How to Calculate Confidence Interval Single Mean

What is a Confidence Interval?

A confidence interval is a range of values that is likely to contain the true population parameter (in this case, the mean) with a certain level of confidence. For example, a 95% confidence interval means that if we took many samples and calculated a 95% confidence interval for each, approximately 95% of those intervals would contain the true population mean.

Confidence intervals are used to estimate the precision of an estimate and to quantify the uncertainty associated with a sample statistic. They are widely used in scientific research, quality control, and decision-making processes.

Confidence Interval Formula

The formula for calculating a confidence interval for a single mean when the population standard deviation (σ) is known is:

Where:

• Sample Mean (x̄) - The mean of the sample data
• Critical Value (z*) - The z-score corresponding to the desired confidence level
• Population Standard Deviation (σ) - The standard deviation of the entire population
• Sample Size (n) - The number of observations in the sample

How to Calculate a Confidence Interval

Step 1: Determine the Sample Mean

Calculate the mean of your sample data. This is the average of all the values in your dataset.

Step 2: Identify the Population Standard Deviation

If you know the standard deviation of the entire population, use that value. If you only have sample data, you can estimate the population standard deviation using the sample standard deviation.

Step 3: Determine the Sample Size

Count the number of observations in your sample. This is your sample size (n).

Step 4: Select the Confidence Level

Choose a confidence level (e.g., 90%, 95%, or 99%). The confidence level represents the probability that the interval will contain the true population mean.

Step 5: Find the Critical Value

Use a z-table or statistical software to find the critical value (z*) corresponding to your chosen confidence level. For example, for a 95% confidence level, the critical value is approximately 1.96.

Step 6: Calculate the Margin of Error

Multiply the critical value by the standard error of the mean (σ / √n). The standard error is the standard deviation of the sampling distribution of the sample mean.

Step 7: Determine the Confidence Interval

Add and subtract the margin of error from the sample mean to get the lower and upper bounds of the confidence interval.

Worked Example

Let's calculate a 95% confidence interval for the mean height of adult males in a city, given the following data:

• Sample Mean (x̄) = 175 cm
• Population Standard Deviation (σ) = 10 cm
• Sample Size (n) = 100

Step-by-Step Calculation

1. Determine the critical value for a 95% confidence level: z* = 1.96
2. Calculate the standard error: σ / √n = 10 / √100 = 1 cm
3. Calculate the margin of error: z* × standard error = 1.96 × 1 = 1.96 cm
4. Determine the confidence interval:
   • Lower bound = x̄ - margin of error = 175 - 1.96 = 173.04 cm
   • Upper bound = x̄ + margin of error = 175 + 1.96 = 176.96 cm

The 95% confidence interval for the mean height of adult males is 173.04 cm to 176.96 cm. This means we are 95% confident that the true population mean height falls within this range.

Interpreting the Results

When interpreting a confidence interval for a single mean, consider the following:

• Confidence Level: The higher the confidence level, the wider the interval. For example, a 99% confidence interval will be wider than a 95% confidence interval.
• Sample Size: Larger sample sizes result in narrower confidence intervals, providing more precise estimates of the population mean.
• Population Standard Deviation: A smaller standard deviation leads to a narrower confidence interval, indicating less variability in the data.

Confidence intervals are not about the probability that the interval contains the true mean. Instead, they represent the long-run proportion of intervals that would contain the true mean if the same study were repeated many times.