A 99% confidence interval uses a procedure with 99% long-run coverage under its assumptions. The parameter is fixed; this does not assign it a 99% probability after an interval is calculated. For a two-sided z interval, the central 99% of the standard normal distribution lies between −z* and +z*.

The critical z-value for a two-sided 99% confidence interval is approximately 2.575829, usually rounded to 2.58.

How to calculate the Z-score for 99% 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 99% confidence interval:

  1. Determine tail areas: The area outside the central 99% is 1%. Split it equally: 0.5% in each tail of the standard normal distribution.
  2. Calculate cumulative area: At the positive critical value, the area to the left is (99% + 0.5%)/100 = 0.995. Equivalently, use (1 + 0.99)/2.
  3. Find the critical value: Look for cumulative area 0.995 in a standard normal table, or calculate its inverse CDF. This gives z* ≈ 2.575829. 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 99% confidence interval.

Calculate Z-score for 99% confidence interval
Table 1: Look up the 0.995 value in the table. It’s the intersection of 2.5 and 0.07 or 0.08, meaning that your Z-score is approximately 2.58.

How Precise Should Your Z-score Be?

Use z* ≈ 2.58 for a quick two-decimal estimate, or retain 2.575829 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 99% confidence interval which is 0.995, but has 0.9949 (Z = 2.57) and 0.9951 (Z = 2.58), we can interpolate to estimate the Z-score.

  1. Identify Lower and Upper Bounds:
    • Lower bound Z-score (Z1) = 2.57 with a cumulative area of 0.9949.
    • Upper bound Z-score (Z2) = 2.58 with a cumulative area of 0.9951.
  2. Determine the Cumulative Area You Need (A):
    • A = 0.98 (the target cumulative area for a 99% CI).
  3. Calculate the Difference in Area and Z-scores:
    • Difference in cumulative area (ΔA) = 0.9951 – 0.9949 = 0.002.
    • Difference in Z-scores (ΔZ) = 2.58 – 2.57= 0.01.
  4. Calculate the Proportion of Your Area Within the Interval:
    • Proportion (P) = (A – 0.9949) / ΔA = (0.9950 – 0.9949) / 0.0002 = 0.5
  5. Estimate the Z-score:
    • Estimated Z = Z1 + (P * ΔZ) = 2.57 + 0.5 * 0.01 = 2.575

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 2.575829 is a useful six-decimal approximation.

What is the Z-score for a 99.5% confidence interval

This exceptionally high level of confidence is seldom used in standard practice but offers near-absolute certainty in the results obtained from statistical analysis. Choosing a 99.5% confidence level underscores an unparalleled commitment to precision, ensuring that the interval is as accurate and representative of the population as possible. Although this level demands a significantly larger sample size than even a 99% confidence interval, the benefit is a profound assurance in the accuracy of the findings, suitable for research where the margin for error must be minimized to the utmost extent.

The Z-score for a 99.5% interval is approximately 2.81.

Find the (1 + 0.995) / 2 = 0.9975 value in the table below to find the Z-score for a 99.5% CI.

Calculate Z-score for 99.5% confidence interval
Table 2: Look up the 0.9975 value in the table. It’s the intersection of 2.8 and 0.01, meaning that your Z-score is approximately 2.81.

What is the Z-score for a 99.7% confidence interval

This level of confidence is exceptionally high, almost guaranteeing the accuracy of the interval in representing the true population parameter.

The Z-score for a 99.7% interval is approximately 2.965.

Find the (1 + 0.995) / 2 = 0.9985 value in the table below to find the Z-score for a 99.7% CI.

Calculate Z-score for 99.7% confidence interval
Table 3: Look up the 0.9985 value in the table. It’s the intersection of 2.9 and 0.06 or 0.07. We’re going to use the median so the Z-score is 2.965.

What is the Z-score for a 99.8% confidence interval

This extraordinarily high level of confidence is rarely employed in everyday statistical analysis but is reserved for scenarios where the implications of uncertainty are especially critical.

The Z-score for a 99.8% interval is approximately 3.09

Find the (1 + 0.998) / 2 = 0.999 value in the table below to find the Z-score for a 99.8% CI.

Calculate Z-score for 99.8% confidence interval
Table 4: Look up the 0.999 value in the table. It’s the intersection of 3.0 and 0.08 or 0.09, and 3.1 and 0.00. We’re going to use the median so the Z-score is 3.09.

What is the Z-score for a 99.9% confidence interval

This level of confidence is among the highest utilized in statistical analysis, reflecting an extreme level of assurance in the interval’s accuracy.

The Z-score for a 99.9% interval is approximately 3.29.

Find the (1 + 0.999) / 2 = 0.9995 value in the table below to find the Z-score for a 99.9% CI.

Calculate Z-score for 99.9% confidence interval
Table 5: Look up the 0.9995 value in the table. It’s the intersection of 3.2 with 0.07, 0.08 and 0.09, but also 3.3 with 0.00 and 0.01 and 0.02. We’re going to use the median so the Z-score is 3.29.