The width of a confidence interval is twice the margin of error. Every confidence interval follows the same structure: a point estimate (like a sample mean) plus or minus a margin of error. The total width is simply the distance from the lower bound to the upper bound, which means you either subtract the lower limit from the upper limit, or multiply the margin of error by two.
That’s the short answer. The more useful skill is understanding how to calculate the margin of error in the first place, what controls how wide or narrow the interval turns out, and how to work backward from a desired width to plan a study.
The General Formula
Every confidence interval follows this pattern:
CI = Point estimate ± (Critical value × Standard error)
The “critical value × standard error” part is your margin of error. Since the interval extends that distance in both directions from the point estimate, the total width is:
Width = 2 × Critical value × Standard error
If you already have the upper and lower bounds calculated, the width is even simpler: just subtract the lower bound from the upper bound. For example, if your 95% confidence interval runs from 12.4 to 18.6, the width is 18.6 − 12.4 = 6.2, and the margin of error is 3.1.
Width for a Mean
When you’re estimating a population mean from sample data, the standard error equals the sample standard deviation divided by the square root of the sample size (s / √n). Plugging that into the width formula gives you:
Width = 2 × z × (s / √n)
Here, z is the critical value for your chosen confidence level, s is the sample standard deviation, and n is the number of observations. If your sample is small (typically under 30) and you’re using the sample standard deviation, you’d swap the z-value for a t-value from the t-distribution, which is slightly larger and produces a wider interval. As n grows, the t-value converges toward the z-value and the distinction matters less.
A Quick Example
Say you measured the blood pressure of 64 patients and found a sample mean of 120 mmHg with a standard deviation of 16. For a 95% confidence interval, you’d use a critical value of 1.96. The standard error is 16 / √64 = 2.0. The margin of error is 1.96 × 2.0 = 3.92, and the total width of the interval is 2 × 3.92 = 7.84 mmHg. Your confidence interval would run from 116.08 to 123.92.
Width for a Proportion
When the data is categorical (like the percentage of people who responded “yes” to a survey), the standard error uses a different formula. For a sample proportion p̂ with sample size n:
Standard error = √(p̂(1 − p̂) / n)
So the full width becomes:
Width = 2 × z × √(p̂(1 − p̂) / n)
Notice that the standard error is largest when p̂ is 0.5 (a 50/50 split), which is why polls and surveys often assume a proportion of 0.5 when planning sample sizes. It gives the widest, most conservative interval.
Critical Values for Common Confidence Levels
The critical value (z) is the piece that changes with your chosen confidence level. The three most common levels and their z-values are:
- 90% confidence: z = 1.645
- 95% confidence: z = 1.96
- 99% confidence: z = 2.576
A higher confidence level means a larger critical value, which directly widens the interval. Going from 95% to 99% increases the critical value by about 31%, so the interval gets roughly a third wider. You’re trading precision for certainty.
Three Factors That Control Width
Looking at the formula, three things determine how wide a confidence interval turns out: the confidence level, the variability in the data, and the sample size. Each one works in a predictable direction.
Confidence level. Raising the confidence level (say, from 95% to 99%) increases the critical value and widens the interval. Lowering it to 90% narrows things. This is the tradeoff between being more certain and being more precise.
Variability. More spread-out data (a larger standard deviation) produces a larger standard error and a wider interval. This makes intuitive sense. If your measurements are all over the place, you’re less sure where the true value sits. You can’t control variability after collecting data, but more precise measurement tools or a more homogeneous sample can reduce it.
Sample size. This is the factor researchers have the most control over, and it has a square-root relationship with width. Because sample size sits under a square root sign in the standard error formula, you need to quadruple your sample size to cut the width in half. Doubling the sample size only reduces the width by about 29% (since 1/√2 ≈ 0.71). That diminishing return is important to keep in mind when planning studies or surveys.
Working Backward From a Target Width
Sometimes you need to figure out how many observations to collect in order to achieve a specific interval width. You can rearrange the width formula to solve for n. For estimating a mean:
n = (2 × z × s / W)²
Where W is the desired total width. For a proportion, the formula becomes:
n = 4 × z² × p̂(1 − p̂) / W²
In both cases, a narrower desired width requires a larger sample. You’ll need an estimate of the standard deviation (or proportion) ahead of time, which usually comes from a pilot study, previous research, or a conservative assumption. For proportions, using 0.5 as your estimate guarantees the sample will be large enough regardless of the actual result.
For example, if you want a 95% confidence interval for a proportion that’s no wider than 6 percentage points total (±3%), and you assume p̂ = 0.5, you’d need n = 4 × (1.96)² × 0.25 / (0.06)² ≈ 1,067 respondents. That’s where the common “about 1,000 people” benchmark in polling comes from.
Reading Width From Published Results
In research papers and reports, confidence intervals are typically written as either a range (45.2, 52.8) or with the ± notation (49.0 ± 3.8). If you see the range format, the width is the upper value minus the lower value. If you see the ± format, the number after ± is the margin of error, and the width is double that. Both give you the same information.
A narrow interval relative to the point estimate signals a precise finding. A wide interval suggests more uncertainty, either because the sample was small, the data was highly variable, or the confidence level was set high. Comparing interval widths across studies on the same topic can tell you which results are more informative, even before looking at statistical significance.