The z-value for a confidence interval is a critical value from the standard normal distribution that corresponds to how confident you want to be. For the most common confidence levels, you can simply memorize or reference these values: 1.645 for 90%, 1.96 for 95%, 2.33 for 98%, and 2.575 for 99%. If you need a z-value for a non-standard confidence level, you can calculate it yourself using a z-table or a calculator.
Why Alpha Gets Split in Half
Every confidence interval is two-sided. It extends both above and below your sample mean, which means the “leftover” probability (the area outside your interval) gets split between two tails of the normal distribution. That leftover probability is called alpha (α), and it equals 1 minus your confidence level expressed as a decimal.
For a 95% confidence interval, α = 0.05. Because the uncertainty is split evenly into both tails, each tail contains 0.025 (half of 0.05). The z-value you need is the one that leaves exactly 0.025 in the upper tail, which is 1.96. This “divide alpha by two” step is the key to finding any z-value for a confidence interval, not just the common ones.
The Three-Step Process
No matter what confidence level you’re working with, the steps are the same:
- Step 1: Calculate alpha. Subtract the confidence level from 1. For 90% confidence, α = 1 − 0.90 = 0.10.
- Step 2: Divide alpha by 2. This gives you the area in one tail. For 90% confidence, α/2 = 0.05.
- Step 3: Find the z-score that corresponds to a cumulative area of 1 − α/2. For 90% confidence, that’s 1 − 0.05 = 0.95. The z-value with a cumulative area of 0.95 is 1.645.
This works for any confidence level. Say your instructor asks for an 92% confidence interval. Alpha is 0.08, so α/2 = 0.04. You need the z-score where the cumulative area to the left equals 1 − 0.04 = 0.96. Looking that up gives you approximately 1.75.
Common Z-Values Worth Memorizing
Most statistics courses and textbooks cycle through the same four confidence levels. Memorizing these saves time on exams and homework:
- 90% confidence: z = 1.645
- 95% confidence: z = 1.96
- 98% confidence: z = 2.33
- 99% confidence: z = 2.575
The 95% confidence level (z = 1.96) appears most often. If you only memorize one, make it that one.
How to Read a Z-Table
A standard normal (z) table shows cumulative probabilities, meaning the total area under the curve to the left of a given z-score. To use it in reverse and find a z-value from a probability, you’ll search the body of the table for the probability you calculated in Step 3, then read the z-score from the row and column headers.
The left column of the table lists the first decimal place of the z-score (for example, 1.9). The top row lists the second decimal place (for example, 0.06). The cell where row 1.9 and column 0.06 intersect contains the cumulative probability for z = 1.96, which is 0.9750. That matches the cumulative area you need for a 95% confidence interval (1 − 0.025 = 0.975), confirming that 1.96 is correct.
When you’re looking up a non-standard confidence level, you may not find your exact probability in the table. In that case, pick the z-score whose cumulative probability is closest to the value you need, or interpolate between the two nearest entries.
Finding Z-Values on a TI-84 Calculator
If you have a TI-84 graphing calculator, the inverse normal function does all the table-lookup work for you. Press 2nd, then VARS to open the distribution menu, and select the inverse normal option (typically the third item in the list).
You’ll be prompted for three inputs: the area (probability), the mean, and the standard deviation. For a standard normal distribution, the mean is 0 and the standard deviation is 1. The area you enter is the cumulative probability from Step 3. For a 95% confidence interval, you’d enter 0.975. Select the left tail option, press Enter, and the calculator returns 1.96.
If your calculator model doesn’t have a tail-selection option, just enter the cumulative left-tail area directly (1 − α/2) and the result will be the positive z-value you need.
Putting the Z-Value Into the Formula
Once you have your z-value, it plugs directly into the confidence interval formula for a population mean:
Confidence interval = sample mean ± z × (standard deviation ÷ √n)
The z-value controls the width of the interval. A higher confidence level produces a larger z-value, which stretches the interval wider. That’s the tradeoff: more confidence means a less precise estimate. Going from 90% to 99% confidence nearly doubles the z-value (from 1.645 to 2.575), so your interval grows substantially even though your data hasn’t changed.
This formula assumes you know the population standard deviation or have a large enough sample (generally n ≥ 30) that the normal approximation holds. For smaller samples where the population standard deviation is unknown, a t-value replaces the z-value. The process for finding it is similar, but you use a t-table with degrees of freedom equal to n − 1 instead of the standard normal table.