Appendix C — Standard Normal Distribution

C.1 Z-Table: Finding Probabilities

The standard normal table (also called a z-table) helps you find probabilities for z-scores. Remember: a z-score tells you how many standard deviations away from the mean a value is. The table tells you what percentage of observations fall below that z-score[1,2].

ImportantWhat the table shows

The number in the table = the proportion (or percentage) of the distribution at or below that z-score.

  • Table value of 0.8413 means 84.13% of observations are at or below that z-score
  • This is also called the cumulative probability: \(P(Z \leq z)\)

C.1.1 How to use the standard normal table

The table works like finding a value in a spreadsheet: you need a row and a column.

C.1.1.1 Finding probabilities for POSITIVE z-scores

Step 1: Break your z-score into a row number and a column number

Your z-score has digits before and after the decimal point. Split it like this:

  • Row: Everything up to the first digit after the decimal (e.g., for 1.23, use row 1.2)
  • Column: The second digit after the decimal (e.g., for 1.23, use column .03)

Example: For z = 1.23: - Row = 1.2 (the 1 and the 2) - Column = .03 (the 3)

Another way to think about it: Round your z-score to one decimal place for the row, then use what’s left over for the column.

  • z = 1.23 → Row: 1.2, Column: .03
  • z = 2.57 → Row: 2.5, Column: .07
  • z = 0.84 → Row: 0.8, Column: .04
  • z = 0.67 → Row: 0.6, Column: .07

Step 2: Find the intersection

Look where your row and column meet—that’s your probability!

Step 3: Interpret the result

  • The value you found = percentage below your z-score
  • Want the percentage above? Subtract from 1.00 (or 100%)
  1. Find the table value: Row 1.2, Column .03 → 0.8907
  2. What this means: 89.07% score at or below z = 1.23
  3. What we want (above): 100% - 89.07% = 10.93% score above z = 1.23

Answer: About 11% of athletes score above z = 1.23

C.1.1.2 Finding probabilities for NEGATIVE z-scores

The table only shows positive z-scores, but you can use symmetry for negative z-scores (the normal curve is perfectly symmetric).

Quick Rule: For a negative z-score, find the positive version in the table, then subtract from 1.00.

Why it works: Because of symmetry, the area below z = -1.50 equals the area above z = +1.50.

  1. Find the positive version: Look up z = +1.50 → 0.9332 (93.32% below +1.50)
  2. Use symmetry: This means 93.32% are above -1.50
  3. What we want (below): 100% - 93.32% = 6.68% below z = -1.50

Answer: About 6.68% fall below z = -1.50

C.1.1.3 Finding probabilities BETWEEN two z-scores

Want the percentage between z = 0.50 and z = 1.50?

  1. Find both cumulative probabilities:
    • z = 1.50 → 0.9332 (93.32% below)
    • z = 0.50 → 0.6915 (69.15% below)
  2. Subtract the smaller from the larger:
    • 93.32% - 69.15% = 24.17% between z = 0.50 and z = 1.50

C.1.2 Interactive Z-Score Calculator

Use this calculator to find probabilities for any z-score without looking up values in the table:

Z-Score Probability Calculator

NoteHow to use the calculator
  1. Type your z-score in the box (can be positive or negative)
  2. Click “Calculate” or press Enter
  3. The calculator shows:
    • Percentage at or below your z-score (left tail)
    • Percentage above your z-score (right tail)
  4. Use this to check your table-reading skills!

C.1.3 Standard normal cumulative probability table

The table below shows cumulative probabilities \(P(Z \leq z)\) for positive z-scores from 0.00 to 3.99[1].

Table C.1: Standard Normal Cumulative Probabilities P(Z ≤ z)
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .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
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998
3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
NotePractice reading the table

Example 1: Find P(Z ≤ 1.23)

  • Row: 1.2
  • Column: .03
  • Answer: 0.8907 → 89.07% of values fall at or below z = 1.23

Example 2: Find P(Z > 1.96)

  • Row: 1.9, Column: .06 → 0.9750
  • This is the area below z = 1.96
  • We want above: 1.0000 - 0.9750 = 0.0250
  • Answer: 2.5% of values fall above z = 1.96

Example 3: Find P(Z ≤ -1.50)

  • First find positive: Row 1.5, Column .00 → 0.9332
  • This is the area below z = +1.50, which equals area above z = -1.50
  • We want below z = -1.50: 1.0000 - 0.9332 = 0.0668
  • Answer: 6.68% of values fall at or below z = -1.50

C.1.4 Common critical values

For quick reference, here are commonly used z-scores and their associated probabilities and percentiles[1]:

Z-score \(P(Z \leq z)\) Percentile Two-tailed probability (
0.00 0.5000 50th 1.0000
0.67 0.7486 75th 0.5028
1.00 0.8413 84th 0.3174
1.28 0.8997 90th 0.2006
1.645 0.9500 95th 0.1000
1.96 0.9750 97.5th 0.0500
2.00 0.9772 97.7th 0.0456
2.326 0.9900 99th 0.0200
2.576 0.9950 99.5th 0.0100
3.00 0.9987 99.87th 0.0026

Note: The z-score of ±1.96 is particularly important because it corresponds to the 95% confidence interval widely used in statistical inference[3].