater then 2. We check out work by going into the <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">Z-Table Calculator</a> to confirm the results. We enter a mean of 0 and a standard deviation of 1 in the top graph and we select "outside" and enter a -2 and a 2. The result is that the proportion of the population with a z-score greater than -2 and less than 2 is equal to .0455 which confirms that we used the z-table correctly. EXPLANATION OF THE PROCEDURES USED FOR THE THREE DIFFERENT TYPES OF Z-TABLES We've used 3 different types of z-tables. All of the tables provide us with a standard method for obtaining the proportion of the population that is to the left of the z-score. The difference is in the z-scores that are carried and in the proportion of the population that is to the left of the z-score. Z-Table Number 1 covered all of the population to the left of the z-score and carried all of the z-scores. This link explains the procedure used for <a href = "http://theo.x10hosting.com/examples/Statistics/ZTableExplanationNumber1.html" target = "_blank">Z-TABLE NUMBER 1</a> Z-Table Number 2 covered all of the population to the left of the z-score but only carried positive z-scores. Special procedures were required to obtain the area to the left of the negative z-scores. This link explains the procedure used for <a href = "http://theo.x10hosting.com/examples/Statistics/ZTableExplanationNumber2.html" target = "_blank">Z-TABLE NUMBER 2</a> Z-Table Number 3 covered all of the population to the left of the z-score TO THE MEAN ONLY. This meant that the bottom half of the population to the left of the mean was not covered. Z-Table Number 3 also only covered positive z-scores. Special procedures were required to obtain ALL of the area to the left of the z-scores. Once that was done, additional special procedures were required to obtain the area to the left of the negative z-scores. This link explains the procedure used for <a href = "http://theo.x10hosting.com/examples/Statistics/ZTableExplanationNumber3.html" target = "_blank">Z-TABLE NUMBER 3</a> here's a problem that was solved using a z-table. it contains a picture on how the z-score was found in the table along with an explanation of how the z-score was used to get the area of the distribution curve that was needed. <a href = "http://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.495964.html" target = "_blank">http://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.495964.html</a>
