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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.

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

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 $z.$

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=σx−μ $

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 $-0…$ 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 $P(z≤-0.5)=0.30854.$

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.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

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 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

$z$ | $.0$ | $.1$ | $.2$ | $.3$ | $.4$ | $.5$ | $.6$ | $.7$ | $.8$ | $.9$ |

$-3$ | $.00135$ | $.00097$ | $.00069$ | $.00048$ | $.00034$ | $.00023$ | $.00016$ | $.00011$ | $.00007$ | $.00005$ |

$-2$ | $.02275$ | $.01786$ | $.01390$ | $.01072$ | $.00820$ | $.00621$ | $.00466$ | $.00347$ | $.00256$ | $.00187$ |

$-1$ | $.15866$ | $.13567$ | $.11507$ | $.09680$ | $.08076$ | $.06681$ | $.05480$ | $.04457$ | $.03593$ | $.02872$ |

$-0$ | $.50000$ | $.46017$ | $.42074$ | $.38209$ | $.34458$ | $.30854$ | $.27425$ | $.24196$ | $.21186$ | $.18406$ |

$0$ | $.50000$ | $.53983$ | $.57926$ | $.61791$ | $.65542$ | $.69146$ | $.72575$ | $.75804$ | $.78814$ | $.81594$ |

$1$ | $.84134$ | $.86433$ | $.88493$ | $.90320$ | $.91924$ | $.93319$ | $.94520$ | $.95543$ | $.96407$ | $.97128$ |

$2$ | $.97725$ | $.98214$ | $.98610$ | $.98928$ | $.99180$ | $.99379$ | $.99534$ | $.99653$ | $.99744$ | $.99813$ |

$3$ | $.99865$ | $.99903$ | $.99931$ | $.99952$ | $.99966$ | $.99977$ | $.99984$ | $.99989$ | $.99993$ | $.99995$ |

$z=σx−μ $

SubstituteValuesSubstitute values

$1.9=13x−40 $

MultEqn$LHS⋅13=RHS⋅13$

$24.7=x−40$

AddEqn$LHS+40=RHS+40$

$64.7=x$

RearrangeEqnRearrange equation

$x=64.7$

RoundIntRound to nearest integer

$x≈65$

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