Determining Probabilities and Percents using z-scores

Although the behavior of normal distributions is predictable, the standard normal distribution and $z$ -scores can be used to determine percents and probabilities for values that do not fall exactly on a standard deviation mark.

Concept

Standard Normal Distribution

For all normal distributions, the percentages in each interval are always the same.

normal distribution with a standard range and percentages

This graph can be used to determine the probability of picking a data point, $x,$ within a certain number of standard deviations from the mean. But what is the probability of choosing a random data point within, say, less than half of one standard deviation of the mean?

This probability is unclear. Here, the standard normal distribution, which is a normal distribution with a mean $μ = 0$ and standard deviation $σ = 1,$ can be useful. The corresponding curve can be drawn.

For a standard normal distribution, the numbers under the horizontal axis are denoted $z .$

Concept

$z$ -score

For a normally distributed data set, $x$ represents any data point. The following formula can be used to translate any $x$ -value into the corresponding $z$ -value or $z$ -score of the standard normal distribution.

$z = \frac{x - μ}{σ}$

When a

z

-score is known, a Standard Normal Table can be used to determine the corresponding area under the curve.

Method

Standard Normal Table

For a randomly chosen $z$ -score of a standard normal distribution, a standard normal table can be used to determine the probability $z$ is greater than or less than a given value. The table below gives the probability that a data point is less than or equal to a specific $z$ -score.

Standard Normal Table
$z$	$. 0$	$. 1$	$. 2$	$. 3$	$. 4$	$. 5$	$. 6$	$. 7$	$. 8$	$. 9$
$- 3$	$. 00135$	$. 00097$	$. 00069$	$. 00048$	$. 00034$	$. 00023$	$. 00016$	$. 00011$	$. 00007$	$. 00005$
$- 2$	$. 02275$	$. 01786$	$. 01390$	$. 01072$	$. 00820$	$. 00621$	$. 00466$	$. 00347$	$. 00256$	$. 00187$
$- 1$	$. 15866$	$. 13567$	$. 11507$	$. 09680$	$. 08076$	$. 06681$	$. 05480$	$. 04457$	$. 03593$	$. 02872$
$- 0$	$. 50000$	$. 46017$	$. 42074$	$. 38209$	$. 34458$	$. 30854$	$. 27425$	$. 24196$	$. 21186$	$. 18406$
$0$	$. 50000$	$. 53983$	$. 57926$	$. 61791$	$. 65542$	$. 69146$	$. 72575$	$. 75804$	$. 78814$	$. 81594$
$1$	$. 84134$	$. 86433$	$. 88493$	$. 90320$	$. 91924$	$. 93319$	$. 94520$	$. 95543$	$. 96407$	$. 97128$
$2$	$. 97725$	$. 98214$	$. 98610$	$. 98928$	$. 99180$	$. 99379$	$. 99534$	$. 99653$	$. 99744$	$. 99813$
$3$	$. 99865$	$. 99903$	$. 99931$	$. 99952$	$. 99966$	$. 99977$	$. 99984$	$. 99989$	$. 99993$	$. 99995$

The left-hand column gives the whole number portion of $z,$ while the top row gives the decimal part of $z .$
Consider finding the probability that a randomly chosen data point is less than or equal to $z = - 0.5 .$ First, locate the whole number of the $z$ -score in the left-hand column. In this case that is $- 0 .$

Standard Normal Table
$z$	$. 0$	$. 1$	$. 2$	$. 3$	$. 4$	$. 5$	$. 6$	$. 7$	$. 8$	$. 9$
$- 3$	$. 00135$	$. 00097$	$. 00069$	$. 00048$	$. 00034$	$. 00023$	$. 00016$	$. 00011$	$. 00007$	$. 00005$
$- 2$	$. 02275$	$. 01786$	$. 01390$	$. 01072$	$. 00820$	$. 00621$	$. 00466$	$. 00347$	$. 00256$	$. 00187$
$- 1$	$. 15866$	$. 13567$	$. 11507$	$. 09680$	$. 08076$	$. 06681$	$. 05480$	$. 04457$	$. 03593$	$. 02872$
$- 0$	$. 50000$	$. 46017$	$. 42074$	$. 38209$	$. 34458$	$. 30854$	$. 27425$	$. 24196$	$. 21186$	$. 18406$
$0$	$. 50000$	$. 53983$	$. 57926$	$. 61791$	$. 65542$	$. 69146$	$. 72575$	$. 75804$	$. 78814$	$. 81594$
$1$	$. 84134$	$. 86433$	$. 88493$	$. 90320$	$. 91924$	$. 93319$	$. 94520$	$. 95543$	$. 96407$	$. 97128$
$2$	$. 97725$	$. 98214$	$. 98610$	$. 98928$	$. 99180$	$. 99379$	$. 99534$	$. 99653$	$. 99744$	$. 99813$
$3$	$. 99865$	$. 99903$	$. 99931$	$. 99952$	$. 99966$	$. 99977$	$. 99984$	$. 99989$	$. 99993$	$. 99995$

The probability that corresponds to a $z$ -score of $- 0 \dots$ appears in the shaded row above. To determine exactly which cell, consider the decimal portion of $z$ in the top row. Here, that is $. 5 .$

Standard Normal Table
$z$	$. 0$	$. 1$	$. 2$	$. 3$	$. 4$	$. 5$	$. 6$	$. 7$	$. 8$	$. 9$
$- 3$	$. 00135$	$. 00097$	$. 00069$	$. 00048$	$. 00034$	$. 00023$	$. 00016$	$. 00011$	$. 00007$	$. 00005$
$- 2$	$. 02275$	$. 01786$	$. 01390$	$. 01072$	$. 00820$	$. 00621$	$. 00466$	$. 00347$	$. 00256$	$. 00187$
$- 1$	$. 15866$	$. 13567$	$. 11507$	$. 09680$	$. 08076$	$. 06681$	$. 05480$	$. 04457$	$. 03593$	$. 02872$
$- 0$	$. 50000$	$. 46017$	$. 42074$	$. 38209$	$. 34458$	$. 30854$	$. 27425$	$. 24196$	$. 21186$	$. 18406$
$0$	$. 50000$	$. 53983$	$. 57926$	$. 61791$	$. 65542$	$. 69146$	$. 72575$	$. 75804$	$. 78814$	$. 81594$
$1$	$. 84134$	$. 86433$	$. 88493$	$. 90320$	$. 91924$	$. 93319$	$. 94520$	$. 95543$	$. 96407$	$. 97128$
$2$	$. 97725$	$. 98214$	$. 98610$	$. 98928$	$. 99180$	$. 99379$	$. 99534$	$. 99653$	$. 99744$	$. 99813$
$3$	$. 99865$	$. 99903$	$. 99931$	$. 99952$	$. 99966$	$. 99977$	$. 99984$	$. 99989$	$. 99993$	$. 99995$

The selected row and column intersect at $0.30854 .$ Thus, the probability of $z$ being lower than $- 0.5$ is $0.30854,$ or $30.854 % .$ This can be written as $P (z \leq - 0.5) = 0.30854 .$

Note that there are other standard normal tables. For instance, one might give the probability of a value being greater than a specific

z

-score.

Use the standard normal table

fullscreen

Exercise

At a university, students take a pre-test on their first day. Last year's result were normally distributed with a mean of $40$ and standard deviation of $13 .$ Use the standard normal table to find the lowest score possible to rank amongst the top $3 % .$

Show Solution

Solution

Note that the data set is not a standard normal distribution. However, we can locate the $z$ -score that corresponds with the top $3 %,$ then translate it into an $x$ -value. The table shows the area under the curve to the left of $z .$ This means, to find the top $3 %,$ we'll use the the bottom $97 % .$ The probability closest to $0.97$ is $0.97128 .$

Standard Normal Table
$z$	$. 0$	$. 1$	$. 2$	$. 3$	$. 4$	$. 5$	$. 6$	$. 7$	$. 8$	$. 9$
$- 3$	$. 00135$	$. 00097$	$. 00069$	$. 00048$	$. 00034$	$. 00023$	$. 00016$	$. 00011$	$. 00007$	$. 00005$
$- 2$	$. 02275$	$. 01786$	$. 01390$	$. 01072$	$. 00820$	$. 00621$	$. 00466$	$. 00347$	$. 00256$	$. 00187$
$- 1$	$. 15866$	$. 13567$	$. 11507$	$. 09680$	$. 08076$	$. 06681$	$. 05480$	$. 04457$	$. 03593$	$. 02872$
$- 0$	$. 50000$	$. 46017$	$. 42074$	$. 38209$	$. 34458$	$. 30854$	$. 27425$	$. 24196$	$. 21186$	$. 18406$
$0$	$. 50000$	$. 53983$	$. 57926$	$. 61791$	$. 65542$	$. 69146$	$. 72575$	$. 75804$	$. 78814$	$. 81594$
$1$	$. 84134$	$. 86433$	$. 88493$	$. 90320$	$. 91924$	$. 93319$	$. 94520$	$. 95543$	$. 96407$	$. 97128$
$2$	$. 97725$	$. 98214$	$. 98610$	$. 98928$	$. 99180$	$. 99379$	$. 99534$	$. 99653$	$. 99744$	$. 99813$
$3$	$. 99865$	$. 99903$	$. 99931$	$. 99952$	$. 99966$	$. 99977$	$. 99984$	$. 99989$	$. 99993$	$. 99995$

The whole part of the $z$ -value is $1,$ and the decimal part is $. 9 .$

Standard Normal Table
$z$	$. 0$	$. 1$	$. 2$	$. 3$	$. 4$	$. 5$	$. 6$	$. 7$	$. 8$	$. 9$
$- 3$	$. 00135$	$. 00097$	$. 00069$	$. 00048$	$. 00034$	$. 00023$	$. 00016$	$. 00011$	$. 00007$	$. 00005$
$- 2$	$. 02275$	$. 01786$	$. 01390$	$. 01072$	$. 00820$	$. 00621$	$. 00466$	$. 00347$	$. 00256$	$. 00187$
$- 1$	$. 15866$	$. 13567$	$. 11507$	$. 09680$	$. 08076$	$. 06681$	$. 05480$	$. 04457$	$. 03593$	$. 02872$
$- 0$	$. 50000$	$. 46017$	$. 42074$	$. 38209$	$. 34458$	$. 30854$	$. 27425$	$. 24196$	$. 21186$	$. 18406$
$0$	$. 50000$	$. 53983$	$. 57926$	$. 61791$	$. 65542$	$. 69146$	$. 72575$	$. 75804$	$. 78814$	$. 81594$
$1$	$. 84134$	$. 86433$	$. 88493$	$. 90320$	$. 91924$	$. 93319$	$. 94520$	$. 95543$	$. 96407$	$. 97128$
$2$	$. 97725$	$. 98214$	$. 98610$	$. 98928$	$. 99180$	$. 99379$	$. 99534$	$. 99653$	$. 99744$	$. 99813$
$3$	$. 99865$	$. 99903$	$. 99931$	$. 99952$	$. 99966$	$. 99977$	$. 99984$	$. 99989$	$. 99993$	$. 99995$

That means that the

z

-score is

z = 1.9 .

We can convert

z

to its corresponding

x

-value as follows.

z = \frac{x - μ}{σ}

SubstituteValues

Substitute values

1.9 = \frac{x - 4 0}{1 3}

MultEqn

$LHS \cdot 13 = RHS \cdot 13$

24.7 = x - 40

AddEqn

$LHS + 40 = RHS + 40$

64.7 = x

RearrangeEqn

Rearrange equation

x = 64.7

RoundInt

Round to nearest integer

x \approx 65

A score of

65

points would put you among the top

3 % .

Example

Use the standard normal table