How to Calculate Sd From Confidence Interval

Calculating standard deviation (SD) from a confidence interval (CI) is a common statistical task when you have interval estimates but need the underlying variability measure. This guide explains the relationship between these two concepts and provides a step-by-step method to derive the standard deviation from a given confidence interval.

What is Standard Deviation?

Standard deviation (SD) is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

Standard deviation is widely used in statistics and probability theory. It's often used to describe the spread of data in a normal distribution, where about 68% of observations fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

What is a Confidence Interval?

A confidence interval (CI) is a range of values that is likely to contain the value of an unknown population parameter. The interval has an associated confidence level that, in many cases, corresponds to the probability that the interval contains the true parameter value.

For example, a 95% confidence interval means that if the same population were sampled multiple times, approximately 95% of the calculated intervals would contain the true parameter value. Common confidence levels are 90%, 95%, and 99%.

Note

The confidence interval is not the probability that the true parameter lies in the interval. Instead, it represents the reliability of the interval estimation method.

Formula to Calculate SD from CI

The relationship between standard deviation and confidence interval can be expressed through the margin of error. The margin of error (ME) is half the width of the confidence interval and is calculated as:

Margin of Error Formula

ME = z × (σ / √n)

Where:

z = z-score corresponding to the confidence level
σ = population standard deviation (what we want to find)
n = sample size

To solve for the standard deviation (σ), we can rearrange the formula:

Standard Deviation from Confidence Interval

σ = (ME × √n) / z

Where:

ME = margin of error (half the width of the confidence interval)
√n = square root of the sample size
z = z-score corresponding to the confidence level

The z-score can be found using standard normal distribution tables or statistical software. Common z-scores for different confidence levels are:

90% confidence: z ≈ 1.645
95% confidence: z ≈ 1.960
99% confidence: z ≈ 2.576

Worked Example

Let's calculate the standard deviation from a confidence interval using the following values:

Confidence interval: 4.5 to 5.5
Sample size (n): 100
Confidence level: 95%

Step 1: Calculate the Margin of Error

The width of the confidence interval is 5.5 - 4.5 = 1.0. The margin of error is half of this width:

ME = 1.0 / 2 = 0.5

Step 2: Determine the Z-Score

For a 95% confidence level, the z-score is approximately 1.960.

Step 3: Calculate the Standard Deviation

Using the formula σ = (ME × √n) / z:

σ = (0.5 × √100) / 1.960

σ = (0.5 × 10) / 1.960

σ = 5 / 1.960 ≈ 2.551

The standard deviation is approximately 2.551.

FAQ

Can I calculate standard deviation from any confidence interval?

Yes, you can calculate standard deviation from any confidence interval, but you need to know the sample size and the confidence level used to create the interval.

What if I don't know the sample size?

Without the sample size, you cannot accurately calculate the standard deviation from a confidence interval. The sample size is essential for determining the margin of error.

Is the standard deviation the same as the standard error?

No, standard deviation measures the dispersion of individual data points, while standard error measures the variability of sample means. The standard error is calculated as σ/√n.