Using the Normal Distribution, Calculate P(X < 8)

Z = | X - μ |

σ/√5 |

Z = | 8 - 8.1 |

0.1/√5 |

Z = | -0.1 |

0.1/2.2360679774998 |

Z = | -0.1 |

0.044721359549996 |

Z = -2.236

From the table below, we see that our Z value is =

The z-table probability runs from 0 to z and -z to 0, so we lookup our value

From the table below, we find our value of 0.487455

Since that represents ½ of the graph, we subtract our value from 0.5 → 0.5 - 0.487455

P(x<-2.236) =

0 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 | |

0 | 0 | 0.004 | 0.008 | 0.012 | 0.016 | 0.0199 | 0.0239 | 0.0279 | 0.0319 | 0.0359 |

0.1 | 0.0398 | 0.0438 | 0.0478 | 0.0517 | 0.0557 | 0.0596 | 0.0636 | 0.0675 | 0.0714 | 0.0753 |

0.2 | 0.0793 | 0.0832 | 0.0871 | 0.091 | 0.0948 | 0.0987 | 0.1026 | 0.1064 | 0.1103 | 0.1141 |

0.3 | 0.1179 | 0.1217 | 0.1255 | 0.1293 | 0.1331 | 0.1368 | 0.1406 | 0.1443 | 0.148 | 0.1517 |

0.4 | 0.1554 | 0.1591 | 0.1628 | 0.1664 | 0.17 | 0.1736 | 0.1772 | 0.1808 | 0.1844 | 0.1879 |

0.5 | 0.1915 | 0.195 | 0.1985 | 0.2019 | 0.2054 | 0.2088 | 0.2123 | 0.2157 | 0.219 | 0.2224 |

0.6 | 0.2257 | 0.2291 | 0.2324 | 0.2357 | 0.2389 | 0.2422 | 0.2454 | 0.2486 | 0.2517 | 0.2549 |

0.7 | 0.258 | 0.2611 | 0.2642 | 0.2673 | 0.2704 | 0.2734 | 0.2764 | 0.2794 | 0.2823 | 0.2852 |

0.8 | 0.2881 | 0.291 | 0.2939 | 0.2967 | 0.2995 | 0.3023 | 0.3051 | 0.3078 | 0.3106 | 0.3133 |

0.9 | 0.3159 | 0.3186 | 0.3212 | 0.3238 | 0.3264 | 0.3289 | 0.3315 | 0.334 | 0.3365 | 0.3389 |

1 | 0.3413 | 0.3438 | 0.3461 | 0.3485 | 0.3508 | 0.3531 | 0.3554 | 0.3577 | 0.3599 | 0.3621 |

1.1 | 0.3643 | 0.3665 | 0.3686 | 0.3708 | 0.3729 | 0.3749 | 0.377 | 0.379 | 0.381 | 0.383 |

1.2 | 0.3849 | 0.3869 | 0.3888 | 0.3907 | 0.3925 | 0.3944 | 0.3962 | 0.398 | 0.3997 | 0.4015 |

1.3 | 0.4032 | 0.4049 | 0.4066 | 0.4082 | 0.4099 | 0.4115 | 0.4131 | 0.4147 | 0.4162 | 0.4177 |

1.4 | 0.4192 | 0.4207 | 0.4222 | 0.4236 | 0.4251 | 0.4265 | 0.4279 | 0.4292 | 0.4306 | 0.4319 |

1.5 | 0.4332 | 0.4345 | 0.4357 | 0.437 | 0.4382 | 0.4394 | 0.4406 | 0.4418 | 0.4429 | 0.4441 |

1.6 | 0.4452 | 0.4463 | 0.4474 | 0.4484 | 0.4495 | 0.4505 | 0.4515 | 0.4525 | 0.4535 | 0.4545 |

1.7 | 0.4554 | 0.4564 | 0.4573 | 0.4582 | 0.4591 | 0.4599 | 0.4608 | 0.4616 | 0.4625 | 0.4633 |

1.8 | 0.4641 | 0.4649 | 0.4656 | 0.4664 | 0.4671 | 0.4678 | 0.4686 | 0.4693 | 0.4699 | 0.4706 |

1.9 | 0.4713 | 0.4719 | 0.4726 | 0.4732 | 0.4738 | 0.4744 | 0.475 | 0.4756 | 0.4761 | 0.4767 |

2 | 0.4772 | 0.4778 | 0.4783 | 0.4788 | 0.4793 | 0.4798 | 0.4803 | 0.4808 | 0.4812 | 0.4817 |

2.1 | 0.4821 | 0.4826 | 0.483 | 0.4834 | 0.4838 | 0.4842 | 0.4846 | 0.485 | 0.4854 | 0.4857 |

2.2 | 0.4861 | 0.4864 | 0.4868 | 0.4871 | 0.4875 | 0.4878 | 0.4881 | 0.4884 | 0.4887 | 0.489 |

2.3 | 0.4893 | 0.4896 | 0.4898 | 0.4901 | 0.4904 | 0.4906 | 0.4909 | 0.4911 | 0.4913 | 0.4916 |

2.4 | 0.4918 | 0.492 | 0.4922 | 0.4925 | 0.4927 | 0.4929 | 0.4931 | 0.4932 | 0.4934 | 0.4936 |

2.5 | 0.4938 | 0.494 | 0.4941 | 0.4943 | 0.4945 | 0.4946 | 0.4948 | 0.4949 | 0.4951 | 0.4952 |

2.6 | 0.4953 | 0.4955 | 0.4956 | 0.4957 | 0.4959 | 0.496 | 0.4961 | 0.4962 | 0.4963 | 0.4964 |

2.7 | 0.4965 | 0.4966 | 0.4967 | 0.4968 | 0.4969 | 0.497 | 0.4971 | 0.4972 | 0.4973 | 0.4974 |

2.8 | 0.4974 | 0.4975 | 0.4976 | 0.4977 | 0.4977 | 0.4978 | 0.4979 | 0.4979 | 0.498 | 0.4981 |

2.9 | 0.4981 | 0.4982 | 0.4982 | 0.4983 | 0.4984 | 0.4984 | 0.4985 | 0.4985 | 0.4986 | 0.4986 |

3 | 0.4987 | 0.4987 | 0.4987 | 0.4988 | 0.4988 | 0.4989 | 0.4989 | 0.4989 | 0.499 | 0.499 |

NOTE: To get in Microsoft Excel using your z-score, enter the formula =NORMSDIST(-2.236)

P(x<-2.236) = **0.012545**

P(x<-2.236) = **0.012545**

Free Normal Distribution Calculator - Calculates the probability that a random variable is less than or greater than a value or between 2 values using the Normal Distribution z-score (z value) method (Central Limit Theorem).

Also calculates the Range of values for the 68-95-99.7 rule, or three-sigma rule, or empirical rule. Calculates z score probability

This calculator has 4 inputs.

Also calculates the Range of values for the 68-95-99.7 rule, or three-sigma rule, or empirical rule. Calculates z score probability

This calculator has 4 inputs.

- distribution
- value range for a variable
- empirical rule
- Provides estimate for the spread of data in a normal distribution. 68% of the data will fall within one standard deviation of the mean. 95% of the data will fall within two standard deviations of the mean. 99.7% of the data will fall within three standard deviations of the mean
- event
- a set of outcomes of an experiment to which a probability is assigned.
- mean
- A statistical measurement also known as the average
- normal distribution
- an arrangement of a data set in which most values cluster in the middle of the range and the rest taper off symmetrically toward either extreme.
- probability
- the likelihood of an event happening. This value is always between 0 and 1.

P(Event Happening) = Number of Ways the Even Can Happen / Total Number of Outcomes - standard deviation
- a measure of the amount of variation or dispersion of a set of values. The square root of variance
- variance
- How far a set of random numbers are spead out from the mean
- z score
- the number of standard deviations from the mean a data point is. Also known as a standard score

Add This Calculator To Your Website