Skip to main content
Statistics LibreTexts

1.1: Table of Critical Values of z

  1. Last updated
  2. Save as PDF
  • Page ID
    67338

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\dsum}{\displaystyle\sum\limits} \)

    \( \newcommand{\dint}{\displaystyle\int\limits} \)

    \( \newcommand{\dlim}{\displaystyle\lim\limits} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \(\newcommand{\longvect}{\overrightarrow}\)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\ket}[1]{\left| #1 \right>}\)
    \(\newcommand{\bra}[1]{\left< #1 \right|}\)
    \(\newcommand{\braket}[2]{\left< #1 \vphantom{#2} \right| \left. #2 \vphantom{#1} \right>}\)
    \(\newcommand{\braopket}[3]{\left< #1 \vphantom{#2}\vphantom{#3} \right| #2 \vphantom{#1}\vphantom{#3} \left| #3 \vphantom{#1}\vphantom{#2} \right>}\)
    \(\newcommand{\qmvec}[1]{\mathbf{\vec{#1}}}\)
    \(\newcommand{\op}[1]{\hat{\mathbf{#1}}}\)
    \(\newcommand{\expect}[1]{\langle #1 \rangle}\)
    \(\newcommand{\dfn}[1]{\emph{\textbf{#1}}}\)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    You can use this table from Wikimedia Commons (website address: https://commons.wikimedia.org/wiki/File:Tabla_z.png) to find the proportion (multiply the p by 100) of a normal distribution that is more extreme than the z-score. Use the table that is easiest for you! They all have the same numbers.

    Table \(\PageIndex{1}\) shows shows positive z-scores, their probability (p-value), and percentage of scores that fall below that proportion of a standard normal curve. The key to using this table is to understand the the percentages are showing the portion of scores that should be below the z-score that you're using. This means that you'll have to do a little bit of math to The figures below the Table \(\PageIndex{1}\) can help to understand what showing that scores below the z-score might mean.

    Table \(\PageIndex{1}\): Positive z-Scores
    z-Score p-value % Below
    0.00 0.5000 50.00%
    0.01 0.5040 50.40%
    0.02 0.5080 50.80%
    0.03 0.5120 51.20%
    0.04 0.5160 51.60%
    0.05 0.5199 51.99%
    0.06 0.5239 52.39%
    0.07 0.5279 52.79%
    0.08 0.5319 53.19%
    0.09 0.5359 53.59%
    0.10 0.5398 53.98%
    0.11 0.5438 54.38%
    0.12 0.5478 54.78%
    0.13 0.5517 55.17%
    0.14 0.5557 55.57%
    0.15 0.5596 55.96%
    0.16 0.5636 56.36%
    0.17 0.5675 56.75%
    0.18 0.5714 57.14%
    0.19 0.5754 57.54%
    0.20 0.5793 57.93%
    0.21 0.5832 58.32%
    0.22 0.5871 58.71%
    0.23 0.5910 59.10%
    0.24 0.5948 59.48%
    0.25 0.5987 59.87%
    0.26 0.6026 60.26%
    0.27 0.6064 60.64%
    0.28 0.6103 61.03%
    0.29 0.6141 61.41%
    0.30 0.6179 61.79%
    0.31 0.6217 62.17%
    0.32 0.6255 62.55%
    0.33 0.6293 62.93%
    0.34 0.6331 63.31%
    0.35 0.6368 63.68%
    0.36 0.6406 64.06%
    0.37 0.6443 64.43%
    0.38 0.6480 64.80%
    0.39 0.6517 65.17%
    0.40 0.6554 65.54%
    0.41 0.6591 65.91%
    0.42 0.6628 66.28%
    0.43 0.6664 66.64%
    0.44 0.6700 67.00%
    0.45 0.6736 67.36%
    0.46 0.6772 67.72%
    0.47 0.6808 68.08%
    0.48 0.6844 68.44%
    0.49 0.6879 68.79%
    0.50 0.6915 69.15%
    0.51 0.6950 69.50%
    0.52 0.6985 69.85%
    0.53 0.7019 70.19%
    0.54 0.7054 70.54%
    0.55 0.7088 70.88%
    0.56 0.7123 71.23%
    0.67 0.7157 71.57%
    0.58 0.7190 71.90%
    0.59 0.7224 72.24%
    0.60 0.7258 72.58%
    0.61 0.7291 72.91%
    0.62 0.7324 73.24%
    0.63 0.7357 73.57%
    0.64 0.7389 73.89%
    0.65 0.7422 74.22%
    0.66 0.7454 74.54%
    0.67 0.7486 74.86%
    0.68 0.7518 75.18%
    0.69 0.7549 75.49%
    0.70 0.7580 75.80%
    0.71 0.7612 76.12%
    0.72 0.7642 76.42%
    0.73 0.7673 76.73%
    0.74 0.7704 77.04%
    0.75 0.7734 77.34%
    0.76 0.7764 77.64%
    0.77 0.7794 77.94%
    0.78 0.7823 78.23%
    0.79 0.7852 78.52%
    0.80 0.7881 78.81%
    0.81 0.7910 79.10%
    0.82 0.7939 79.39%
    0.83 0.7967 79.67%
    0.84 0.7996 79.96%
    0.85 0.8023 80.23%
    0.86 0.8051 80.51%
    0.87 0.8079 80.79%
    0.88 0.8106 81.06%
    0.89 0.8133 81.33%
    0.90 0.8159 81.59%
    0.91 0.8186 81.86%
    0.92 0.8212 82.12%
    0.93 0.8238 82.38%
    0.94 0.8264 82.64%
    0.95 0.8289 82.89%
    0.96 0.8315 83.15%
    0.97 0.8340 83.40%
    0.98 0.8365 83.65%
    0.99 0.8389 83.89%
    1.00 0.8413 84.13%
    1.01 0.8438 84.38%
    1.02 0.8461 84.61%
    1.03 0.8485 84.85%
    1.04 0.8508 85.08%
    1.05 0.8531 85.31%
    1.06 0.8554 85.54%
    1.07 0.8577 85.77%
    1.08 0.8599 85.99%
    1.09 0.8621 86.21%
    1.10 0.8643 86.43%
    1.11 0.8665 86.65%
    1.12 0.8686 86.86%
    1.13 0.8708 87.08%
    1.14 0.8729 87.29%
    1.15 0.8749 87.49%
    1.16 0.8770 87.70%
    1.17 0.8790 87.90%
    1.18 0.8810 88.10%
    1.19 0.8830 88.30%
    1.20 0.8849 88.49%
    1.21 0.8869 88.69%
    1.22 0.8838 88.38%
    1.23 0.8907 89.07%
    1.24 0.8925 89.25%
    1.25 0.8944 89.44%
    1.26 0.8562 85.62%
    1.27 0.8980 89.80%
    1.28 0.8997 89.97%
    1.29 0.9015 90.15%
    1.30 0.9032 90.32%
    1.31 0.9049 90.49%
    1.32 0.9066 90.66%
    1.33 0.9082 90.82%
    1.34 0.9099 90.99%
    1.35 0.9115 91.15%
    1.36 0.9131 91.31%
    1.37 0.9147 91.47%
    1.38 0.9162 91.62%
    1.39 0.9177 91.77%
    1.40 0.9192 91.92%
    1.41 0.9207 92.07%
    1.42 0.9222 92.22%
    1.43 0.9236 92.36%
    1.44 0.9251 92.51%
    1.45 0.9265 92.65%
    1.46 0.9279 92.79%
    1.47 0.9292 92.92%
    1.48 0.9306 93.06%
    1.49 0.9319 93.19%
    1.50 0.9332 93.32%
    1.51 0.9345 93.45%
    1.52 0.9357 93.57%
    1.53 0.9370 93.70%
    1.54 0.9382 93.82%
    1.55 0.9394 93.94%
    1.56 0.9406 94.06%
    1.57 0.9418 94.18%
    1.58 0.9430 94.30%
    1.59 0.9441 94.41%
    1.60 0.9452 94.52%
    1.61 0.9463 94.63%
    1.62 0.9474 94.74%
    1.63 0.9485 94.85%
    1.64 0.9495 94.95%
    1.65 0.9505 95.05%
    1.66 0.9515 95.15%
    1.67 0.9525 95.25%
    1.68 0.9535 95.35%
    1.69 0.9545 95.45%
    1.70 0.9554 95.54%
    1.71 0.9564 95.64%
    1.72 0.9573 95.73%
    1.73 0.9582 95.82%
    1.74 0.9591 95.91%
    1.75 0.9599 95.99%
    1.76 0.9608 96.08%
    1.77 0.9616 96.16%
    1.78 0.9625 96.25%
    1.79 0.9633 96.33%
    1.80 0.9641 96.41%
    1.81 0.9649 96.49%
    1.82 0.9656 96.56%
    1.83 0.9664 96.64%
    1.84 0.9671 96.71%
    1.85 0.9678 96.78%
    1.86 0.9686 96.86%
    1.87 0.9693 96.93%
    1.88 0.9700 97.00%
    1.89 0.9706 97.06%
    1.90 0.9713 97.13%
    1.91 0.9719 97.19%
    1.92 0.9726 97.26%
    1.93 0.9732 97.32%
    1.94 0.9738 97.38%
    1.95 0.9744 97.44%
    1.96 0.9750 97.50%
    1.97 0.9756 97.56%
    1.98 0.9762 97.62%
    1.99 0.9767 97.67%
    2.00 0.9773 97.73%
    2.01 0.9778 97.78%
    2.02 0.9783 97.83%
    2.03 0.9788 97.88%
    2.04 0.9793 97.93%
    2.05 0.9798 97.98%
    2.06 0.9803 98.03%
    2.07 0.9808 98.08%
    2.08 0.9812 98.12%
    2.09 0.9817 98.17%
    2.10 0.9821 98.21%
    2.11 0.9826 98.26%
    2.12 0.9830 98.30%
    2.13 0.9834 98.34%
    2.14 0.9838 98.38%
    2.15 0.9842 98.42%
    2.16 0.9846 98.46%
    2.17 0.9850 98.50%
    2.18 0.9854 98.54%
    2.19 0.9857 98.57%
    2.20 0.9861 98.61%
    2.21 0.9865 98.65%
    2.22 0.9868 98.68%
    2.23 0.9871 98.71%
    2.24 0.9875 98.75%
    2.25 0.9878 98.78%
    2.26 0.9881 98.81%
    2.27 0.9884 98.84%
    2.28 0.9887 98.87%
    2.29 0.9890 98.90%
    2.30 0.9893 98.93%
    2.31 0.9896 98.96%
    2.32 0.9898 98.98%
    2.33 0.9901 99.01%
    2.34 0.9904 99.04%
    2.35 0.9906 99.06%
    2.36 0.9909 99.09%
    2.37 0.9911 99.11%
    2.38 0.9913 99.13%
    2.39 0.9916 99.16%
    2.40 0.9918 99.18%
    2.41 0.9920 99.20%
    2.42 0.9922 99.22%
    2.43 0.9925 99.25%
    2.44 0.9927 99.27%
    2.45 0.9929 99.29%
    2.46 0.9931 99.31%
    2.47 0.9932 99.32%
    2.48 0.9934 99.34%
    2.49 0.9936 99.36%
    2.50 0.9938 99.38%
    2.51 0.9940 99.40%
    2.52 0.9941 99.41%
    2.53 0.9943 99.43%
    2.54 0.9945 99.45%
    2.55 0.9946 99.46%
    2.56 0.9948 99.48%
    2.57 0.9949 99.49%
    2.58 0.9951 99.51%
    2.59 0.9952 99.52%
    2.60 0.9953 99.53%
    2.61 0.9955 99.55%
    2.62 0.9956 99.56%
    2.63 0.9957 99.57%
    2.64 0.9959 99.59%
    2.65 0.9960 99.60%
    2.66 0.9961 99.61%
    2.67 0.9962 99.62%
    2.68 0.9963 99.63%
    2.69 0.9964 99.64%
    2.70 0.9965 99.65%
    2.71 0.9968 99.68%
    2.72 0.9967 99.67%
    2.73 0.9968 99.68%
    2.74 0.9969 99.69%
    2.75 0.9970 99.70%
    2.76 0.9971 99.71%
    2.77 0.9972 99.72%
    2.78 0.9973 99.73%
    2.79 0.9974 99.74%
    2.80 0.9974 99.74%
    2.81 0.9975 99.75%
    2.82 0.9976 99.76%
    2.83 0.9977 99.77%
    2.84 0.9977 99.77%
    2.85 0.9978 99.78%
    2.86 0.9979 99.79%
    2.87 0.9980 99.80%
    2.88 0.9980 99.80%
    2.89 0.9981 99.81%
    2.90 0.9981 99.81%
    2.91 0.9982 99.82%
    2.92 0.9983 99.83%
    2.93 0.9983 99.83%
    2.94 0.9984 99.84%
    2.95 0.9984 99.84%
    2.96 0.9985 99.85%
    2.97 0.9985 99.85%
    2.98 0.9986 99.86%
    2.99 0.9986 99.86%
    3.00 0.9987 99.87%

    Table \(\PageIndex{1}\): Positive z-Scores. (CC-BY-SA; modified by Michelle Oja from Jsmura via Wikimedia Commons)

    If Table \(\PageIndex{1}\) is too unwieldy, here is a PDF of a z-score table with three sets of each columns (z-score, percent).

    Again, this probabilities (p-values) and percentage below show the z-score that you calculated. Figure \(\PageIndex{1}\) and Figure \(\PageIndex{2}\) show what this might look like in a standard normal curve.

    clipboard_e5132d40de683de0f1a19fcd3c0066b97.png
    Figure \(\PageIndex{1}\): Example of z=1.00 (CC-BY-SA, Ashley Baumann via Wikipedia)
    clipboard_e9bba0b8e728e97c613592ed834420e30.png
    Figure \(\PageIndex{2}\): Example of z=0.60 (CC-BY-SA, Ashley Baumann via Wikipedia)

    In Figure \(\PageIndex{1}\), the z-score was calculated to be 1.00. A z-score of 1.00 is one standard deviation above the mean because the z-score is positive. Using the z-table, this shows us that the percentage below that z-score of 1.00 is 84.13%. This percentage is shaded in Figure \(\PageIndex{1}\). Similarly, the shaded area in Figure \(\PageIndex{2}\) for a z-score of 0.60 is almost exactly the "% Below" shown in Table \(\PageIndex{1}\), which is 72.58%. More details on this example can be found on the Standard Score page of Wikipedia.

    Negative Scores

    Figure \(\PageIndex{1}\) shows a positive z-score of 1.00, but what if the mean was bigger than the target score and we end up with a negative z-score? You can see a z-score for -1 in Figure \(\PageIndex{1}\), but how can you find the percentage of scores that should be below a negative z-score using Table \(\PageIndex{1}\)? Well, one way is to use a different z-table that includes negative z-scores (such as this PDF of very long columns, but includes both positive and negative z-scores). Another option is to think about the Standard Normal Distribution, and realize that 100% of all scores will fall somewhere within it. And if we know that 84.13% of scores should be below a z-score of 1.00, then we could figure out that 100% minus 84.13% is 15.87%. This is the percentage of scores that should fall below z=-1.00.

    Example \(\PageIndex{1}\)

    What portion of scores should fall below a z-score of -0.60 (z=-0.60)?

    Solution

    Looking at Table \(\PageIndex{1}\), we see that the percentage of scores that should score below z=0.60 (a positive z-score) was 72.58%. If we subract that from 100%, then we get the percentage of scores that should fall below a z-score of -0.60.

    \[100-72.58=27.42\nonumber\]

    The portion of scores that should score below z=-0.60 is 27.42%.

    Attributions & Contributors

    1. Back to top
    • Was this article helpful?

    Recommended articles

    1. Article type
      Section or Page
      Author
      Michelle Oja
      License
      CC BY-SA
      Show TOC
      no
    2. Tags
      1. source[1]-stats-18684
      2. source[1]-stats-44838