A standardized z score table (standard normal table) reports values of \(\Phi(z)=P(Z\le z)\) for the standard normal random variable \(Z\sim N(0,1)\), linking z-scores to cumulative probability, tail probability, and normal-distribution intervals.
Meaning of a standardized z score
A z-score standardizes a value \(x\) from a normal distribution \(X\sim N(\mu,\sigma^2)\) by expressing it in units of standard deviations: \[ z=\frac{x-\mu}{\sigma}. \] Under this standardization, \(Z=\frac{X-\mu}{\sigma}\) follows the standard normal distribution \(N(0,1)\).
Meaning of the standardized z score table
The most common standardized z score table lists the cumulative distribution function (CDF) of \(Z\): \[ \Phi(z)=P(Z\le z). \] Many printed tables display only \(z\ge 0\); negative z-scores are handled using symmetry of the normal curve.
Probability relationships used with z tables
Many standardized z score tables use the left-tail cumulative convention \(\Phi(z)=P(Z\le z)\). When a table uses a different convention (right tail or mean-to-z area), simple conversions map back to \(\Phi(z)\).
| Quantity | Expression in terms of \(\Phi(z)\) | Interpretation |
|---|---|---|
| Cumulative (left tail) | \(\Phi(z)\) | Area to the left of \(z\) |
| Right tail | \(P(Z\ge z)=1-\Phi(z)\) | Area to the right of \(z\) |
| Mean to z (for \(z\ge 0\)) | \(P(0\le Z\le z)=\Phi(z)-\tfrac12\) | Area between 0 and \(z\) |
| Symmetry for negatives | \(\Phi(-z)=1-\Phi(z)\) | Reflection about 0 |
| Central two-sided probability | \(P(-z\le Z\le z)=2\Phi(z)-1\) | Area between \(-z\) and \(z\) |
Worked probabilities using the standardized z score table
For \(Z\sim N(0,1)\) and \(z=1.23\), the table entry \(\Phi(1.23)=0.8907\) gives \(P(Z\le 1.23)=0.8907\). The complementary right-tail probability satisfies \(P(Z\ge 1.23)=1-0.8907=0.1093\).
For a nonstandard normal variable \(X\sim N(\mu,\sigma^2)\), the standardization \(z=(x-\mu)/\sigma\) converts a probability statement about \(X\) into one about \(Z\). As an illustration, if \(X\sim N(50,10^2)\) and \(x=62.3\), then \[ z=\frac{62.3-50}{10}=1.23, \qquad P(X\le 62.3)=P\!\left(Z\le 1.23\right)=\Phi(1.23)=0.8907. \]
Interval probabilities use two z-scores. If \(X\sim N(50,10^2)\) and the event is \(45
Many hypothesis tests and confidence intervals require a critical value \(z_\alpha\) satisfying \(P(Z\ge z_\alpha)=\alpha\), equivalently \(\Phi(z_\alpha)=1-\alpha\).
A common two-sided 95% level corresponds to \(\alpha/2=0.025\), giving \(\Phi(z)=0.975\) and \(z\approx 1.96\).
The table below lists \(\Phi(z)=P(Z\le z)\) for \(Z\sim N(0,1)\) and nonnegative z-scores.
Negative values use \(\Phi(-z)=1-\Phi(z)\).
Row labels give \(z\) to one decimal place. Column labels add the second decimal place. Example: row 1.2 and column 0.03 correspond to \(z=1.23\).
For large \(|z|\), the normal tail probability becomes very small. Table rounding can mask differences in extreme tails; higher-precision computation is preferred when \(1-\Phi(z)\) is near the table’s rounding threshold.
Critical z values and inverse lookup
Level
Cumulative probability \(\Phi(z)\)
Approximate z
Right tail \(1-\Phi(z)\)
90% one-sided
0.9000
1.282
0.1000
95% one-sided
0.9500
1.645
0.0500
95% two-sided (central)
0.9750
1.960
0.0250
99% two-sided (central)
0.9950
2.576
0.0050
Standardized z score table for \(\Phi(z)\) with \(z\ge 0\)
Standardized z score table (cumulative \(\Phi(z)\))
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 Numerical caution for extreme z-scores