Confidence Interval to Standard Deviation Calculator
Convert a confidence interval percentage into a z-score, the number of standard deviations from the mean of a normal distribution, with formula and example.
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Confidence interval to standard deviation calculator
A confidence interval to standard deviation calculator converts a confidence level, such as 95%, into a z-score. A z-score is the number of standard deviations a value sits from the mean of a normal distribution. This conversion is used across statistics to turn a stated confidence level into the multiplier needed for margin-of-error and interval calculations.
What is a confidence interval?
A confidence interval is a range of values believed to contain a true population value, such as an average, with a stated level of confidence. A 95% confidence interval means that if the same study were repeated many times, about 95% of the calculated intervals would contain the true value. Under a normal distribution, each confidence level maps to a fixed number of standard deviations on either side of the mean.
Confidence interval to z-score formula
The conversion uses the inverse of the standard normal cumulative distribution function, also called the quantile function or probit function:
z = Φ⁻¹((1 + CI / 100) / 2)
Here, CI is the confidence interval written as a percentage between 0 and 100, and Φ⁻¹ is the inverse standard normal CDF. The result, z, is the number of standard deviations above the mean that marks the upper edge of the interval. The lower edge sits the same distance below the mean, so the full interval spans ±z standard deviations.
How to calculate standard deviations from a confidence interval
- Take the confidence level as a percentage, for example 95.
- Divide by 100 to get a fraction, then apply the formula: p = (1 + 0.95) / 2 = 0.975.
- Look up the z-score where the standard normal curve reaches that cumulative probability: z = Φ⁻¹(0.975).
- Round the result. For 95%, z ≈ 1.9600.
Worked example
For a 99% confidence interval:
- p = (1 + 0.99) / 2 = 0.995
- z = Φ⁻¹(0.995) ≈ 2.5758
So a 99% confidence interval extends about 2.5758 standard deviations above and below the mean.
Common confidence intervals and their z-scores
| Confidence interval | Standard deviations (z) |
|---|---|
| 50% | ±0.6745 |
| 68.27% | ±1.0000 |
| 90% | ±1.6449 |
| 95% | ±1.9600 |
| 99% | ±2.5758 |
| 99.73% | ±3.0000 |
| 99.9% | ±3.2905 |
These are the standard textbook values for a normal distribution. The 68.27%, 95.45%, and 99.73% levels correspond to exactly 1, 2, and 3 standard deviations.
Special cases
- 0% confidence: z = 0. A 0% interval has no width, so it collapses to the mean itself, the midpoint of the distribution.
- 100% confidence: z is infinite. Capturing every possible value under a normal distribution requires an interval of infinite width, since the curve never fully reaches zero.
- 50% confidence: z ≈ 0.6745, sometimes called the probable error, splitting the distribution into equal halves around the mean.
Valid input is any percentage from 0 to 100. Values outside that range are not meaningful confidence levels and are rejected.
Where this conversion is used
- Hypothesis testing: choosing a critical z-value for a chosen significance level.
- Margin of error: the margin of error in a confidence interval equals z multiplied by the standard error, so the z-score is a required input.
- A/B testing: interpreting whether a measured difference is likely real or due to chance.
- Quality control: setting control limits a fixed number of standard deviations from a process average.
These uses assume the underlying data, or the sampling distribution of a statistic like a mean, is approximately normal. For small samples or skewed data, other methods such as the t-distribution are more appropriate.
Frequently asked questions
How do I convert a 95% confidence interval to standard deviations? A 95% confidence interval corresponds to about ±1.9600 standard deviations from the mean. It comes from solving z = Φ⁻¹(0.975).
What confidence interval corresponds to 1 standard deviation? One standard deviation on each side of the mean (±1σ) covers about 68.27% of a normal distribution.
Why is 1.96 used for 95% confidence intervals? Because a 95% interval leaves 2.5% of the area in each tail of the normal curve. The z-score that marks the 97.5th percentile is about 1.96.
What happens at 0% or 100% confidence? A 0% confidence interval has zero width and equals 0 standard deviations, the mean itself. A 100% confidence interval must cover the whole distribution, which requires an infinite number of standard deviations.
Does this conversion work for non-normal data? No. It assumes the data, or the relevant sampling distribution, is normal. Skewed or small-sample data usually calls for a t-distribution or a resampling method instead.
How do I get the margin of error from a z-score? Margin of error equals z times the standard error of the estimate: z × (σ / √n), where σ is the standard deviation and n is the sample size.