A 80% confidence interval uses a procedure with 80% long-run coverage under its assumptions. The parameter is fixed; this does not assign it a 80% probability after an interval is calculated. For a two-sided z interval, the central 80% of the standard normal distribution lies between −z* and +z*.

The critical z-value for a two-sided 80% confidence interval is approximately 1.281552, usually rounded to 1.28.

How to calculate the Z-score for 80% confidence interval

A Z-table provides the area (probability) to the left of a Z-score in a standard normal distribution. Here’s how to use it for an 80% confidence interval:

  1. Determine tail areas: The area outside the central 80% is 20%. Split it equally: 10% in each tail of the standard normal distribution.
  2. Calculate cumulative area: At the positive critical value, the area to the left is (80% + 10%)/100 = 0.9. Equivalently, use (1 + 0.8)/2.
  3. Find the critical value: Look for cumulative area 0.9 in a standard normal table, or calculate its inverse CDF. This gives z* ≈ 1.281552. Use −z* and +z* as the two critical bounds.

Below is a Z-table and how it was used to identify the Z-score for an 80% confidence interval.

Calculate the Z-score for 80% confidence interval
Table 1: Look up the 0.9 value in the table. It’s closer to the intersection of 1.2 and 0.08, meaning that your Z-score is approximately 1.28.

Should I just use an approximation?

Use z* ≈ 1.28 for a quick two-decimal estimate, or retain 1.281552 for more precision. These are rounded representations of the same critical value; table interpolation is also an approximation.

You can use an arithmetic method to estimate the Z-score between two values in a Z-table if the exact value for your desired confidence level isn’t listed. This approach, often called linear interpolation, can give you a closer approximation of the Z-score than simply selecting the closest value.

Calculating the Z-score using linear interpolation

The Z-table doesn’t list 0.9 exactly but has values for 0.8997 (Z = 1.28) and 0.9015 (Z = 1.29). You can interpolate to estimate the Z-score for a cumulative area of 0.9.

  1. Identify Lower and Upper Bounds:
    • Lower bound Z-score (Z1) = 1.28 with a cumulative area of 0.8997.
    • Upper bound Z-score (Z2) = 1.29 with a cumulative area of 0.9015.
  2. Determine the Cumulative Area You Need (A):
    • A = 0.9 (the target cumulative area for an 80% CI).
  3. Calculate the Difference in Area and Z-scores:
    • Difference in cumulative area (ΔA) = 0.9015 – 0.8997 = 0.0018.
    • Difference in Z-scores (ΔZ) = 1.29 – 1.28 = 0.01.
  4. Calculate the Proportion of Your Area Within the Interval:
    • Proportion (P) = (A – 0.8997) / ΔA = (0.9 – 0.8997) / 0.0018 = 0.16666666666
  5. Estimate the Z-score:
    • Estimated Z = Z1 + (P * ΔZ) = 1.28 + 0.16666666666 * 0.01 = 1.28166666667

A table or linear interpolation gives an approximation, not an exact decimal value. For confidence-interval calculations, retain more digits in z* and round only the final bounds. The value 1.281552 is a useful six-decimal approximation.