## Compute a P-value from a Z-score

P(𝑍 < 𝑥) | 0.50000 | |
---|---|---|

P(𝑍 > 𝑥) | 0.50000 | |

P(0 < 𝑍 < 𝑥) | 0.00000 | |

P(−𝑥 < 𝑍 < 𝑥) | 0.00000 | |

P(𝑍 < −𝑥 or 𝑍 > 𝑥) | 1.00000 |

## Probability between two Z-Scores

P(𝑥_{min} < 𝑍 < 𝑥_{max}) |
0.34134 | |
---|---|---|

P(𝑍 < 𝑥_{min} or 𝑍 > 𝑥_{max}) |
0.65866 | |

P(𝑍 < 𝑥_{min}) |
0.50000 | |

P(𝑍 > 𝑥_{max}) |
0.15866 |

## Z Score Calculator

## Negative Z Table

Z | .00 | .01 | .02 | .03 | .04 | .05 | .06 | .07 | .08 | .09 |
---|---|---|---|---|---|---|---|---|---|---|

-3.9 | 0.00005 | 0.00005 | 0.00004 | 0.00004 | 0.00004 | 0.00004 | 0.00004 | 0.00004 | 0.00003 | 0.00003 |

-3.8 | 0.00007 | 0.00007 | 0.00007 | 0.00006 | 0.00006 | 0.00006 | 0.00006 | 0.00005 | 0.00005 | 0.00005 |

-3.7 | 0.00011 | 0.0001 | 0.0001 | 0.0001 | 0.00009 | 0.00009 | 0.00008 | 0.00008 | 0.00008 | 0.00008 |

-3.6 | 0.00016 | 0.00015 | 0.00015 | 0.00014 | 0.00014 | 0.00013 | 0.00013 | 0.00012 | 0.00012 | 0.00011 |

-3.5 | 0.00023 | 0.00022 | 0.00022 | 0.00021 | 0.0002 | 0.00019 | 0.00019 | 0.00018 | 0.00017 | 0.00017 |

-3.4 | 0.00034 | 0.00032 | 0.00031 | 0.0003 | 0.00029 | 0.00028 | 0.00027 | 0.00026 | 0.00025 | 0.00024 |

-3.3 | 0.00048 | 0.00047 | 0.00045 | 0.00043 | 0.00042 | 0.0004 | 0.00039 | 0.00038 | 0.00036 | 0.00035 |

-3.2 | 0.00069 | 0.00066 | 0.00064 | 0.00062 | 0.0006 | 0.00058 | 0.00056 | 0.00054 | 0.00052 | 0.0005 |

-3.1 | 0.00097 | 0.00094 | 0.0009 | 0.00087 | 0.00084 | 0.00082 | 0.00079 | 0.00076 | 0.00074 | 0.00071 |

-3.0 | 0.00135 | 0.00131 | 0.00126 | 0.00122 | 0.00118 | 0.00114 | 0.00111 | 0.00107 | 0.00104 | 0.001 |

-2.9 | 0.00187 | 0.00181 | 0.00175 | 0.00169 | 0.00164 | 0.00159 | 0.00154 | 0.00149 | 0.00144 | 0.00139 |

-2.8 | 0.00256 | 0.00248 | 0.0024 | 0.00233 | 0.00226 | 0.00219 | 0.00212 | 0.00205 | 0.00199 | 0.00193 |

-2.7 | 0.00347 | 0.00336 | 0.00326 | 0.00317 | 0.00307 | 0.00298 | 0.00289 | 0.0028 | 0.00272 | 0.00264 |

-2.6 | 0.00466 | 0.00453 | 0.0044 | 0.00427 | 0.00415 | 0.00402 | 0.00391 | 0.00379 | 0.00368 | 0.00357 |

-2.5 | 0.00621 | 0.00604 | 0.00587 | 0.0057 | 0.00554 | 0.00539 | 0.00523 | 0.00508 | 0.00494 | 0.0048 |

-2.4 | 0.0082 | 0.00798 | 0.00776 | 0.00755 | 0.00734 | 0.00714 | 0.00695 | 0.00676 | 0.00657 | 0.00639 |

-2.3 | 0.01072 | 0.01044 | 0.01017 | 0.0099 | 0.00964 | 0.00939 | 0.00914 | 0.00889 | 0.00866 | 0.00842 |

-2.2 | 0.0139 | 0.01355 | 0.01321 | 0.01287 | 0.01255 | 0.01222 | 0.01191 | 0.0116 | 0.0113 | 0.01101 |

-2.1 | 0.01786 | 0.01743 | 0.017 | 0.01659 | 0.01618 | 0.01578 | 0.01539 | 0.015 | 0.01463 | 0.01426 |

-2.0 | 0.02275 | 0.02222 | 0.02169 | 0.02118 | 0.02068 | 0.02018 | 0.0197 | 0.01923 | 0.01876 | 0.01831 |

-1.9 | 0.02872 | 0.02807 | 0.02743 | 0.0268 | 0.02619 | 0.02559 | 0.025 | 0.02442 | 0.02385 | 0.0233 |

-1.8 | 0.03593 | 0.03515 | 0.03438 | 0.03362 | 0.03288 | 0.03216 | 0.03144 | 0.03074 | 0.03005 | 0.02938 |

-1.7 | 0.04457 | 0.04363 | 0.04272 | 0.04182 | 0.04093 | 0.04006 | 0.0392 | 0.03836 | 0.03754 | 0.03673 |

-1.6 | 0.0548 | 0.0537 | 0.05262 | 0.05155 | 0.0505 | 0.04947 | 0.04846 | 0.04746 | 0.04648 | 0.04551 |

-1.5 | 0.06681 | 0.06552 | 0.06426 | 0.06301 | 0.06178 | 0.06057 | 0.05938 | 0.05821 | 0.05705 | 0.05592 |

-1.4 | 0.08076 | 0.07927 | 0.0778 | 0.07636 | 0.07493 | 0.07353 | 0.07215 | 0.07078 | 0.06944 | 0.06811 |

-1.3 | 0.0968 | 0.0951 | 0.09342 | 0.09176 | 0.09012 | 0.08851 | 0.08691 | 0.08534 | 0.08379 | 0.08226 |

-1.2 | 0.11507 | 0.11314 | 0.11123 | 0.10935 | 0.10749 | 0.10565 | 0.10383 | 0.10204 | 0.10027 | 0.09853 |

-1.1 | 0.13567 | 0.1335 | 0.13136 | 0.12924 | 0.12714 | 0.12507 | 0.12302 | 0.121 | 0.119 | 0.11702 |

-1.0 | 0.15866 | 0.15625 | 0.15386 | 0.15151 | 0.14917 | 0.14686 | 0.14457 | 0.14231 | 0.14007 | 0.13786 |

-0.9 | 0.18406 | 0.18141 | 0.17879 | 0.17619 | 0.17361 | 0.17106 | 0.16853 | 0.16602 | 0.16354 | 0.16109 |

-0.8 | 0.21186 | 0.20897 | 0.20611 | 0.20327 | 0.20045 | 0.19766 | 0.19489 | 0.19215 | 0.18943 | 0.18673 |

-0.7 | 0.24196 | 0.23885 | 0.23576 | 0.2327 | 0.22965 | 0.22663 | 0.22363 | 0.22065 | 0.2177 | 0.21476 |

-0.6 | 0.27425 | 0.27093 | 0.26763 | 0.26435 | 0.26109 | 0.25785 | 0.25463 | 0.25143 | 0.24825 | 0.2451 |

-0.5 | 0.30854 | 0.30503 | 0.30153 | 0.29806 | 0.2946 | 0.29116 | 0.28774 | 0.28434 | 0.28096 | 0.2776 |

-0.4 | 0.34458 | 0.3409 | 0.33724 | 0.3336 | 0.32997 | 0.32636 | 0.32276 | 0.31918 | 0.31561 | 0.31207 |

-0.3 | 0.38209 | 0.37828 | 0.37448 | 0.3707 | 0.36693 | 0.36317 | 0.35942 | 0.35569 | 0.35197 | 0.34827 |

-0.2 | 0.42074 | 0.41683 | 0.41294 | 0.40905 | 0.40517 | 0.40129 | 0.39743 | 0.39358 | 0.38974 | 0.38591 |

-0.1 | 0.46017 | 0.4562 | 0.45224 | 0.44828 | 0.44433 | 0.44038 | 0.43644 | 0.43251 | 0.42858 | 0.42465 |

-0.0 | 0.5 | 0.49601 | 0.49202 | 0.48803 | 0.48405 | 0.48006 | 0.47608 | 0.4721 | 0.46812 | 0.46414 |

## Positive Z Table

Z | .00 | .01 | .02 | .03 | .04 | .05 | .06 | .07 | .08 | .09 |
---|---|---|---|---|---|---|---|---|---|---|

0.0 | 0.5 | 0.50399 | 0.50798 | 0.51197 | 0.51595 | 0.51994 | 0.52392 | 0.5279 | 0.53188 | 0.53586 |

0.1 | 0.53983 | 0.5438 | 0.54776 | 0.55172 | 0.55567 | 0.55962 | 0.56356 | 0.56749 | 0.57142 | 0.57535 |

0.2 | 0.57926 | 0.58317 | 0.58706 | 0.59095 | 0.59483 | 0.59871 | 0.60257 | 0.60642 | 0.61026 | 0.61409 |

0.3 | 0.61791 | 0.62172 | 0.62552 | 0.6293 | 0.63307 | 0.63683 | 0.64058 | 0.64431 | 0.64803 | 0.65173 |

0.4 | 0.65542 | 0.6591 | 0.66276 | 0.6664 | 0.67003 | 0.67364 | 0.67724 | 0.68082 | 0.68439 | 0.68793 |

0.5 | 0.69146 | 0.69497 | 0.69847 | 0.70194 | 0.7054 | 0.70884 | 0.71226 | 0.71566 | 0.71904 | 0.7224 |

0.6 | 0.72575 | 0.72907 | 0.73237 | 0.73565 | 0.73891 | 0.74215 | 0.74537 | 0.74857 | 0.75175 | 0.7549 |

0.7 | 0.75804 | 0.76115 | 0.76424 | 0.7673 | 0.77035 | 0.77337 | 0.77637 | 0.77935 | 0.7823 | 0.78524 |

0.8 | 0.78814 | 0.79103 | 0.79389 | 0.79673 | 0.79955 | 0.80234 | 0.80511 | 0.80785 | 0.81057 | 0.81327 |

0.9 | 0.81594 | 0.81859 | 0.82121 | 0.82381 | 0.82639 | 0.82894 | 0.83147 | 0.83398 | 0.83646 | 0.83891 |

1.0 | 0.84134 | 0.84375 | 0.84614 | 0.84849 | 0.85083 | 0.85314 | 0.85543 | 0.85769 | 0.85993 | 0.86214 |

1.1 | 0.86433 | 0.8665 | 0.86864 | 0.87076 | 0.87286 | 0.87493 | 0.87698 | 0.879 | 0.881 | 0.88298 |

1.2 | 0.88493 | 0.88686 | 0.88877 | 0.89065 | 0.89251 | 0.89435 | 0.89617 | 0.89796 | 0.89973 | 0.90147 |

1.3 | 0.9032 | 0.9049 | 0.90658 | 0.90824 | 0.90988 | 0.91149 | 0.91309 | 0.91466 | 0.91621 | 0.91774 |

1.4 | 0.91924 | 0.92073 | 0.9222 | 0.92364 | 0.92507 | 0.92647 | 0.92785 | 0.92922 | 0.93056 | 0.93189 |

1.5 | 0.93319 | 0.93448 | 0.93574 | 0.93699 | 0.93822 | 0.93943 | 0.94062 | 0.94179 | 0.94295 | 0.94408 |

1.6 | 0.9452 | 0.9463 | 0.94738 | 0.94845 | 0.9495 | 0.95053 | 0.95154 | 0.95254 | 0.95352 | 0.95449 |

1.7 | 0.95543 | 0.95637 | 0.95728 | 0.95818 | 0.95907 | 0.95994 | 0.9608 | 0.96164 | 0.96246 | 0.96327 |

1.8 | 0.96407 | 0.96485 | 0.96562 | 0.96638 | 0.96712 | 0.96784 | 0.96856 | 0.96926 | 0.96995 | 0.97062 |

1.9 | 0.97128 | 0.97193 | 0.97257 | 0.9732 | 0.97381 | 0.97441 | 0.975 | 0.97558 | 0.97615 | 0.9767 |

2.0 | 0.97725 | 0.97778 | 0.97831 | 0.97882 | 0.97932 | 0.97982 | 0.9803 | 0.98077 | 0.98124 | 0.98169 |

2.1 | 0.98214 | 0.98257 | 0.983 | 0.98341 | 0.98382 | 0.98422 | 0.98461 | 0.985 | 0.98537 | 0.98574 |

2.2 | 0.9861 | 0.98645 | 0.98679 | 0.98713 | 0.98745 | 0.98778 | 0.98809 | 0.9884 | 0.9887 | 0.98899 |

2.3 | 0.98928 | 0.98956 | 0.98983 | 0.9901 | 0.99036 | 0.99061 | 0.99086 | 0.99111 | 0.99134 | 0.99158 |

2.4 | 0.9918 | 0.99202 | 0.99224 | 0.99245 | 0.99266 | 0.99286 | 0.99305 | 0.99324 | 0.99343 | 0.99361 |

2.5 | 0.99379 | 0.99396 | 0.99413 | 0.9943 | 0.99446 | 0.99461 | 0.99477 | 0.99492 | 0.99506 | 0.9952 |

2.6 | 0.99534 | 0.99547 | 0.9956 | 0.99573 | 0.99585 | 0.99598 | 0.99609 | 0.99621 | 0.99632 | 0.99643 |

2.7 | 0.99653 | 0.99664 | 0.99674 | 0.99683 | 0.99693 | 0.99702 | 0.99711 | 0.9972 | 0.99728 | 0.99736 |

2.8 | 0.99744 | 0.99752 | 0.9976 | 0.99767 | 0.99774 | 0.99781 | 0.99788 | 0.99795 | 0.99801 | 0.99807 |

2.9 | 0.99813 | 0.99819 | 0.99825 | 0.99831 | 0.99836 | 0.99841 | 0.99846 | 0.99851 | 0.99856 | 0.99861 |

3.0 | 0.99865 | 0.99869 | 0.99874 | 0.99878 | 0.99882 | 0.99886 | 0.99889 | 0.99893 | 0.99896 | 0.999 |

3.1 | 0.99903 | 0.99906 | 0.9991 | 0.99913 | 0.99916 | 0.99918 | 0.99921 | 0.99924 | 0.99926 | 0.99929 |

3.2 | 0.99931 | 0.99934 | 0.99936 | 0.99938 | 0.9994 | 0.99942 | 0.99944 | 0.99946 | 0.99948 | 0.9995 |

3.3 | 0.99952 | 0.99953 | 0.99955 | 0.99957 | 0.99958 | 0.9996 | 0.99961 | 0.99962 | 0.99964 | 0.99965 |

3.4 | 0.99966 | 0.99968 | 0.99969 | 0.9997 | 0.99971 | 0.99972 | 0.99973 | 0.99974 | 0.99975 | 0.99976 |

3.5 | 0.99977 | 0.99978 | 0.99978 | 0.99979 | 0.9998 | 0.99981 | 0.99981 | 0.99982 | 0.99983 | 0.99983 |

3.6 | 0.99984 | 0.99985 | 0.99985 | 0.99986 | 0.99986 | 0.99987 | 0.99987 | 0.99988 | 0.99988 | 0.99989 |

3.7 | 0.99989 | 0.9999 | 0.9999 | 0.9999 | 0.99991 | 0.99991 | 0.99992 | 0.99992 | 0.99992 | 0.99992 |

3.8 | 0.99993 | 0.99993 | 0.99993 | 0.99994 | 0.99994 | 0.99994 | 0.99994 | 0.99995 | 0.99995 | 0.99995 |

3.9 | 0.99995 | 0.99995 | 0.99996 | 0.99996 | 0.99996 | 0.99996 | 0.99996 | 0.99996 | 0.99997 | 0.99997 |

## How to use a Z Table

A **z-table**, also called **standard normal table**, is a table used to find the percentage of values below a given z-score in a standard normal distribution.

A **z-score**, also known as **standard score**, indicates how many standard deviations away a data point is above (or below) the mean.
A positive z-score implies that the data point is above the mean, while a negative z-score indicates that the data point falls below the mean.

It is calculated with the following formula: **𝑧 = (𝑋 − 𝜇) ⁄ 𝜎** (where **𝑋** is the data point, **𝜇** is the population mean, and **𝜎** is the population standard deviation). A z-score is basically a standardized variable that has been rescaled to have a mean ยต of 0 and a standard deviation ฯ of 1 (which ultimately provides a standard set of z-values - from the z-table - that can be used for easy calculations).

### How to read a Z Table

- Check the sign of your z-score. If it's negative, use a negative z-score table. If it's positive, use a positive z-score table.
- Split the z-score into two parts. The first part goes up to the first digit after the decimal (i.e.
**2.3**4 → 2.3). The second part is made up of the remaining digit (i.e. 2.3**4**→ 0.04). - Look at the leftmost column of the table and find the row that matches the first part of your z-score (e.g. 2.3).
- Look at the topmost row of the table and find the column that matches the second part of your z-score (e.g. 0.04).
- The intersection between the column and the row corresponds to the p-value. For a z-score of 2.34, the p-value is 0.99036 (or 99.036%).

### Example 1

Suppose the scores on a college exam are normally distributed with a mean 𝜇 of 70 and a standard deviation 𝜎 of 4. What is the z-score of the value 75? In other words, what proportion of students scored **less** than 75 points?

- Compute the z-score: 𝑧 = (75 - 70) / 4 = 1.25 (this result means that a score of 75 points is 1.25 standard deviations above from the mean).
- Since the z-score is positive, look for the value 1.25 in the positive z-table: 0.89435 (89.435% of the students scored less than 75 points).

### Example 2

Following the previous example (𝜇 = 70 and 𝜎 = 4), what proportion of students scored **more** than 64 points?

- Compute the z-score: 𝑧 = (64 - 70) / 4 = -1.5 (this result means that a score of 64 points is 1.5 standard deviations below the mean).
- Since the z-score is negative, look for the value -1.5 in the negative z-table: 0.06681 (or 6.681%).
- At this point, we've obtained the proportion of students who scored less than 64 points (P(Z < -1.5) = 0.06681), but since we want to know the proportion of those who scored more than 64 points, we just need to calculate the complement of P(Z < -1.5): 1 - P(Z < -1.5) = 1 - 0.06681 = 0.93319 (93.319%).

### Example 3

Still following the same example (𝜇 = 70 and 𝜎 = 4), what proportion of students scored **between** 68 and 73 points?

- Compute the first z-score: 𝑧 = (68 - 70) / 4 = -0.5.
- Compute the second z-score: 𝑧 = (73 - 70) / 4 = 0.75.
- Look for the value -0.5 in the negative z-table: 0.30854
- Look for the value 0.75 in the positive z-table: 0.77337
- The proportion of students who scored between 68 and 73 points is: P(68 < X < 73) = P(-0.5 < Z < 0.75) = P(Z < 0.75) - P(Z < -0.5) = 0.46483 (46.483%).

### Example 4

What score corresponds to the 90th percentile? In other words, what is the data point 𝑋 for which we get a p-value of 0.9?

- Find a z-score that is the closest of the p-value 0.9: z = 1.28 (with a p-value of 0.89973).
- Compute 𝑋:
**𝑋 = ๐ + ๐ง๐**. 𝑋 ≈ 70 + (1.28 x 4) = 75.12.