Normal distribution, which is also called Gaussian distribution, is a probability distribution, which is given by the probability density function, coinciding with a Gaussian function. A special z score table was invented to present the values of the distribution.

The history of the z score table dates back almost 300 years ago. It is said that Abraham de Moivre was the first who opened the z score table. He gave his views on the matter back in 1733. Then the z score table was known as a theoretical approximation of the binomial distribution with a large number of observations. However, his works were not appreciated and Abraham is often unfairly overlooked when it comes to normal distribution. Z score table became widespread due to selective data analysis.

The importance of the z score table in many areas of science (e.g., mathematical statistics and statistical physics) comes from the central limit theorem of probability theory. If the result of observations is the sum of many random weakly interdependent variables, each of which makes a relatively small contribution to the total, then when the number of terms increases, the distribution of centered and normalized result tends to be normal. This law of probability theory is the result of wide dissemination of the normal distribution, and that was one reason for its name.

Below you will find standard tables of the distribution functions. Such a traditional view has its own advantages over the probability calculator, as the tables contain a large number of values at the same time, and the user can fast enough explore the wide range of probability values.

- Z score Table.

The standard normal distribution (z score table) is used in testing of various hypotheses, including the mean value, the difference between the average and the proportionality of values. This distribution has a mean value of 0 and standard deviation of 1. The values in the z score table represent the value of the area under the standard normal (Gaussian) curve from 0 to the corresponding z-score. For example, the value of the square values between 0 and 2.36 is shown in the cell located at the intersection of column lines 2.30 and 0.06, and is 0.4909. The value of the area between 0 and a negative value is at the intersection of rows and columns, which together correspond to the absolute value of a predetermined value. For example, the area under the curve from 0 to -1.3 is equal to the area under the curve between 0 and 1.3, so its value is at the intersection of the row 1.3 and column 0.00 and is 0.4032. - Student’s distribution.

The form of Student’s distribution depends on the number of degrees of freedom. When the degree of freedom increases, the distribution form changes. The top of the table gives the probability to obtain values greater than those specified in the corresponding cell. The critical value corresponding to the probability of 0.05 t-distribution with 6 degrees of freedom, is located at the intersection of column 0.05 and row 6: t (.05.6) = 1.943180. - The chi-squared distribution.

As in the case of the Student’s t-distribution, the form of the chi-square distribution is determined by the number of degrees of freedom. The table shows the critical value of the chi-square distribution with a given number of degrees of freedom. The desired value is at the intersection of a column with the corresponding value of probability and row with many degrees of freedom. For example, the chi-square distribution’s critical value with 4 freedom degrees for the probability of 0.25 is 5.38527. This means that the area under the curve of the density of the chi-square distribution with 4 degrees of freedom on the right of the value 5.38527 is 0.25. - F-distribution.

F-distribution is asymmetric and is commonly used in the analysis of variance. This density function has values that are the ratio of two quantities that have chi-square distribution, and the corresponding F-distribution is defined by two values of the number of degrees of freedom. The first index always corresponds to the number of freedom degrees for the numerator, and this order is significant, because F (10,12) is not equal to F (12,10). The column shows the amount of freedom degrees of the numerator, and the lines show the number of the degrees of freedom for the denominator. The title of the table has the value of probability. For example, the critical value of F-distribution for the probability of.05 and the degrees of freedom 10 and 12 is located at the intersection of a column with a value of 10 (the numerator) and the line with the value of 12 (denominator).

The whole family of distributions, each of which is defined by its parameters (expectation value and variance) gets under normal law of distribution. Among one of such distributions is the so-called standard normal distribution, which is used as a probabilistic-statistical model, a kind of standard. When having such a model (formula), it is possible to obtain the probability of events that of interest to us. However, it is difficult to make necessary calculations in mind or even on a calculator. Therefore, to make the task easier, the probability for different values of a variable have been calculated long time ago and put in a special z score table.

Professors at universities teach to use z score table like this: take the value of the variable (z), and then at the intersection of the corresponding row and column find the desired probability.

There are two types of the z score table:

- A table of density values of the standard normal distribution.
- A table of values of the standard normal distribution (the integral of the density).

Suppose you need to find the density value for z = 1, i.e. the density value that is 1 sigma away from the mathematical expectation. Before using the z score table, make sure it is the right table. To check that, look at the top of the table with the name of the function. In this case it should be the z score table of the Gaussian function.

Then, depending on the organization of the data, find the desired value according to the title of the column and the row. In our example, we take the line 1.0 and look at the first column of data, as we don’t have any hundredths. The desired value is 0.2420 (0 is omitted before 2420). Do not be afraid of different designations of the variable, most often only x is indicated in the tables. The main thing is the formula above the table.

One of the main properties of the Gaussian function is that it is symmetrical to the y axis. Therefore F (z) = F (-z), that is, density to 1 is identical to the density of -1.

However, the standard normal distribution function (a Laplace function) is of the biggest interest to any analysist. These tables are also usually made only for positive values. Therefore, for correct use of this z score table and the right finding of any relevant probabilities it is recommended to first introduce yourself with some key features of the standard normal distribution.

The function F (z) is symmetric with respect to its value of 0.5 (not ordinate axis as a Gaussian function). Hence the equality: F (z) + F (-z) = 1.

The values of the function F (-z) and F (z) divide the chart into 3 parts. The upper and the lower parts are equal. To complement the probability F (z) up to 1, it is enough to add the missing value F (-z). This way, you will obtain the equality, described above.

If you need to find the probability of getting into the interval (0; z), i.e. the probability of deviation from zero in the positive direction to a certain number of standard deviations, it is enough to subtract 0.5 from the value of the standard normal distribution function.

If you need to find the probability of deviation on each side of zero, the value for this function in the z score table is simply multiplied by 2.

Values in the z score table represent the area under the curve part, which describes the normal standard function (Gausse) in the range from 0 to Z. For example, the area bounded by the values of 0 and 2.36, is located at the intersection of the line 2.30 and the column 0.06. It is equal to 0.4909. If the value of Z is negative, then due to the symmetry of the function relatively to the average, the required probability (area) will be located at the intersection of line and column that correspond to the absolute value of Z (|Z|). For example, the area between 0 and -1.3 equal to the area between 0 and 1.3; that is, it is at the intersection of line 1.3 and the column 0 and is equal to 0.4032.

The fact that almost all of the values in the z score table are usually within the ± 3 sigma from the mean value is called the rule of three sigma. This rule is often used for various examples, as it can be used in advance to outline the range of possible values, beyond which the data is almost never fall. And if such a value is detected, most likely it belongs to a different set of data or an abnormal deviation (ejection).

The understanding of the meaning of the z score table allows to better imagine the mathematical formalism of statistical methods.

To better understand how the probability of hypotheses is calculated with the help of the z score table of the normal distribution of values, try to build your own z score table in Excel using the built-in functions of the tabulator.

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