Z Score Table for Confidence Intervals
For a confidence interval you need one z value: the one that leaves (1 + confidence level) / 2 of the distribution to its left. For 95% that is 1.96; for 90% it is 1.645; for 99% it is 2.576. This page gives the full table, the method for any level in between, and the interval formula the value plugs into.
Try the method yourself in the finder: for a 93% interval, (1 + 0.93) / 2 = 0.965, and typing 0.965 in the probability box returns z ≈ 1.81. No printed table of "common levels" can do that — the reverse lookup can.
The z value for every common confidence level
| Confidence level | Alpha | Look up in table | z value |
|---|---|---|---|
| 80% | 0.20 | 0.9000 | 1.282 |
| 85% | 0.15 | 0.9250 | 1.440 |
| 90% | 0.10 | 0.9500 | 1.645 |
| 92% | 0.08 | 0.9600 | 1.751 |
| 95% | 0.05 | 0.9750 | 1.960 |
| 96% | 0.04 | 0.9800 | 2.054 |
| 98% | 0.02 | 0.9900 | 2.326 |
| 99% | 0.01 | 0.9950 | 2.576 |
| 99.5% | 0.005 | 0.9975 | 2.807 |
| 99.9% | 0.001 | 0.9995 | 3.291 |
The (1 + CL) / 2 method, step by step
- Turn the confidence level into a decimal. 95% becomes 0.95.
- Apply the formula: (1 + 0.95) / 2 = 0.975. This is the cumulative probability your z must have to its left — the middle 95% plus the 2.5% lower tail.
- Read the table backwards. Find 0.975 in the body of the z score table (it sits at row 1.9, column 0.06) and read off z = 1.96.
- Use ±z in the interval. The interval is symmetric: estimate ± 1.96 × standard error.
Why the formula works: a confidence interval keeps the middle slice of the distribution and discards both tails equally. If the level is CL, the discarded share is 1 − CL, half in each tail, so the lower fence of the kept slice has (1 − CL)/2 below it and the upper fence has (1 − CL)/2 + CL = (1 + CL)/2 below it. Every row of the table above is that one idea at a different level — which is why a level nobody printed, like 93% or 97%, is a ten-second job instead of a dead end.
Worked intervals
Example 1 — a 95% interval for a mean
A sample of 64 measurements has mean 112 and the population standard deviation is known to be 16. The standard error is 16 / √64 = 2. Margin of error: 1.96 × 2 = 3.92. The 95% confidence interval is 112 ± 3.92, that is 108.08 to 115.92. Meaning, stated carefully: the method used to build this interval captures the true mean in 95% of repeated samples.
Example 2 — the same data at 99%
Swap 1.96 for 2.576: margin of error 2.576 × 2 = 5.15, interval 106.85 to 117.15. Higher confidence is bought with width — the interval grows by about 31% — which is the honest price of being wrong less often. If a report shows a suspiciously narrow interval at 99.9%, check the arithmetic before admiring the precision.
Example 3 — a 90% interval for a proportion
In a survey of 400 people, 46% support a proposal. The standard error of the proportion is √(0.46 × 0.54 / 400) = 0.0249. At 90%, z = 1.645, so the margin of error is 1.645 × 0.0249 = 0.0410. Interval: 41.9% to 50.1% — wide enough to include 50%, which is why the headline "majority supports" would not survive this interval. Pollsters quoting ±3 points at 95% are using samples near 1,100; the z value is never the hard part, the sample size is.
The confidence interval formula
One warning that markers love to test: the z-based interval assumes the population standard deviation is known, or the sample is large enough (a common rule of thumb is n ≥ 30) for the sample version to behave. With a small sample and an estimated standard deviation, the correct multiplier comes from the t distribution and is larger than z — see z score vs t score for the boundary. Using 1.96 where a t value belongs understates your uncertainty, the one sin interval estimation cannot forgive.
Frequently asked questions
What z value should I use for a 95% confidence interval?
1.96. It is the z score with 0.9750 of the distribution to its left: the middle 95% plus the lower 2.5% tail. In the interval formula it appears as ±1.96 multiplied by the standard error.
How do I find the z value for a confidence level that is not in the table?
Compute (1 + CL) / 2 and run a reverse lookup. For 97%: (1.97)/2 = 0.985, and the z with 0.985 to its left is 2.17. The finder at the top of this page does it in one step — type 0.985 in the probability box.
Why do the tails split equally?
Because a standard two-sided interval is built symmetrically: it can miss high or miss low, and the convention is to risk both equally. One-sided intervals exist, but they answer a different question and use the one-tailed critical values instead.
What is the difference between these z values and critical z values?
They are the same numbers seen from two jobs. In hypothesis testing they are called critical values (the fence a statistic must cross); in interval estimation they are the multiplier that sets the width. The critical z values page lists them by alpha with the testing framing.
Does a 95% confidence interval mean there is a 95% chance the true mean is inside?
Strictly, no — the true mean is fixed and the interval either contains it or not. The 95% describes the procedure: intervals built this way contain the truth in 95% of repeated samples. Most working statisticians say "we are 95% confident", and knowing the precise meaning keeps you honest in reports.