A 96% confidence interval uses a procedure with 96% long-run coverage under its assumptions. The parameter is fixed; this does not assign it a 96% probability after an interval is calculated. For a two-sided z interval, the central 96% of the standard normal distribution lies between −z* and +z*.

The critical z-value for a two-sided 96% confidence interval is approximately 2.053749, usually rounded to 2.05.

How to calculate the Z-score for 96% 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 a 96% confidence interval:

  1. Determine tail areas: The area outside the central 96% is 4%. Split it equally: 2% in each tail of the standard normal distribution.
  2. Calculate cumulative area: At the positive critical value, the area to the left is (96% + 2%)/100 = 0.98. Equivalently, use (1 + 0.96)/2.
  3. Find the critical value: Look for cumulative area 0.98 in a standard normal table, or calculate its inverse CDF. This gives z* ≈ 2.053749. 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 a 96% confidence interval.

Calculate Z-score for 96% confidence interval
Table 1: Look up the 0.98 value in the table. It’s the intersection of 2.0 and 0.05, meaning that your Z-score is approximately 2.05.

Deciding Between Z-score Approximations

Use z* ≈ 2.05 for a quick two-decimal estimate, or retain 2.053749 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

Given that the Z-table does not list the exact cumulative area for a 96% confidence interval which is 0.98, but has 0.9798 (Z = 2.05) and 0.9803 (Z = 2.06), we can interpolate to estimate the Z-score.

  1. Identify Lower and Upper Bounds:
    • Lower bound Z-score (Z1) = 2.05 with a cumulative area of 0.9798.
    • Upper bound Z-score (Z2) = 2.06 with a cumulative area of 0.9803.
  2. Determine the Cumulative Area You Need (A):
    • A = 0.98 (the target cumulative area for a 96% CI).
  3. Calculate the Difference in Area and Z-scores:
    • Difference in cumulative area (ΔA) = 0.9803 – 0.9798 = 0.005.
    • Difference in Z-scores (ΔZ) = 2.06 – 2.05= 0.01.
  4. Calculate the Proportion of Your Area Within the Interval:
    • Proportion (P) = (A – 0.9798) / ΔA = (0.98 – 0.9798) / 0.0005 = 0.4
  5. Estimate the Z-score:
    • Estimated Z = Z1 + (P * ΔZ) = 2.05 + 0.4 * 0.01 = 2.054

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 2.053749 is a useful six-decimal approximation.