What is statistics? – Standard Normal Table – z scores – Rules – Part III

Explanation on Rules of z / Φ(z) calculations
Many publications or Internet sites give some rules of Φ(z). These rules are based on the Symmetry that the Standard Normal Distribution has. In order to fully explain these rules:

Normal Distribution Examples Percentages

Explanation of Starting and End point of Φ(z) / cdf in General
—The Starting point is the point that the “Area under the Curve” begins.
—The End point is the point that the “Area under the Curve” finishes.
—The Starting point and The End point refers to points on the Horizontal X axis.
—The Starting point and The End point defines Φ(z).
—The values of Φ(z) starts from Zero “0” and they are increasing until One “1”.
—Τhe Total “Area under the Curve” is Equal to One “1”.
—The Total “Area under the Curve” has as a Starting and End point the Negative and Positive Infinity
—Usually, Standard Normal Tables include z values from -4 to +4.

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Explanation of Starting and End point of Φ(z) in Standard Normal Tables
—The Starting point of Φ(z) in relation to its given Table cell values can DIFFER from the Starting point of a related Φ(z) question.
—The starting point of Φ(z) on X Axis may differ to each publication: i) Negative Infinity (most cases) or ii) Positive Infinity.
—The publications may implicit or explicit state the “starting point” of the Table Φ(z) cell values as well of the related Φ(z) question.
—Some Normal (z/Φ(z)) Tables may present only the Positive z values, from 0.00 to +4.00, and its corresponding Φ(z) values, from 0.50 until 1.

Explanation of Starting and End point of Φ(z) in Questions
—Note that some authors use as symbols “the big Z” or “the big X” which both refer to the Horizontal Axis of the Standard Normal Distribution.
—Therefore, the “small z” or the “small x” refers to some specific value on this axis.
—Intervals on the X Axis of the Standard Normal Distribution can be denoted with “a” and “b” symbols.

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Explanation of Probability symbolism
—This notion Pr(Z\leq z) requests the probability -that is a Φ(z) value- that will have as:
i) Starting point, the Negative Infinity -\infty.
ii) End point, any point that is higher than the Negative Infinity -\infty.
iii) The End point is included in calculations.

—This notion Pr(Z > z) requests the probability -that is a Φ(z) value- that will have as:
i) Starting point, the Positive Infinity +\infty.
ii) End point, any point that is Lower than the Positive Infinity +\infty.
iii) The End point is not included in calculations.

The following rules are based on z / Φ(z tables that have as a Starting point the Negative Infinity -\infty
—IF the starting point of the Φ(z) on this Table is the Positive Infinity +\infty
—Then the guides for RULE I must replaced from the guides of RULE II and
—The guides for RULE II must replaced from the guides of RULE I
—The Table included in this page, it contains both Negative and Positive z values and its corresponding Φ(z) values with starting point the -\infty.

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Note that the “Left Table” includes the Negative z values and the “Right Table” the Positive z values. The cells inside the Table are the Φ(z) values (Probabilities). For more information about how a Standard Normal Table is used, please, see Part II
Z values - Z Table Short from +4 to -4

Rule I
What is the Probability of a “Z value” to be lower or equal to a specific “z value”?
—The starting point of the Question is -\infty and the end point is any value Higher than Negative Infinity -\infty.

A) Tables that provide both Positive and Negative z values – Φ(z)

Probabilities for Positive z values and Negative z values
Pr(Z\leq z) then: \Phi(z)=z or \Phi(-z)=z
—This notion “Φ(z)=z” or “Φ(-z)=z” actually means “Φ(z) and Φ(-z) =Corresponding Table cell value”.

—The corresponding Φ(z)/Φ(-z) value of a Positive or Negative z value can be found by:
a) Using the given z value and then
b) You find the corresponding Φ(z) value

Standard_Normal_Distribution_pdf_cdf_area_under_curve_Starting_end_point_Negative_positive_infinity_z_Fz_values_Table_A_1

B) Tables that provide ONLY the Positive z values – Φ(z)

Probabilities for Positive z values
Pr(Z\leq z) then: \Phi(z)=z
—This notion “Φ(z)=z” actually means “Φ(z)=Corresponding Table cell value”.

—The corresponding Φ(z) value of a Positive z value can be found by:
a) Using the given z value
b) You find the corresponding Φ(z) value

Probabilities for Negative z values
Pr(Z\leq z) then: \Phi(-z)=1-\Phi(z)=1-z
—This notion “Φ(-z)=1-Φ(z)=z” actually means “Φ(-z)=1-Φ(z)=1-Corresponding Table cell value”.

—The corresponding Φ(z) value of a Negative z value can be found by:
a) Using this “Negative z value” as a positive one and then
b) You find the corresponding Φ(z) value and then
c) you must subtract this Φ(z) value from One (1) – Remember, the total “Area under the Curve” is equal to one (1).

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Rule II
What is the Probability of a “Z value” to be Higher to a specific “z value”?
—The starting point of the Question is +\infty and the end point is any value Lower than Positive Infinity +\infty.

A) Tables that provide both Positive and Negative z values – Φ(z)

Probabilities for Positive z values and Negative z values
Pr(Z > z) then: 1-\Phi(z)=1-z or 1-\Phi(-z)=1-z
—This notion “1-Φ(z)=1-z” or “1-Φ(-z)=1-z” actually means “1-Φ(z or -z)=1-Corresponding Table cell value”.

—The corresponding Φ(z) value of a Positive or Negative z value can be found by:
a) Using the given z value and then
b) You find the corresponding Φ(z) value and then
c) you must subtract this Φ(z) value from One (1)

Standard_Normal_Distribution_pdf_cdf_area_under_curve_Starting_end_point_Negative_positive_infinity_z_Fz_values_Table_A_5

B) Tables that provide ONLY Positive z values – Φ(z)

Probabilities for Positive z values
Pr(Z > z) then: 1-\Phi(z)=1-z
—This notion “1-Φ(z)=1-z” actually means “1-Φ(z)=1-Corresponding Table cell value”.

—The corresponding Φ(z) value of a Positive z value can be found by:
a) Using the given z value and then
b) you must find the corresponding Φ(z) value and then
c) You must subtract this Φ(z) value from One (1)

Probabilities for Negative z values
Pr(Z > z) then: \Phi(-z)=\Phi(z)=z
—This notion “Φ(-z)=Φ(z)=z” actually means “Φ(-z)=Φ(z)=Corresponding Table cell value”.

—The corresponding Φ(z) value of a Negative z value can be found by:
a) Using this “Negative z value” as a positive one and then
b) You must find the corresponding Φ(z) value

Standard_Normal_Distribution_pdf_cdf_area_under_curve_Starting_end_point_Negative_positive_infinity_z_Fz_values_Table_A_6

Rule ΙΙΙ
What is the Probability of a “Z value” to be i) Lower or Equal to a specific “z value” (a) as well Higher than a specific “z value” (b):
—The Starting and The End point of the Question can be any TWO points “a and b” on X axis, representing Z values.

Therefore: Pr(b<Z\leq a) then: \Phi(b)-\Phi(a)

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—This Statistical expression of Probabilities is consisted of TWO main parts:
a) Pr(b<Z) which has been fully analyzed in Rule ΙΙ
b) Pr(Z\leq a) which has been fully analyzed in Rule Ι
c) When you find the Φ(z) values of the “a and b” points according to the Rules I and II, then,
d) You must subtract the Lowest Φ(z) value (a) from the Highest Φ(z) value (b) , that is: “Φ(b)-Φ(a)”

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The result -as in all the previous cases- shows the Percentage of the Data that are represented from the “Area under the Curve” between these two points (a and b) of a Random Variable X.

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