An 86% confidence interval indicates that there is an 86% probability the true population parameter is captured within our calculated range. This level provides more certainty than an 80% confidence interval but less than the more commonly used 95% level. Choosing an 86% confidence level strikes a balance between confidence and efficiency, allowing for slightly smaller sample sizes compared to higher confidence levels, thereby saving time and resources while still offering a reasonable degree of certainty.

The Z-score for an 86% confidence interval is approximately 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: Since an 86% confidence interval means 86% of the data is within the interval, there’s 7% of the data in each tail of the distribution (100% – 86% = 14%, divided by 2 because there are two tails).
2. Calculate Cumulative Area: For an 86% 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 0.93 (86% confidence + 7% tail).
3. Find the Z-score in the Z-table: Look up the area closest to 0.93 in the Z-table. The Z-score that corresponds to this area is approximately 1.48. This is the value that indicates our data point is 1.48 standard deviations from the mean.

Below is a Z-table and how it was used to identify the Z-score for an 86% confidence interval.

## Is a Z-score approximation enough?

If you need a quick estimate that is easy to calculate and understand, using an approximation like 1.48 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 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

For most general purposes, educational contexts, and preliminary analyses, using an approximation like 1.48 for an 86% 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.