Z-score tables, also known as standard normal tables or Z-tables, provide a way to find the probability of a particular Z-score occurring within a normal distribution. There are two types of Z-score tables: left-tailed (negative Z-scores) and right-tailed (positive Z-scores). These tables help in determining how unusual or typical a particular score is within a dataset.

Reading the Left Z-score Table (Negative)

The left Z-score table typically lists Z-scores on the leftmost column, representing data points below the mean. The body of the table shows the probability of a score falling at or below a given Z-score. For instance, a Z-score of -1.96 corresponds to a probability of approximately 0.025. This means there’s a 2.5% chance of a score falling below this Z-score in a standard normal distribution.

Left Z-score Table

Reading the Right Z-score Table (Positive)

Conversely, the right Z-score table deals with positive Z-scores, which are scores above the mean. Reading this table involves understanding the probability of a score falling at or above a particular Z-score. For example, a Z-score of 1.96 in a right-tailed table also indicates a 2.5% chance, but this time it’s the likelihood of a score falling above this value.

Right Z-score Table

Practical Examples and Applications

In a college entrance exam with a mean score of 500 and a standard deviation of 100, what is the probability of a student scoring less than 450?

Solution: To find this, we first calculate the Z-score.

The formula for Z-score is \( Z = \frac { X – mean}{standard_deviation} \)

For our example Z-score = \( \frac {(450−500)}{100} = -0.5\)

A Z-score of -0.5 indicates that the score of 450 is half a standard deviation below the mean. Now, using the left Z-score table, we look up the probability corresponding to a Z-score of -0.5. In the table, this value is 0.3085. This means there’s a 30.85% chance of a student scoring below 450 on this exam.

Note: Since a Z-score of -0.5 is relatively close to the mean it suggests that scoring below 450 is not highly unusual in this distribution.

Limitations and Considerations

While Z-score tables are incredibly useful, they have limitations. They are only accurate for normally distributed data. In datasets with significant skew or kurtosis, Z-scores might not provide the correct insights. That’s why it’s crucial to understand the context and nature of the data when applying Z-scores.

Below we explain why context matters, along with some “on the contrary” examples:

a) Normal Distribution Assumption: Z-scores are based on the assumption that the data follows a normal distribution. If the data is heavily skewed or has a significant departure from normality, using Z-scores might lead to misleading interpretations.

  • Deviant Scenario: In a dataset representing annual incomes in a region with a large income disparity, the distribution might be right-skewed with a few very high incomes. Using Z-scores here could suggest that moderately high incomes are more ‘unusual’ than they actually are, due to the influence of the extremely high incomes.

b) Outliers Impact: Z-scores are sensitive to outliers. In datasets with extreme values, the mean and standard deviation (which are used to calculate Z-scores) can be heavily influenced by these outliers, leading to distorted Z-scores.

  • Deviant Scenario: In a small class’s test scores, if one student scores exceptionally high or low, it can skew the average and standard deviation, thus affecting the Z-scores of all other students. A score that would normally be considered average might appear as an outlier when it is actually not.

c) Scale and Units of Measurement: The meaning and implications of a Z-score can vary depending on the scale and units of measurement in the dataset. This necessitates understanding the data’s context to interpret Z-scores correctly.

  • Deviant Scenario: In psychological testing, a Z-score might indicate a clinically significant deviation from the norm. However, the same Z-score in a different context, like height measurements in a population, might be perfectly normal and not indicate any unusual characteristic.

d) Misinterpretation of Probabilities: Z-scores correspond to probabilities in a normal distribution. Misinterpreting these probabilities can lead to incorrect conclusions, especially in non-normal distributions.

  • Deviant Scenario: In a distribution of a rare event, like major natural disasters, using Z-scores might suggest a higher probability of extreme values than what is realistically the case, as rare events do not typically follow a normal distribution.

Z-table in HTML Format

Below you can see the HTML versions of the left Z-table and the right Z-table, which allows copy and pasting.

Left Z-table

Z-score0.000.010.020.030.040.050.060.070.080.09
-3.40.00030.00030.00030.00030.00030.00030.00030.00030.00030.0002
-3.30.00050.00050.00050.00040.00040.00040.00040.00040.00040.0003
-3.20.00070.00070.00060.00060.00060.00060.00060.00050.00050.0005
-3.10.00100.00090.00090.00090.00080.00080.00080.00080.00070.0007
-3.00.00130.00130.00130.00120.00120.00110.00110.00110.00100.0010
-2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014
-2.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.0019
-2.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.0026
-2.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.0036
-2.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.0048
-2.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.0064
-2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084
-2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.0110
-2.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.0143
-2.00.02280.02220.02170.02120.02070.02020.01970.01920.01880.0183
-1.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.0233
-1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294
-1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367
-1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455
-1.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.0559
-1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681
-1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823
-1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985
-1.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.1170
-1.00.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379
-0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611
-0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867
-0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148
-0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451
-0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776
-0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121
-0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483
-0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859
-0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247
-0.00.50000.49600.49200.48800.48400.48010.47610.47210.46810.4641

Right Z-table

Z-score0.000.010.020.030.040.050.060.070.080.09
0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359
0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753
0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141
0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517
0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879
0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224
0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549
0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852
0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133
0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389
1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621
1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830
1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015
1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177
1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319
1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441
1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545
1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633
1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706
1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767
2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817
2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857
2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890
2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916
2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936
2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952
2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964
2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974
2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981
2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986
3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990
3.10.99900.99910.99910.99910.99920.99920.99920.99920.99930.9993
3.20.99930.99930.99940.99940.99940.99940.99940.99950.99950.9995
3.30.99950.99950.99950.99960.99960.99960.99960.99960.99960.9997
3.40.99970.99970.99970.99970.99970.99970.99970.99970.99970.9998

Related: Check out our Z-score calculator for easier computations.

How do I make my own Z-score tables in Google Sheets?

Start by creating the header column: 0.00 to 0.09 (increments of 0.01). The create the vertical ones 0.0 to whatever you choose (increments of 0.1).

Then use the NORMDIST() function and sum up (for the right Z-table) the two cells you need. In our case D4 is “=NORMDIST($B4 + D$2)”. Blocking the B column and the $2 row in the formula allos for dragging it to the whole table and generate the needed Z-score values.

Right Z-score table in Google Sheets

You do similarly with the left Z-table except that you need to substract instead of summing the two cells.

Left Z-score table in Google Sheets