A 90% confidence interval conveys a 90% likelihood that the true population parameter is encompassed within our determined range. This level yields greater assurance compared to an 80% confidence interval but is less stringent than the 95% confidence level often used. Selecting a 90% confidence level achieves a compromise, enhancing the reliability of findings with a moderate increase in sample size requirements. This balance makes it an efficient choice for studies seeking a higher degree of confidence without the extensive demands associated with a 95% interval.

The Z-score for a 90% confidence interval is roughly 1.64.

How to calculate the Z-score for 90% 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 90% confidence interval:

1. Determine Tail Areas: Since a 90% confidence interval means 90% of the data is within the interval, there’s 5% of the data in each tail of the distribution (100% – 90% = 10%, divided by 2 because there are two tails).
2. Calculate Cumulative Area: For a 90% confidence interval, we want the area to the left of the Z-score to include half of the tail area, plus the confidence interval itself. This equals 0.95 (90% confidence + 5% tail).
3. Find the Z-score in the Z-table: Look up the area closest to 0.95 in the Z-table. The Z-score that corresponds to this area is approximately 1.64. This is the value that indicates our data point is 1.64 standard deviations from the mean.

Below is a Z-table and how it was used to identify the Z-score for a 90% confidence interval.

Is Approximating the Z-score Sufficient?

If you need a quick estimate that is easy to calculate and understand, using an approximation like 1.64 is perfectly acceptable. It’s especially useful in educational settings or initial analyses where the exact precision isn’t critical. Whether to use an approximation or the exact Z-score for a confidence interval calculation depends on the context and precision requirements of your analysis.

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 90% confidence interval which is 0.94, but has 0.9495 (Z = 1.64) and 0.9505 (Z = 1.65), we can interpolate to estimate the Z-score.

1. Identify Lower and Upper Bounds:
• Lower bound Z-score (Z1) = 1.64 with a cumulative area of 0.9495.
• Upper bound Z-score (Z2) = 1.65 with a cumulative area of 0.9505.
2. Determine the Cumulative Area You Need (A):
• A = 0.95 (the target cumulative area for a 90% CI).
3. Calculate the Difference in Area and Z-scores:
• Difference in cumulative area (ΔA) = 0.9505 – 0.9495 = 0.0010.
• Difference in Z-scores (ΔZ) = 1.65 – 1.64 = 0.01.
4. Calculate the Proportion of Your Area Within the Interval:
• Proportion (P) = (A – 0.9495) / ΔA = (0.95 – 0.9495) / 0.0010 = 0.5
5. Estimate the Z-score:
• Estimated Z = Z1 + (P * ΔZ) = 1.64 + 0.5 * 0.01 = 1.645

For most general purposes, educational contexts, and preliminary analyses, using an approximation like 1.64 for a 90% confidence interval is entirely adequate. However, for rigorous scientific research, publishing, or any situation where the accuracy of your results is paramount, taking the extra step to calculate the exact Z-score is the best approach.