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# Understanding the Z-Table The Z-table, also known as the standard normal table, is a mathematical table that tells us the percentage of values below a Z-score in a standard normal distribution. ## Parts of a Z-Table A Z-table has rows and columns. The rows represent the Z-score up to the first...

# Understanding the Z-Table The Z-table, also known as the standard normal table, is a mathematical table that tells us the percentage of values below a Z-score in a standard normal distribution. ## Parts of a Z-Table A Z-table has rows and columns. The rows represent the Z-score up to the first decimal place. The columns represent the second decimal place. For example, say you are trying to find the area below $z = 1.15$. You would look at the row labeled 1.1 and the column labeled 0.05. | | 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 | The value where the row and column intersect is the area below that Z-score. Therefore, the area below $z = 1.15$ is 0.8749. In other words, 87.49% of the data in a standard normal distribution is below $z = 1.15$. ## How to Use a Z-Table 1. State your conclusion. 2. Standardize your X-value to a Z-score. * $z = \frac{x - \mu}{\sigma}$ 3. Look up your Z-score on the Z-table. 4. Solve the problem based on the area found. ## Example A coffee shop determines that the average customer spends \$13.50 with a standard deviation of \$4. Suppose a particular customer spends \$16.00. What percentage of customers spend less than them? 1. We want to determination the area of customers below x = 16. 2. $z = \frac{16 - 13.50}{4} = 0.625 \approx 0.63$ 3. Look up z = 0.63 on the Z-table. 4. 0. 7357 Therefore, 73.57% of customers spend less than them.