A 93% confidence interval uses a procedure with 93% long-run coverage under its assumptions. The parameter is fixed; this does not assign it a 93% probability after an interval is calculated. For a two-sided z interval, the central 93% of the standard normal distribution lies between −z* and +z*.

The critical z-value for a two-sided 93% confidence interval is approximately 1.811911, usually rounded to 1.81.

How to calculate the Z-score for 93% 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 93% confidence interval:

  1. Determine tail areas: The area outside the central 93% is 7%. Split it equally: 3.5% in each tail of the standard normal distribution.
  2. Calculate cumulative area: At the positive critical value, the area to the left is (93% + 3.5%)/100 = 0.965. Equivalently, use (1 + 0.93)/2.
  3. Find the critical value: Look for cumulative area 0.965 in a standard normal table, or calculate its inverse CDF. This gives z* ≈ 1.811911. 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 93% confidence interval.

Calculate Z-score for 93% confidence interval
Table 1: Look up the 0.965 value in the table. It’s the intersection of 1.8 and 0.01, meaning that your Z-score is approximately 1.81.

Considering Z-score Approximations: Are They Enough?

Use z* ≈ 1.81 for a quick two-decimal estimate, or retain 1.811911 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 93% confidence interval which is 0.965, but has 0.9649 (Z = 1.81) and 0.9656 (Z = 1.82), we can interpolate to estimate the Z-score.

  1. Identify Lower and Upper Bounds:
    • Lower bound Z-score (Z1) = 1.81 with a cumulative area of 0.9649.
    • Upper bound Z-score (Z2) = 1.82 with a cumulative area of 0.9656.
  2. Determine the Cumulative Area You Need (A):
    • A = 0.96 (the target cumulative area for a 93% CI).
  3. Calculate the Difference in Area and Z-scores:
    • Difference in cumulative area (ΔA) = 0.9656 – 0.9649= 0.007.
    • Difference in Z-scores (ΔZ) = 1.82- 1.81= 0.01.
  4. Calculate the Proportion of Your Area Within the Interval:
    • Proportion (P) = (A – 0.9649) / ΔA = (0.965 – 0.9649) / 0.0007 = 0.14285714285
  5. Estimate the Z-score:
    • Estimated Z = Z1 + (P * ΔZ) = 1.81+ 0.14285714285 * 0.01 = 1.81142857143

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 1.811911 is a useful six-decimal approximation.