Confidence Interval Formula

While calculating the mean or proportion for certain data populations, using the samples and confidence intervals we can make the calculation more manageable. A confidence interval is the implication of the uncertainty available with any particular statistic. Confidence intervals are mainly used with a margin of error. These tell us how confident we can be from the results of a poll or survey. This article will explain the concept of confidence interval and confidence interval formula with examples. Let us learn this statistical concept!

What is the Confidence Interval?

A confidence interval means CI refers to the amount of uncertainty that may be associated with a sample population estimate of a true population. For example, if we wanted to determine the average age of victims of chain snatching in our city last year. While there is a true answer, say 32 years old, but the best we can do is that to find an interval of age group says, 25-40 years old.

Source: en.wikipedia.org

Therefore, the confidence interval is the sample mean or proportion with \pm the margin of error i.e. ME. This ME is the value used to calculate the upper limit and lower limit of the sample statistic given. In this example, the upper limit is 40, and the lower limit is 25.

Confidence intervals are naturally connected to the confidence levels. In statistics, a confidence interval proposes a range of plausible values for the unknown parameter like mean. So, the interval has a linked confidence level that the true parameter is in the proposed range.

The confidence interval formula in statistics helps to describe the amount of uncertainty in a sample estimate. Thus it describes the uncertainty associated with the sampling method.

The Formula for Confidence Interval

\(CI = \bar{X} \pm z (\frac{\sigma}{\sqrt{n}}) for n \geq 30\)

Or

\(CI = \bar{X} \pm t (\frac{\sigma}{\sqrt{n}}) for n < 30\)

Where,

n	Number of terms
\(\bar{X}\)	Sample Mean
\(\sigma\(	Standard Deviation
z	Value corresponding in the z table
t	Value corresponding in the t table
CI	confidence interval

Calculation of CI:

Steps to calculate a confidence interval using the mean is as follows:

1. Identify a population, and then select a representative sample. Note down the number of the sample (n).
2. Calculate the mean value by adding all of the sample values, divided by n.
3. Select the value of CL (typically 95%) and then locate it in the t-table of z-table, whichever is applicable.
4. Noe, calculate the standard deviations by subtracting each value in the sample from the mean value. Then square each result and calculate the mean of all of those squared differences. This will be the variance.
5. Find the square root of the variance.
6. Finally, calculate the CI using the given formula.

Solved Examples for Confidence Interval Formula

Q.1: Determine the average age of victims of chain snatching in any city during last year for a sample of size 100 with mean age 34.25 years.

Solution: n = 100.

Mean = 34.25 years.

Now, use the 95% CL here. It has z value of 1.96.

Now, calculate the standard deviation. With a mean value of 34.25 and a standard deviation value of 10, a margin of error will be

\(CI = \bar{X} \pm z (\frac{\sigma}{\sqrt{n}})\\\)

\(CI = 34.25 \pm 1.96 (\frac{10}{\sqrt{100}})\\\)

Thus Upper limit of CI will be 34.25 + 1.96 = 36.21

And Lower limit of CI will be 34.25 – 1.96 = 32.29

Thus we may say with 95% confidence, average age will fall between 32.29 and 36.21 years.