What is statistics? – Standard Normal Table – z scores – Part II

Φ(z) values and Z – Standard Normal Distribution Table
As it was already told, the Standard Normal Distribution (pdf) has an absolute symmetry around the Mean. Also, its Mean is equal to Zero (0) and its SD to one (1). Moreover, a z value shows the distance of a Raw score that has relative to its Mean, expressed in Population SD points (σ).

Normal Distribution Examples Percentages

The Bell Curve represents the f(x) – pdf – function and the “Area under the Curve” represents the Φ(z) – cdf – function. The Normal Table or the z / Φ(z) table is used to transform a z value to Φ(z) value.

Standard_Normal_Distribution (cdf_Cumulative_Density_Function)_functions2

How z / Φ(z) table is working
In this page, a short edition of this table was created. This short z table has index values from -4 to +4 in steps of 0.5. The full edition of the z / Φ(z) table in steps of 0.1 can be found here, in a PDF form.

Table explanation
A z / Φ(z) table usually includes:

i) The values of Φ(z) – cdf which are found inside the Table
—These values shows how much data are contained in a specified area of Φ(z) “under the curve”.
—The total “Area under the curve” equals to 1.
—These values also can be expressed in percentages.
—These values are increasing as the Area Coverage is increasing.
—In that Table, the Φ(z) Area has as a starting point the most Left point on X axis -\infty.
—As the “Area under the Curve” increases its Coverage to +\infty point, its value is going to 1 (100%), in this Table.
—The middle point is the Zero “0”, therefore, this point splits the total “Area under the curve” in Half
—Therefore, 50% of this Area is Left and 50% is Right of this point.
—Due to Symmetry of this Area, the Area properties in one of these halves can be reflected to the other Half Area.

Standard_Normal_Distribution_pdf_area_under_curve_Symmetry

ii) The z values in a special way:
—The first part of z value is found on the First column (the gray one) that includes the first digit and the first decimal digit.
—The second decimal digit of the Z value is found on the First Row (gray one)
—The intersection of these two values indicate the corresponding Φ(z) – cdf value for the z value.

Note that the “Left Table” includes the Negative z values and the “Right Table” the positive z values. The cells inside the Table includes the Φ(z) values.
Z values - Z Table Short from +4 to -4

z / Φ(z) Table: Example
Note that any “Area” cannot be described by a single point, but it needs at least two points. Let’s say that you need to find what is the value that describes the “Area under the curve” – that is, the Φ(z) value – between -\infty and +3.55, always expressed in SD points: 3.55σ. Also, let’s suggest that the “Area under the curve” describes the data of a random X variable.

Standard_Normal_Distribution_pdf_area_under_curve_example

—The first part of z value is found on the First column: you must find the value “+3.5″.
—The second decimal digit of the Z value is found on the First Row: you must find the value “.05″
—Then, you look at the cell that these two values are intersecting, which is Φ(+3.55)=.9998.

z_table_z_example_standard_normal_table

Conclusion
—This result shows that the “Area under the curve” that is defined by -\infty to +3.55 on X axis includes the 99.98% of the total data.
—The rest 0.02% of the data are described by the “Area under the curve” that is defined by +3.55 to +\infty on X axis.

Notes
—The “Area under the curve” that is defined by -\infty to 0 includes the 50% of the values and
—The “Area under the curve” that is defined by 0 to +\infty includes the rest 50% of the values
—This can be seen if you find the intersecting point of “0” and “.00″ values on the Table.
—Then, you can suggest that the “Area under the curve” that is defined by 0 to +3.55 on X axis includes the 99.98\%-50\%=49.98\% of the total data
—Note that the 0 point is not included in 49.98% because it was included in “50%”.

Standard_Normal_Distribution_pdf_area_under_curve_Symmetry_2

Z values / Φ(z) Table for Standard Normal Distribution from -4 to +4 in steps of .01