Convert confidence intervals (95%, 99%, 90%) to standard deviations and z-scores instantly. Free calculator for statistical analysis, hypothesis testing, and research data interpretation.
A confidence interval to standard deviations converter transforms confidence interval percentages into corresponding z-scores or standard deviations. This statistical tool is essential for researchers, data scientists, and analysts who need to interpret confidence intervals and understand the spread of data in a normal distribution.
A confidence interval represents the range of values within which a population parameter is likely to fall with a certain level of confidence. Common confidence intervals include 95% (±1.96σ), 99% (±2.576σ), and 68.27% (±1σ).
To convert a confidence interval to standard deviations, use the inverse normal cumulative distribution function (quantile function):
z = Φ⁻¹((1 + CI) / 2)
Where:
| Confidence Interval | Standard Deviations (z-score) |
|---|---|
| 68.27% | ±1.0σ |
| 90% | ±1.645σ |
| 95% | ±1.96σ |
| 99% | ±2.576σ |
| 99.73% | ±3.0σ |
The following diagram illustrates the relationship between confidence intervals and standard deviations in a normal distribution:
This conversion assumes:
For non-normal distributions, use appropriate transformations or alternative methods.
Here are code examples to convert confidence intervals to standard deviations in various programming languages:
1' Excel VBA Function for Confidence Interval to Standard Deviations
2Function ConfidenceToStdDev(CI As Double) As Double
3 ConfidenceToStdDev = Application.NormSInv(1 - (1 - CI) / 2)
4End Function
5' Usage:
6' =ConfidenceToStdDev(0.95)
71confidence_to_std_dev <- function(confidence_interval) {
2 qnorm((1 + confidence_interval) / 2)
3}
4
5# Example usage:
6ci <- 0.95 # 95% confidence interval
7z_score <- confidence_to_std_dev(ci)
8cat(sprintf("%.2f%% confidence interval corresponds to %.4f standard deviations\n", ci*100, z_score))
91function z = confidenceToStdDev(confidenceInterval)
2 z = norminv((1 + confidenceInterval) / 2);
3end
4
5% Example usage:
6ci = 0.95; % 95% confidence interval
7zScore = confidenceToStdDev(ci);
8fprintf('%.2f%% confidence interval corresponds to %.4f standard deviations\n', ci*100, zScore);
91import scipy.stats as stats
2
3def confidence_to_std_dev(confidence_interval):
4 return stats.norm.ppf((1 + confidence_interval) / 2)
5
6# Example usage:
7ci = 0.95 # 95% confidence interval
8z_score = confidence_to_std_dev(ci)
9print(f"{ci*100}% confidence interval corresponds to {z_score:.4f} standard deviations")
101function confidenceToStdDev(confidenceInterval) {
2 // Using an approximation for the inverse error function
3 function erfInv(x) {
4 const a = 0.147;
5 const y = Math.log(1 - x*x);
6 const z = 2/(Math.PI * a) + y/2;
7 return Math.sign(x) * Math.sqrt(Math.sqrt(z*z - y/a) - z);
8 }
9
10 return Math.sqrt(2) * erfInv(confidenceInterval);
11}
12
13// Example usage:
14const ci = 0.95;
15const zScore = confidenceToStdDev(ci);
16console.log(`${ci*100}% confidence interval corresponds to ${zScore.toFixed(4)} standard deviations`);
171public class ConfidenceIntervalConverter {
2 public static double confidenceToStdDev(double confidenceInterval) {
3 // Using Moro's algorithm for inverse normal CDF approximation
4 double p = (1 + confidenceInterval) / 2;
5 double t = Math.sqrt(-2 * Math.log(1 - p));
6 double c0 = 2.515517;
7 double c1 = 0.802853;
8 double c2 = 0.010328;
9 double d1 = 1.432788;
10 double d2 = 0.189269;
11 double d3 = 0.001308;
12
13 return t - ((c0 + c1 * t + c2 * t * t) / (1 + d1 * t + d2 * t * t + d3 * t * t * t));
14 }
15
16 public static void main(String[] args) {
17 double ci = 0.95;
18 double zScore = confidenceToStdDev(ci);
19 System.out.printf("%.2f%% confidence interval corresponds to %.4f standard deviations%n", ci*100, zScore);
20 }
21}
221#include <iostream>
2#include <cmath>
3
4double confidenceToStdDev(double confidenceInterval) {
5 // Using Moro's algorithm for inverse normal CDF approximation
6 double p = (1 + confidenceInterval) / 2;
7 double t = std::sqrt(-2 * std::log(1 - p));
8 double c0 = 2.515517, c1 = 0.802853, c2 = 0.010328;
9 double d1 = 1.432788, d2 = 0.189269, d3 = 0.001308;
10
11 return t - ((c0 + c1 * t + c2 * t * t) / (1 + d1 * t + d2 * t * t + d3 * t * t * t));
12}
13
14int main() {
15 double ci = 0.95;
16 double zScore = confidenceToStdDev(ci);
17 printf("%.2f%% confidence interval corresponds to %.4f standard deviations\n", ci*100, zScore);
18 return 0;
19}
201def confidence_to_std_dev(confidence_interval)
2 # Using an approximation for the inverse error function
3 p = (1 + confidence_interval) / 2
4 t = Math.sqrt(-2 * Math.log(1 - p))
5 c0, c1, c2 = 2.515517, 0.802853, 0.010328
6 d1, d2, d3 = 1.432788, 0.189269, 0.001308
7
8 t - ((c0 + c1 * t + c2 * t * t) / (1 + d1 * t + d2 * t * t + d3 * t * t * t))
9end
10
11# Example usage:
12ci = 0.95
13z_score = confidence_to_std_dev(ci)
14puts "#{ci*100}% confidence interval corresponds to #{z_score.round(4)} standard deviations"
151<?php
2function confidenceToStdDev($confidenceInterval) {
3 // Using an approximation for the inverse error function
4 $p = (1 + $confidenceInterval) / 2;
5 $t = sqrt(-2 * log(1 - $p));
6 $c0 = 2.515517; $c1 = 0.802853; $c2 = 0.010328;
7 $d1 = 1.432788; $d2 = 0.189269; $d3 = 0.001308;
8
9 return $t - (($c0 + $c1 * $t + $c2 * $t * $t) / (1 + $d1 * $t + $d2 * $t * $t + $d3 * $t * $t * $t));
10}
11
12// Example usage:
13$ci = 0.95;
14$zScore = confidenceToStdDev($ci);
15printf("%.2f%% confidence interval corresponds to %.4f standard deviations\n", $ci*100, $zScore);
16?>
171fn confidence_to_std_dev(confidence_interval: f64) -> f64 {
2 // Using an approximation for the inverse error function
3 let p = (1.0 + confidence_interval) / 2.0;
4 let t = (-2.0 * (1.0 - p).ln()).sqrt();
5 let c0 = 2.515517;
6 let c1 = 0.802853;
7 let c2 = 0.010328;
8 let d1 = 1.432788;
9 let d2 = 0.189269;
10 let d3 = 0.001308;
11
12 t - ((c0 + c1 * t + c2 * t * t) / (1.0 + d1 * t + d2 * t * t + d3 * t * t * t))
13}
14
15fn main() {
16 let ci = 0.95;
17 let z_score = confidence_to_std_dev(ci);
18 println!("{:.2}% confidence interval corresponds to {:.4} standard deviations", ci*100.0, z_score);
19}
201using System;
2
3class ConfidenceIntervalConverter
4{
5 static double ConfidenceToStdDev(double confidenceInterval)
6 {
7 // Using an approximation for the inverse error function
8 double p = (1 + confidenceInterval) / 2;
9 double t = Math.Sqrt(-2 * Math.Log(1 - p));
10 double c0 = 2.515517, c1 = 0.802853, c2 = 0.010328;
11 double d1 = 1.432788, d2 = 0.189269, d3 = 0.001308;
12
13 return t - ((c0 + c1 * t + c2 * t * t) / (1 + d1 * t + d2 * t * t + d3 * t * t * t));
14 }
15
16 static void Main()
17 {
18 double ci = 0.95;
19 double zScore = ConfidenceToStdDev(ci);
20 Console.WriteLine($"{ci*100:F2}% confidence interval corresponds to {zScore:F4} standard deviations");
21 }
22}
231package main
2
3import (
4 "fmt"
5 "math"
6)
7
8func confidenceToStdDev(confidenceInterval float64) float64 {
9 // Using an approximation for the inverse error function
10 p := (1 + confidenceInterval) / 2
11 t := math.Sqrt(-2 * math.Log(1 - p))
12 c0, c1, c2 := 2.515517, 0.802853, 0.010328
13 d1, d2, d3 := 1.432788, 0.189269, 0.001308
14
15 return t - ((c0 + c1*t + c2*t*t) / (1 + d1*t + d2*t*t + d3*t*t*t))
16}
17
18func main() {
19 ci := 0.95
20 zScore := confidenceToStdDev(ci)
21 fmt.Printf("%.2f%% confidence interval corresponds to %.4f standard deviations\n", ci*100, zScore)
22}
231import Foundation
2
3func confidenceToStdDev(_ confidenceInterval: Double) -> Double {
4 // Using an approximation for the inverse error function
5 let p = (1 + confidenceInterval) / 2
6 let t = sqrt(-2 * log(1 - p))
7 let c0 = 2.515517, c1 = 0.802853, c2 = 0.010328
8 let d1 = 1.432788, d2 = 0.189269, d3 = 0.001308
9
10 return t - ((c0 + c1 * t + c2 * t * t) / (1 + d1 * t + d2 * t * t + d3 * t * t * t))
11}
12
13// Example usage:
14let ci = 0.95
15let zScore = confidenceToStdDev(ci)
16print(String(format: "%.2f%% confidence interval corresponds to %.4f standard deviations", ci*100, zScore))
17To ensure the accuracy of the conversion function across different confidence intervals, here are some test cases:
1import unittest
2import math
3
4def confidence_to_std_dev(confidence_interval):
5 return stats.norm.ppf((1 + confidence_interval) / 2)
6
7class TestConfidenceToStdDev(unittest.TestCase):
8 def test_common_confidence_intervals(self):
9 self.assertAlmostEqual(confidence_to_std_dev(0.6827), 1.0, places=4)
10 self.assertAlmostEqual(confidence_to_std_dev(0.95), 1.96, places=2)
11 self.assertAlmostEqual(confidence_to_std_dev(0.99), 2.576, places=3)
12 self.assertAlmostEqual(confidence_to_std_dev(0.9973), 3.0, places=4)
13
14 def test_edge_cases(self):
15 self.assertAlmostEqual(confidence_to_std_dev(0.5), 0.6745, places=4)
16 self.assertTrue(math.isinf(confidence_to_std_dev(1.0)))
17 self.assertEqual(confidence_to_std_dev(0.0), -float('inf'))
18
19if __name__ == '__main__':
20 unittest.main()
21A 95% confidence interval corresponds to approximately ±1.96 standard deviations from the mean. Use the formula z = Φ⁻¹(0.975) to get the exact value.
The z-score (standard deviations) determines the width of a confidence interval. Higher confidence levels require more standard deviations: 90% = ±1.645σ, 95% = ±1.96σ, 99% = ±2.576σ.
This converter assumes a normal distribution. For non-normal data, use bootstrapping methods, t-distribution (small samples), or distribution-specific quantile functions.
Standard deviation measures data spread, while standard error measures sampling distribution spread. Confidence intervals use standard error: SE = σ/√n.
A 99% confidence interval equals approximately ±2.576 standard deviations, meaning 99% of data falls within this range in a normal distribution.
The value 1.96 comes from the inverse normal CDF at 97.5% (since 95% CI leaves 2.5% in each tail). It's the z-score that captures 95% of the area under the normal curve.
Margin of error = z × σ/√n, where z is the standard deviation multiplier (1.96 for 95% CI), σ is standard deviation, and n is sample size.
One standard deviation (±1σ) corresponds to approximately a 68.27% confidence interval in a normal distribution.
This confidence interval to standard deviations converter simplifies statistical analysis by transforming confidence levels into z-scores. Whether you're conducting hypothesis tests, analyzing survey data, or calculating sample sizes, understanding this relationship is fundamental to interpreting statistical results accurately.
Use the calculator above to quickly convert any confidence interval percentage to its corresponding standard deviation value for your statistical analysis needs.
Discover more tools that might be useful for your workflow