Normal Distribution, Z-Score and P-Value Calculator

Q: Why does my textbook say 0.4750 and the cumulative table 0.9750 for z = 1.96?

They are different table types. The 0.4750 one measures the area from the mean (z = 0) to 1.96; the 0.9750 one measures the area from minus infinity to 1.96, which includes the 0.5 of the left half. The difference is always 0.5 for positive z.

Q: How do I find a z with three decimals, like 1.965?

The table stops at the second decimal. Type the value in the lookup field to see the exact Φ (for 1.965 it is 0.9753), or interpolate between columns 0.06 and 0.07. The calculator does it without interpolation.

Q: Can I use the z table for the t distribution?

No. The t depends on the degrees of freedom and has heavier tails: the 95% two-tailed critical value is 1.96 for z but 2.228 with 10 df and 2.042 with 30 df. Use the P-value tab, which accepts z and t.

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z table standard normal distribution

Z table for the standard normal distribution

Open the Z table tab of the tool, pick the type, type a z to highlight the cell and copy or print. The four variations exist because each textbook adopts a different convention, and mixing them up is the most common exam mistake.

How to read the z table without picking the wrong type

The table row carries z to the first decimal and the column carries the second decimal. For z = 1.96, find row 1.9 and column 0.06. What sits inside the cell depends on the type. In the cumulative table, the most common, it is the area to the left, Φ(1.96) = 0.9750. In the "mean to z" table, like the NIST one, it is the area between 0 and z, 0.4750. In the right-tail table it is the area to the right, 0.0250. Look at the shaded figure at the top of your textbook table: it tells you which is which.
For a negative z, the symmetry of the curve does the job: Φ(−z) = 1 − Φ(z). That is why there is a cumulative table just for z < 0, where Φ(−1.5) = 0.0668. If you only have the positive-z table, compute 1 − 0.9332 and reach the same number. The tool table covers z from 0.00 to 3.99; beyond that the cumulative area already exceeds 0.99996, and the card above the table shows the exact value for any z, including 5 or 6, where the tail is 2.9e-7 and 9.9e-10.
The values come from the error function computed at machine precision, not from a typed-in table, so there are no transcription errors or accumulated rounding. Below the table you find the most requested critical values: 1.2816 (80% two-tailed), 1.6449 (90%), 1.9600 (95%), 2.3263 (98%), 2.5758 (99%) and 3.2905 (99.9%). Use the copy button to take the table to a spreadsheet, or print for an A4 sheet with 11 readable columns.

Checked examples

Cumulative area for z = 1.96

Input: z = 1,96
Expected output: Φ = 0,9750

Row 1.9, column 0.06.

Mean to z = 1

Input: z = 1,00
Expected output: 0,3413

Φ(1) − 0.5 = 0.8413 − 0.5.

Negative z

Input: z = −1,5
Expected output: Φ = 0,0668

1 − 0.9332 by symmetry.

Right tail of z = 3

Input: z = 3,00
Expected output: 0,0013

1 − 0.9987; the two-tailed value is 0.0027.

Critical value at 99% two-tailed

Input: 99%
Expected output: z = 2,5758

Φ(2.5758) = 0.995.

Full tool FAQ

How do you calculate a probability in the normal distribution?

Standardize the value with z = (x − μ) / σ and read the cumulative area Φ(z). For P(X < x) the answer is Φ(z); for P(X > x) it is 1 − Φ(z); for P(a < X < b) it is Φ(zb) − Φ(za). With μ = 100, σ = 15 and x = 115, z = 1 and P(X < 115) = 0.8413. In the calculator, pick the type, type the values and read the result with its step-by-step work.

What is a z-score and how do I interpret it?

How do I find the x value that matches a percentile?

What is the difference between a one-tailed and a two-tailed p-value?

What does the 68-95-99.7 empirical rule say?

How do I read the z table?

When should I use t instead of z for the p-value?

Why is P(X = x) zero?

How accurate are the calculations?

Can I use it for data that is not normal?

Frequently asked questions

Why does my textbook say 0.4750 and the cumulative table 0.9750 for z = 1.96?

They are different table types. The 0.4750 one measures the area from the mean (z = 0) to 1.96; the 0.9750 one measures the area from minus infinity to 1.96, which includes the 0.5 of the left half. The difference is always 0.5 for positive z.

How do I find a z with three decimals, like 1.965?

The table stops at the second decimal. Type the value in the lookup field to see the exact Φ (for 1.965 it is 0.9753), or interpolate between columns 0.06 and 0.07. The calculator does it without interpolation.

Can I use the z table for the t distribution?

No. The t depends on the degrees of freedom and has heavier tails: the 95% two-tailed critical value is 1.96 for z but 2.228 with 10 df and 2.042 with 30 df. Use the P-value tab, which accepts z and t.

Next steps

Share of values within k standard deviations of the mean
Band	Range	Rule	Exact
μ ± 1σ	85 – 115	68%	68.2689%
μ ± 2σ	70 – 130	95%	95.45%
μ ± 3σ	55 – 145	99.7%	99.73%

Share of values within k standard deviations of the mean

Band

Range

Rule

Exact

μ ± 1σ

85 – 115

68%

68.2689%

μ ± 2σ

70 – 130

95%

95.45%

μ ± 3σ

55 – 145

99.7%

99.73%

How to read the z table without picking the wrong type

The table row carries z to the first decimal and the column carries the second decimal. For z = 1.96, find row 1.9 and column 0.06. What sits inside the cell depends on the type. In the cumulative table, the most common, it is the area to the left, Φ(1.96) = 0.9750. In the "mean to z" table, like the NIST one, it is the area between 0 and z, 0.4750. In the right-tail table it is the area to the right, 0.0250. Look at the shaded figure at the top of your textbook table: it tells you which is which.

For a negative z, the symmetry of the curve does the job: Φ(−z) = 1 − Φ(z). That is why there is a cumulative table just for z < 0, where Φ(−1.5) = 0.0668. If you only have the positive-z table, compute 1 − 0.9332 and reach the same number. The tool table covers z from 0.00 to 3.99; beyond that the cumulative area already exceeds 0.99996, and the card above the table shows the exact value for any z, including 5 or 6, where the tail is 2.9e-7 and 9.9e-10.

The values come from the error function computed at machine precision, not from a typed-in table, so there are no transcription errors or accumulated rounding. Below the table you find the most requested critical values: 1.2816 (80% two-tailed), 1.6449 (90%), 1.9600 (95%), 2.3263 (98%), 2.5758 (99%) and 3.2905 (99.9%). Use the copy button to take the table to a spreadsheet, or print for an A4 sheet with 11 readable columns.

Checked examples

Cumulative area for z = 1.96

Input: z = 1,96
Expected output: Φ = 0,9750

Row 1.9, column 0.06.

Mean to z = 1

Input: z = 1,00
Expected output: 0,3413

Φ(1) − 0.5 = 0.8413 − 0.5.

Negative z

Input: z = −1,5
Expected output: Φ = 0,0668

1 − 0.9332 by symmetry.

Right tail of z = 3

Input: z = 3,00
Expected output: 0,0013

1 − 0.9987; the two-tailed value is 0.0027.

Critical value at 99% two-tailed

Input: 99%
Expected output: z = 2,5758

Φ(2.5758) = 0.995.

Full tool FAQ

How do you calculate a probability in the normal distribution?

What is a z-score and how do I interpret it?

How do I find the x value that matches a percentile?

What is the difference between a one-tailed and a two-tailed p-value?

What does the 68-95-99.7 empirical rule say?

How do I read the z table?

When should I use t instead of z for the p-value?

Why is P(X = x) zero?

How accurate are the calculations?

Can I use it for data that is not normal?

Frequently asked questions