Z-Tables
This page follows the JoeCMath video Everything You Need to Know About Z-Tables. The video focuses on left-tail z-tables: tables that give the area below a z-score in the standard normal distribution.
Main Idea
The video starts with the standard normal distribution. It has mean $0$ and standard deviation $1$:
\[\mu=0,\qquad \sigma=1.\]After a normal distribution value is converted to a z-score, the z-table helps find the area under the standard normal curve. In this video, the table style gives the area to the left of the z-score.
So when the table gives a value like $0.23270$, it means $23.270\%$ of the standard normal distribution is below that z-score.
Watch this section: introduction at 0:00.
How The Table Is Read
The video uses separate negative and positive z-score tables. The row gives the z-score to the tenths place, and the column gives the hundredths place.
For example, to find $z=-0.73$, use the row for $-0.7$ and the column for $.03$. The value at the intersection is the table output for $z=-0.73$.
The key detail is that the table is left-tail. The number in the table is the area below the z-score, not automatically the area above or between values.
Watch this section: z-score and z-score table at 0:08.
Example 1: Area Below $z=-0.73$
The first worked example asks for the percent of the distribution below
\[z=-0.73.\]Since the z-score is negative, the video uses the negative z-score table. The row is $-0.7$, and the column is $.03$.
The table value is
\[0.23270.\]That means
\[P(Z<-0.73)=0.23270.\]In percent form, about $23.27\%$ of the standard normal distribution falls below $z=-0.73$.
Watch this section: area below a z-score at 0:55.
Example 2: Area Above $z=1.35$
The second example asks for the percent of the distribution above
\[z=1.35.\]The table still gives the area below the z-score. For $z=1.35$, the positive z-table gives
\[0.91149.\]So
\[P(Z<1.35)=0.91149.\]But the question asks for the area above. Since the entire area under the normal curve is $1$, subtract the left-tail area from $1$:
\[P(Z>1.35)=1-0.91149=0.08851.\]So about $8.85\%$ of the distribution falls above $z=1.35$.
Watch this section: area above a z-score at 1:55.
Example 3: Area Between Two Z-Scores
The last example asks for the area between
\[z_1=-2.31\]and
\[z_2=0.40.\]The video finds the left-tail area for each z-score first.
For $z_1=-2.31$, the table gives
\[P(Z<-2.31)=0.01044.\]For $z_2=0.40$, the table gives
\[P(Z<0.40)=0.65542.\]To find the area between the two z-scores, subtract the smaller left-tail area from the larger one:
\[P(-2.31<Z<0.40)=0.65542-0.01044=0.64498.\]So about $64.50\%$ of the distribution lies between $z=-2.31$ and $z=0.40$.
Watch this section: area between two z-scores at 3:25.
Timestamp Guide
Related Statistics Work
If you need the step before using a z-table, review why we convert to z-scores.
You can also browse the Normal Distribution notes as this topic grows.
The video description links to the full JoeCMath normal distribution playlist: watch the normal distribution playlist on YouTube.