Why Convert to a Z-Score?
This page follows the JoeCMath video Why Convert to a Z-Score?. The video explains why a value from any normal distribution can be translated into the standard normal distribution, where the mean is $0$ and the standard deviation is $1$.
Main Idea
The video starts with the reason z-scores are useful. A z-score translates a value from its original normal distribution into the standard normal distribution.
The standard normal distribution has
\[\mu=0\]and
\[\sigma=1.\]Once a value has been converted to a z-score, the video uses the standard normal table to connect that z-score to an area below or above the value.
Watch this section: the value of z-scores at 0:00.
The Z-Score Formula
The formula used in the video is
\[z=\frac{x-\mu}{\sigma}.\]Here, $x$ is the value from the original normal distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation.
The numerator, $x-\mu$, measures how far the value is from the mean. Dividing by $\sigma$ converts that distance into standard deviations.
Watch this section: setting up the formula at 1:12.
Example: Find The Area Below 62
The first example in the video uses a normal distribution with mean $50$ and standard deviation $10$. The question asks for the percent of the distribution below $62$.
That means
\[x=62,\qquad \mu=50,\qquad \sigma=10.\]Substitute those values into the z-score formula:
\[z=\frac{62-50}{10}.\]Simplify:
\[z=\frac{12}{10}=1.2.\]So $62$ in this normal distribution matches $z=1.2$ in the standard normal distribution.
Watch this section: area under the distribution example at 0:47.
Using The Z-Table
After finding $z=1.2$, the video uses the positive z-table. Since the z-score is $1.20$, look for the row $1.2$ and the column $.00$.
The table value shown in the video is
\[0.88493.\]That value means about $88.49\%$ of the standard normal distribution falls below $z=1.2$. Because $62$ was translated into $z=1.2$, the same percent applies to the original normal distribution:
\[P(X<62)\approx 0.8849.\]In words, about $88.49\%$ of the normal distribution with mean $50$ and standard deviation $10$ falls below $62$.
Watch this section: looking up $z=1.20$ at 1:39.
Z-Scores Measure Distance From The Mean
The second example shifts the purpose of the z-score. Instead of asking for area, the video asks how many standard deviations a value is from the mean.
The example uses
\[x=29,\qquad \mu=34,\qquad \sigma=3.\]Substitute into the formula:
\[z=\frac{29-34}{3}.\]Then simplify:
\[z=\frac{-5}{3}\approx -1.67.\]The negative sign tells you the value is below the mean. The size, $1.67$, tells you the distance in standard deviations. So $29$ is about $1.67$ standard deviations below the mean.
Watch this section: standard deviation measurement example at 2:39.
Why The Conversion Matters
The video closes by naming the two main uses.
First, z-scores let you use one standard normal table instead of needing a different table for every possible normal distribution.
Second, the z-score itself tells how far a value is from the mean in standard-deviation units:
\[z>0\]means the value is above the mean, while
\[z<0\]means the value is below the mean.
Watch this section: summary of the two uses at 3:35.
Timestamp Guide
Related Statistics Work
For the broader topic, use the Normal Distribution notes.
For statistics fundamentals, review the mean notes, since the z-score formula depends on comparing a value to the mean.
The video description also links to the JoeCMath normal distribution playlist: watch the normal distribution playlist on YouTube.