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Normal Distribution

Determining Probabilities and Percents using z-scores

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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?

normal distribution with a standard range and percentages

This probability is unclear. Here, the standard normal distribution, which is a normal distribution with a mean and standard deviation 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.

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 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 This can be written as
P(z-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.

Example

Use the standard normal table

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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

Show Solution expand_more

Note that the data set is not a standard normal distribution. However, we can locate the z-score that corresponds with the top 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 we'll use the the bottom 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.
24.7=x40
64.7=x
x=64.7
A score of 65 points would put you among the top
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