Confidence Interval vs. Standard Deviation: What's the Difference?

When analyzing data, it's easy to confuse confidence intervals and standard deviations. But is a confidence interval the same as a standard deviation? No, and in this article, we'll explain why.

Here's a table summarizing the respective features of the standard deviation and the confidence interval, to help you understand their differences better:

Standard deviation (SD): It is a positive real quantity (sometimes infinite) measuring the distribution of a random variable around its mean. A few essential points about SD:

• A large standard deviation indicates that the data are dispersed around the mean. This means that there is a lot of variance in the observed data.
• Conversely, the smaller the standard deviation, the more values are clustered around the mean.
• If the standard deviation is close to zero, then the data are only slightly dispersed in relation to the mean. The standard deviation cannot be negative. Visit our standard deviation calculator 🇺🇸 to learn more!

For example, suppose that the mean of a math test taken by 50 students is 75/100 and the standard deviation is 5 points. Based on the standard deviation, you can deduce that most students scored 5 points below or above the average (between 70 and 80).

Confidence interval (CI): It is a range of values most likely to contain a population parameter (like a mean or proportion). Confidence intervals come with confidence levels expressed as percentages, like 90%, 95%, 99%, etc.

For example, a 95% confidence interval of [10, 15] for average days off taken means that if we repeated this study 100 times using proper methods, about 95 out of 100 intervals would contain the true population mean.

Although confidence intervals and standard deviation may not be the same, they are somewhat connected. Indeed, standard deviation is used to calculate the standard error of the mean, which is then used to build the confidence interval. You can write the confidence interval as CI = sample mean ± margin of error, with the margin of error 🇺🇸 depending on the standard deviation and sample size.

If you're working with normally distributed data or a large enough sample, you can use this formula:

where:

• $\bar{x}$ — Sample mean;
• $z^*$ — z-score 🇺🇸 for your confidence level;
• $\sigma$ — Population standard deviation. In practice, we rarely know the true population standard deviation. We usually use the standard deviation of our sample as an estimate; and
• $\sqrt{n}$ — Square root of the sample size.

Suppose you want to study the IQ scores of a sample of 100 people. You know that the mean is 105, the standard deviation is 15, and the confidence level is 95%.

What can you deduce from this?

The standard deviation of 15 tells us that individual IQ scores generally vary by 15 points from the mean, meaning most scores are between 90 and 120.

The 95% confidence interval 🇺🇸 estimates the true average IQ in the population. Based on the sample data, using a standard deviation of 15 and a 95% confidence level, the mean IQ is probably between 102.06 and 107.94.

To learn more about interpreting confidence intervals, check out our article: How to Interpret Confidence Intervals: A Complete Guide.