A 86% confidence interval uses a procedure with 86% long-run coverage under its assumptions. The parameter is fixed; this does not assign it a 86% probability after an interval is calculated. For a two-sided z interval, the central 86% of the standard normal distribution lies between −z* and +z*.

The critical z-value for a two-sided 86% confidence interval is approximately 1.475791, usually rounded to 1.48.

How to calculate the Z-score for 86% 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 86% confidence interval:

  1. Determine tail areas: The area outside the central 86% is 14%. Split it equally: 7% in each tail of the standard normal distribution.
  2. Calculate cumulative area: At the positive critical value, the area to the left is (86% + 7%)/100 = 0.93. Equivalently, use (1 + 0.86)/2.
  3. Find the critical value: Look for cumulative area 0.93 in a standard normal table, or calculate its inverse CDF. This gives z* ≈ 1.475791. 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 86% confidence interval.

Calculate Z-score for 86% confidence interval
Table 1: Look up the 0.93 value in the table. It’s closer to the intersection of 1.4 and 0.08, meaning that your Z-score is approximately 1.48.

Is a Z-score approximation enough?

Use z* ≈ 1.48 for a quick two-decimal estimate, or retain 1.475791 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 an 86% confidence interval which is 0.93, but has 0.9292 (Z = 1.47) and 0.9306 (Z = 1.48), we can interpolate to estimate the Z-score.

  1. Identify Lower and Upper Bounds:
    • Lower bound Z-score (Z1) = 1.47 with a cumulative area of 0.9292.
    • Upper bound Z-score (Z2) = 1.48 with a cumulative area of 0.9306.
  2. Determine the Cumulative Area You Need (A):
    • A = 0.93 (the target cumulative area for an 86% CI).
  3. Calculate the Difference in Area and Z-scores:
    • Difference in cumulative area (ΔA) = 0.9292 – 0.9306= 0.0014.
    • Difference in Z-scores (ΔZ) = 1.48 – 1.47 = 0.01.
  4. Calculate the Proportion of Your Area Within the Interval:
    • Proportion (P) = (A – 0.9292) / ΔA = (0.93 – 0.9292) / 0.0014 = 0.57142857142
  5. Estimate the Z-score:
    • Estimated Z = Z1 + (P * ΔZ) = 1.47 + 0.57142857142 * 0.01 = 1.47571428571

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 1.475791 is a useful six-decimal approximation.