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Are All Z Tables The Same

author
James Smith
• Tuesday, 18 May, 2021
• 7 min read

Use the negative Z score table below to find values on the left of the mean as can be seen in the graph alongside. Corresponding values which are less than the mean are marked with a negative score in the z -table and represent the area under the bell curve to the left of z.

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Contents

Use the positive Z score table below to find values on the right of the mean as can be seen in the graph alongside. Corresponding values which are greater than the mean are marked with a positive score in the z -table and represent the area under the bell curve to the left of z.

Q: 300 college student’s exam scores are tallied at the end of the semester. The average score for the batch was 700 (µ) and the standard deviation was 180 ().

First, traverse horizontally down the Y-Axis on the leftmost column to find the value of the first two digits that is -1.3 Once we have that, we will traverse along the X axis in the topmost row to map the second decimal (0.05 in the case) and find the corresponding column for it.

The intersection of the row of the first two digits and column of the second decimal value in the above Z table is the answer we’re looking for which in case of our example is 0.08851 or 8.85% That is because for a standard normal distribution table, both halves of the curves on the other side of the mean are identical.

Sure it can be combined into one single larger Z -table but that can be a bit overwhelming for a lot of beginners, and it also increases the chance of human errors during calculations. If you want to know the area between the mean and a negative value you will use the first table (1.1) shown above which is the left-hand/negative Z -table.

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De Moiré came about to create the normal distribution through his scientific and math based approach to the gambling. He was trying to come up with a mathematical expression for finding the probabilities of coin flips and various inquisitive aspects of gambling.

He discovered that although data sets can have a wide range of values, we can ‘standardize’ it using a bell shaped distribution curve which makes it easier to analyze data by setting it to a mean of zero and a standard deviation of one. It was realized that normal distribution applied to many mathematical and real life phenomena.

For example, Belgian astronomer, Lambert Outlet (22nd February 1796 to 17th February 1874) discovered that despite people’s height, weight and strength presents a big range of datasets with people’s height ranging from 3 to 8 feet and with weight’s ranging from few pounds too few hundred pounds, there was a strong link between people’s height, weight and strength following a standard normal distribution curve. For example, the normal curve was used to analyze errors in astronomical observation measurements.

Whereas in probability theory a special case of the central limit theorem known as the DE Moivre-Laplace theorem states that the normal distribution may be used as an approximation to the binomial distribution under certain conditions. This theorem appears in the second edition pf the book published in 1738 by Abraham de Moivre titled ‘Doctrine of Chances’.

The label for rows contains the integer part and the first decimal place of Z. Because the normal distribution curve is symmetrical, probabilities for only positive values of Z are typically given.

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The user has to use a complementary operation on the absolute value of Z, as in the example below. Since this is the portion of the area above Z, the proportion that is greater than Z is found by subtracting Z from 1.

The values correspond to the shaded area for given Z This table gives a probability that a statistic is between 0 (the mean) and Z. F(z)=(z)12{\display style f(z)=\Phi (z)-{\franc {1}{2}}} Note that for z = 1, 2, 3, one obtains (after multiplying by 2 to account for the interval) the results f(z) = 0.6827, 0.9545, 0.9974, characteristic of the 68–95–99.7 rule.

The values are calculated using the cumulative distribution function of a standard normal distribution with mean of zero and standard deviation of one, usually denoted with the capital Greek letter {\display style \Phi} (phi), is the integral F(z)=1(z){\display style f(z)=1-\Phi (z)} z +0.00+0.01+0.02+0.03+0.04+0.05+0.06+0.07+0.08+0.09 0.0 0.500000.496010.492020.488030.484050.480060.476080.472100.468120.46414 0.1 0.460170.456200.452240.448280.444330.440380.436400.432510.428580.42465 0.2 0.420740.416830.412940.409050.405170.401290.397430.393580.389740.38591 0.3 0.382090.378280.374480.370700.366930.363170.359420.355690.351970.34827 0.4 0.344580.340900.337240.333600.329970.326360.322760.319180.315610.31207 0.5 0.308540.305030.301530.298060.294600.291160.287740.284340.280960.27760 0.6 0.274250.270930.267630.264350.261090.257850.254630.251430.248250.24510 0.7 0.241960.238850.235760.232700.229650.226630.223630.220650.217700.21476 0.8 0.211860.208970.206110.203270.200450.197660.194890.192150.189430.18673 0.9 0.184060.181410.178790.176190.173610.171060.168530.166020.163540.16109 1.0 0.158660.156250.153860.151510.149170.146860.144570.142310.140070.13786 1.1 0.135670.133500.131360.129240.127140.125070.123020.121000.119000.11702 1.2 0.115070.113140.111230.109350.107490.105650.103830.102040.100270.09853 1.3 0.096800.095100.093420.091760.090120.088510.086920.085340.083790.08226 1.4 0.080760.079270.077800.076360.074930.073530.072150.070780.069440.06811 1.5 0.066810.065520.064260.063010.061780.060570.059380.058210.057050.05592 1.6 0.054800.053700.052620.051550.050500.049470.048460.047460.046480.04551 1.7 0.044570.043630.042720.041820.040930.040060.039200.038360.037540.03673 1.8 0.035930.035150.034380.033620.032880.032160.031440.030740.030050.02938 1.9 0.028720.028070.027430.026800.026190.025590.025000.024420.023850.02330 2.0 0.022750.022220.021690.021180.020680.020180.019700.019230.018760.01831 2.1 0.017860.017430.017000.016590.016180.015780.015390.015000.014630.01426 2.2 0.013900.013550.013210.012870.012550.012220.011910.011600.011300.01101 2.3 0.010720.010440.010170.009900.009640.009390.009140.008890.008660.00842 2.4 0.008200.007980.007760.007550.007340.007140.006950.006760.006570.00639 2.5 0.006210.006040.005870.005700.005540.005390.005230.005080.004940.00480 2.6 0.004660.004530.004400.004270.004150.004020.003910.003790.003680.00357 2.7 0.003470.003360.003260.003170.003070.002980.002890.002800.002720.00264 2.8 0.002560.002480.002400.002330.002260.002190.002120.002050.001990.00193 2.9 0.001870.001810.001750.001690.001640.001590.001540.001490.001440.00139 3.0 0.001350.001310.001260.001220.001180.001140.001110.001070.001040.00100 3.1 0.000970.000940.000900.000870.000840.000820.000790.000760.000740.00071 3.2 0.000690.000660.000640.000620.000600.000580.000560.000540.000520.00050 3.3 0.000480.000470.000450.000430.000420.000400.000390.000380.000360.00035 3.4 0.000340.000320.000310.000300.000290.000280.000270.000260.000250.00024 3.5 0.000230.000220.000220.000210.000200.000190.000190.000180.000170.00017 3.6 0.000160.000150.000150.000140.000140.000130.000130.000120.000120.00011 3.7 0.000110.000100.000100.000100.000090.000090.000080.000080.000080.00008 3.8 0.000070.000070.000070.000060.000060.000060.000060.000050.000050.00005 3.9 0.000050.000050.000040.000040.000040.000040.000040.000040.000030.00003 4.0 0.000030.000030.000030.000030.000030.000030.000020.000020.000020.00002 This table gives a probability that a statistic is greater than Z, for large integer Z values.

Functions based on the normal distribution are easy to retrieve in code or excel, so we do not really need tables anymore, in practice. Because the popular exam calculators (TI BA II+ and HP 12c) do not include z table functionality, so we do need to use them to lookup values on the exam (yes, the z table has been provided in recent FRM exams).

I hope you noticed the phrase “normally distributed?” It comes up often in exams. The normal distribution is rarely realistic, but it is popular for learning purposes due to its special properties and what is called parsimony.

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Parsimony here refers to the normal conveniently has only two parameters, mean and variance. The Z value of 1.350 means “The value of 5.0 is 1.350 standard deviations above the mean of 2.30.” Now we can use the common Z table to retrieve the associated probability.

The first function says “The probability that X is less than or equal to 5.0 conditional on a mean of X equal to 2.3 and standard deviation of X equal to 2.0.” The second function, Pr(Z 1.350), reflects the normalization (translation) from the normal X to the standard normal Z, and we don’t need to specify the mean or standard deviation of the Z. The values inside the Z table are probabilities, so they must lie between 0% and 100% inclusive.

The later function simply makes explicit the zero mean and unit standard deviation. The plot on the right above gives the area under the curve that is between zero and 1.35; some Tables employ this format instead.

Instead of a cumulative probability, the table returns the probability that the standard random normal variable will lie between zero and the critical value. Now let’s alter the question a bit in order to test our understanding.

They have plastic, metal, wood and even concrete picnic tables to choose from. -The Twelve Tables spelled out the Roman code of laws.

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-The Twelve Tables protected all citizens, including the plebeians. Things like the forms, reports and queries are all using data that originally comes from the tables.

Forms, queries and reports can process data to get information, like getting totals. By good programming practice, deadlocks can be avoided (but not altogether eliminated) by locking tables in the same order each time.

This table is very useful for finding probabilities when the event in question follows a normal distribution. The green shaded area in the diagram represents the area that is within `1.45` standard deviations above the mean.

To get this area of `0.4265`, we read down the left side of the table for the standard deviation's first 2 digits (the whole number and the first number after the decimal point, in this case `1.4`), then we read across the table for the “`0.05`” part (the top row represents the 2nd decimal place of the standard deviation that we are interested in.) (left column) `1.4\ +` (top row) `0.05 = 1.45` standard deviations The area represented by `1.45` standard deviations to the right of the mean is shaded in green in the standard normal curve above.

Every student learns how to look up areas under the normal curve using Z -Score tables in their first statistics class. You get the z -score by evaluating the integral of the equation for the bell-shaped normal curve, usually from -Inf to the z -score of interest.

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Here is the slide presentation I put together to explain the use and origin of the Z -Score table, and how it relates to norm and norm (the command that lets you input an area to find the z -score at which the area to the left is swiped out). It’s free to use under Creative Commons, and is part of the course materials that is available for use with this 2017 book.

Now that we have slots for all the z -scores, we can use norm to transform all those values into the areas that are swiped out to the left of that z -score.

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Sources
1 factsandtrends.net - https://factsandtrends.net/2017/09/29/10-traits-of-generation-z/
2 growingleaders.com - https://growingleaders.com/blog/six-defining-characteristics-of-generation-z/
3 money.usnews.com - https://money.usnews.com/money/blogs/outside-voices-careers/articles/generation-z-characteristics-in-the-workplace
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