Standard normal distribution (mean 0, SD 1) · areas from the z-table · z-table.com
Empirical rule: 68.27% of a normal distribution is within 1 standard deviation of the mean, 95.45% within 2, and 99.73% within 3. The dashed lines mark z = ±1.96, which holds exactly 95.00%.
Areas are for the standard normal distribution (mean 0, standard deviation 1), rounded to 5 decimals (percentages to 2). Full table: z-table.com.
The horizontal axis is z, the number of standard deviations from the mean. The shaded area over any stretch of the axis is the share of the distribution in that range. The z-table lists the area to the left of each z; the chart turns those numbers into a picture so you can see the middle area and the tails at once.
Question: exam scores are normal with mean 70 and standard deviation 10. What share of students score between 50.4 and 89.6, and what share score above 89.6?
Step 1. Convert to z: (89.6 − 70) / 10 = 1.96 and (50.4 − 70) / 10 = −1.96.
Step 2. Tap ±1.96 (95%) on the chart. The middle band is 95.00%.
Step 3. The right tail is 1 − 0.97500 = 0.02500, so 2.50% score above 89.6.
Rule-of-thumb check: ±2 SD (50 to 90) holds 95.45%, slightly more than 95% because 2 is a bit wider than 1.96.
For a normal distribution, 68.27% lies within ±1 standard deviation (z = −1 to +1), 95.45% within ±2, and 99.73% within ±3. That is the 68-95-99.7 empirical rule, with the exact areas shown on the chart.
Because the area between −1.96 and +1.96 is 95.00%, while the area between −2 and +2 is 95.45%. Use 1.96 when you need exactly 95%; z = 2 is only the rounded rule of thumb.
Pick the band you want, then tap Print this chart. The print layout hides the site menu and ads and prints the curve plus the reference table in landscape. Choose “Save as PDF” in the print dialog for a PDF copy.
Type it into the “Your z” box (0 to 3.99). The chart shades ±z and shows the left area, the middle area and each tail, taken from the z-table. For example, z = 1.28 has a left area of 0.89973 and a right tail of 0.10027.
No. This chart is the standard normal (z) curve. Student’s t has fatter tails, so its critical values are larger for small samples. For t values, use the t-table on ttable.org and its t vs z critical value chart.