What is Confidence Interval and How to Calculate it (with Examples)

One use of statistical analysis is to make inferences from data samples to larger populations. Point estimation uses a sample statistic to estimate an unknown population parameter, such as the mean, proportion, or standard deviation. The confidence interval is an estimate that contains a range of values for the population parameter and represents the uncertainty of a sample statistic compared to the true population value.1 They are essential tools in statistics for estimating population parameters, quantifying uncertainty, making decisions, hypothesis testing, comparisons, communicating results, and sample size determination. They provide a comprehensive framework for statistical inference and contribute to the robustness and reliability of statistical analyses. 

What is Confidence Interval?  

A confidence interval is a range of values used to estimate an unknown population parameter based on sample data. It provides an idea of where the true population parameter might lie, along with a degree of confidence in that estimate. The confidence level (e.g., 95%, 90%) represents the percentage of times the interval would contain the true population parameter in multiple samples. Confidence intervals are essential for estimating population parameters and quantifying uncertainty in statistical analyses. 

Understanding Confidence Intervals 

Understanding confidence intervals involves grasping several key concepts: 

• Population Parameter: This describes a population, such as the mean or proportion. 
• Sample Statistics: Numerical measures calculated from sample data to estimate population parameters, including the sample mean, sample proportion, and sample standard deviation. 
• Sampling Distribution: The probability distribution of a sample statistic, assuming repeated random sampling from the population. 
• Standard Error: Measures the variability of a sample statistic across different samples. 
• Critical Values: Specific values from a probability distribution that define the boundaries of a confidence interval based on the desired confidence level. 
• Confidence Level: The probability that a confidence interval will contain the true population parameter, commonly 90%, 95%, or 99%. 
• Margin of Error: The amount by which the sample statistic may differ from the population parameter within the confidence interval. 

By understanding these concepts and how they relate to confidence intervals, you can interpret confidence intervals accurately and use them effectively in statistical analysis and inference. 

Calculating the Confidence Interval 

To understand how to calculate the confidence interval, let us consider the following example:2 

Suppose we have collected data on the systolic blood pressure (in mmHg) of a sample of 50 individuals. The sample mean systolic blood pressure is 120 mmHg, and the sample standard deviation is 10 mmHg. We want to calculate a 95% confidence interval for the population mean systolic blood pressure. 

Follow the steps below to calculate the confidence interval: 

Therefore, the 95% confidence interval for the population mean systolic blood pressure is (117.23, 122.77) mmHg. This interval suggests that we are 95% confident that the true population mean systolic blood pressure falls within the range of 117.23 to 122.77 mmHg based on the sample data collected. 

Why are Confidence Intervals Used? 

Confidence intervals are used in statistical analysis to estimate population parameters, quantify uncertainty, compare groups, complement hypothesis testing, communicate results, assess precision, and support predictive modeling. They provide a nuanced understanding of data for evidence-based decision-making. 

Confidence Interval for the Mean of Normally Distributed Data 

Normally-distributed data forms a bell shape when plotted on a graph, with the sample mean in the middle and the rest of the data distributed fairly evenly on either side of the mean. 

The confidence interval for data which follows a standard normal distribution is 

The formula for the confidence interval in the t distribution is the same as for the z distribution, but it replaces Z* with t*. 

In the real world, uncovering the true values of the population can be quite challenging unless a comprehensive census is carried out. Consequently, sample data values step in to power the formula, driving accurate insights, so the formula becomes: 

Confidence Interval for Non-Normally Distributed Data 

When dealing with non-normally distributed data and the need to calculate a confidence interval around the mean, there are two constructive options available:3  

1. Identify similar distribution: Identify a distribution that aligns with the shape of the data and utilize it to calculate the confidence interval.  
2. Perform data transformation: Transform the data to fit a normal distribution and then determine the confidence interval for the transformed data. 

Confidence Interval for Proportions 

To build the confidence interval for population proportion 𝑝, we use the following formula: 

Reporting Confidence Intervals

Reporting confidence intervals involves clearly communicating the results of statistical analyses in a manner that is informative and understandable to the intended audience. Here’s how you can report confidence intervals, with examples: 

a) Specify the Parameter: Clearly state the population parameter for which the confidence interval is being reported. This provides context and ensures clarity.  

Example: “We calculated a 90% confidence interval for the mean systolic blood pressure of the population.” 

b) Provide Sample Information: Include details about the sample data used to calculate the confidence interval. This allows readers to assess the reliability of the estimate.  

Example: “Using a sample of 𝑛 = 100 individuals, we obtained a sample mean systolic blood pressure of x̄ = 120 mmHg, with a sample standard deviation of 𝑠 = 10 mmHg.” 

c) Specify Confidence Level: Indicate the confidence level associated with the interval, such as 90%, 95%, or 99%. This helps readers understand the level of certainty in the estimate.  

Example: “We calculated a 90% confidence interval for the mean systolic blood pressure.” 

d) Present the Interval: Report the confidence interval itself, typically in the form of an interval with lower and upper bounds.  

Example: “The 90% confidence interval for the mean systolic blood pressure is [115, 125] mmHg.” 

e) Assumptions: If any assumptions were made in calculating the confidence interval (e.g., normality of data, independence of observations), mention them. This helps readers understand the limitations of the analysis.  

Example: “We assumed that the systolic blood pressure measurements were normally distributed and independent.” 

By following these steps and providing clear and detailed information, you can effectively report confidence intervals in a manner that enhances understanding and facilitates interpretation by stakeholders. 

All You Should Know When Using Confidence Intervals

When using confidence intervals, there are several key considerations to keep in mind to ensure accurate interpretation and application: 

1. Sample Size: Larger sample sizes generally result in more precise estimates and narrower confidence intervals. However, even with smaller sample sizes, confidence intervals can still provide valuable information if calculated correctly. 
2. Level of Confidence: The chosen confidence level (e.g., 90%, 95%, 99%) determines the probability that the interval contains the true population parameter in repeated sampling. Higher confidence levels provide wider intervals but offer greater confidence in capturing the true parameter. 
3. Population Distribution: While confidence intervals are robust to violations of normality for large sample sizes due to the Central Limit Theorem, it’s essential to consider the distribution of the population when interpreting the results. For small sample sizes or non-normal data, alternative methods may be more appropriate. 
4. Standard Errors: Understanding the concept of standard errors is crucial. The standard error quantifies the variability of a sample statistic (e.g., mean or proportion) across different samples. It’s used to calculate the margin of error in constructing confidence intervals. 
5. Critical Values: Depending on the confidence level and distribution assumptions, critical values are obtained from the standard normal distribution (Z) or t-distribution. These values define the boundaries of the confidence interval and are essential for accurate calculation. 
6. Assumptions: Be aware of any assumptions underlying the calculation of confidence intervals, such as the independence of observations, random sampling, and the appropriateness of the chosen statistical method. 
7. Interpretation: When interpreting confidence intervals, avoid common misconceptions, such as treating the interval as a range of plausible values for a single sample or confusing the confidence level with the probability that a particular interval contains the true parameter. 
8. Comparisons: When comparing confidence intervals between groups or over time, consider overlap. If two intervals overlap, it suggests that there may not be a significant difference between the corresponding population parameters. 
9. Communicating Results: When reporting confidence intervals, clearly state the parameter of interest, sample data, confidence level, interval values, and any relevant assumptions. Provide context and interpretation to facilitate understanding by stakeholders. 

By keeping these factors in mind, you can effectively use confidence intervals to estimate population parameters, quantify uncertainty, and make informed decisions based on statistical inference. 

Key Takeaways 

1. Estimation: Confidence intervals provide a range of plausible values for population parameters, such as means, proportions, or differences between means. 
2. Uncertainty Quantification: They quantify the uncertainty associated with estimating population parameters from sample data. The wider the interval, the greater the uncertainty. 
3. Level of Confidence: Confidence intervals are associated with a chosen confidence level, typically 90%, 95%, or 99%. This represents the probability that the interval contains the true population parameter in repeated sampling. 
4. Sampling Variation: Confidence intervals are based on sample data and are subject to sampling variation. Different samples may yield slightly different intervals. 
5. Interpretation: The interval should be interpreted as a range of plausible values for the population parameter, not as a prediction interval for individual observations. 
6. Critical Values: Critical values from the appropriate distribution (e.g., standard normal distribution or t-distribution) define the boundaries of the confidence interval. 
7. Assumptions: Assumptions such as random sampling, independence of observations, and normality of data may be required for accurate calculation of confidence intervals. 
8. Reporting: When reporting confidence intervals, provide clear information about the parameter of interest, sample data, confidence level, and any relevant assumptions. 
9. Decision Making: Confidence intervals aid in evidence-based decision-making by providing a range of values within which the true population parameter is estimated to lie with a specified level of confidence. 

Understanding these key points enables researchers, analysts, and decision-makers to use confidence intervals effectively in statistical analysis and inference. 

Frequently Asked Questions 

1. What does the level of confidence mean? 

The level of confidence typically refers to the degree of certainty or assurance associated with a statement, prediction, measurement, or decision. In statistical terms, it often relates to the reliability of an estimate or inference made from a sample of data. In statistical inference, confidence level is commonly used in constructing confidence intervals and conducting hypothesis tests. It represents the probability that a parameter (such as a population mean or proportion) falls within a specified range. For example, if a 95% confidence interval for a population mean is calculated from a sample, it means that if the same sampling procedure were repeated many times, 95% of the resulting intervals would contain the true population mean. 

In general, a higher confidence level indicates greater certainty, but it’s important to understand that it does not guarantee absolute certainty. The level of confidence is typically chosen based on the desired balance between precision and reliability, as well as the consequences of potential errors. 

2. How is a confidence interval calculated? 

The following steps are used to calculate the confidence interval: 

Step 1: Identify the sample mean (x̄), sample size (n), and sample standard deviation (s): 

Step 2: Find the degrees of freedom (d) and critical value (t): 

• Degrees of Freedom: These reflect the number of independent pieces of information available in the sample. It is as 𝑑 = 𝑛−1. 
• Critical Value: Look up the critical value 𝑡 from the Student’s t-distribution table based on the desired confidence level (𝛼) and degrees of freedom (𝑑). The confidence level determines the probability that the interval contains the true population parameter, and the degrees of freedom account for the variability in the sample. 

Step 4: Write the confidence interval: 

• Confidence Interval (L, U): Present the confidence interval as a range of values, typically in the form (L, U). For example, if L=50 and U=70, the confidence interval would be (50,70).

3. When should I use a confidence interval? 

Confidence intervals are essential in statistics for estimating population parameters based on sample data. They are useful in various scenarios such as estimating population parameters, comparing groups, prediction, hypothesis testing, and communicating results. Confidence intervals provide a range of plausible values for a population parameter with a degree of confidence, offering a more nuanced understanding of the data than point estimates alone. 

4. How do I interpret a confidence interval? 

Interpreting a confidence interval involves considering the range of values, the level of confidence, and the precision of the estimate, while also recognizing the inherent uncertainty due to sampling variation. A specific confidence interval gives a range of plausible values for the parameter of interest. Here’s how you interpret a confidence interval: 

1. Range of Values: A confidence interval provides a range of values. For example, if you have a 90% confidence interval for the population mean of [50, 60], it means that you are 90% confident that the true population mean falls somewhere between 50 and 60. 
2. Level of Confidence: The “90%” in the example above represents the confidence level. It indicates the probability that the true parameter lies within the interval. In this case, if you were to take many samples and compute a confidence interval for each sample, approximately 90% of those intervals would contain the true population mean. 

5. What factors affect the width of a confidence interval? 

The width of a confidence interval depends on several factors: 

1. Sample Size: Larger samples result in narrower intervals, providing more precise estimates. 
2. Data Variability: Highly variable data results in wider confidence intervals. 
3. Level of Confidence: Higher confidence levels lead to wider intervals. 
4. Population Variability: More variable populations lead to wider intervals. 
5. Distribution Assumptions: The choice of statistical distribution influences interval width. 
6. Estimation Method: Different methods can result in different interval widths. 
7. Sample Design: The method used to collect the sample can impact the interval width. 

Understanding these factors helps in making informed decisions about obtaining precise confidence intervals. 

6. What are the limitations of confidence intervals? 

Confidence intervals quantify uncertainty in estimates but should be used with caution. 

1. Sample Size Dependency: Smaller samples yield wider intervals and less precise estimates, while larger samples result in narrower intervals.  
2. Assumption of Normality: Many methods for constructing confidence intervals rely on the assumption of an approximately normal population distribution. 
3. Bias and Variability: Confidence intervals are susceptible to bias and variability.  
4. Limited Coverage: Confidence intervals do not guarantee that the true population parameter falls within the interval with a specific probability.  
5. Misinterpretation: There is a risk of misinterpreting confidence intervals, particularly by conflating the confidence level with the probability that a specific interval contains the true parameter. 
6. Sensitivity to Outliers: Confidence intervals can be sensitive to outliers or extreme values in the data, especially with small sample sizes.  
7. Population Assumptions: Constructing confidence intervals often necessitates assumptions about the population distribution.  
8. Precision vs. Accuracy: A narrow confidence interval signifies high precision but does not necessarily imply accuracy.  

7. What is the formula for calculating a confidence interval for a population mean?        

The general formula for the confidence interval is given below: 

8. How do you interpret a 95% confidence interval? 

Interpreting a 95% confidence interval involves understanding the level of confidence and the range of values it represents for estimating a population parameter. 

1. Level of Confidence: A 95% confidence interval means that if you were to take many samples from the same population and compute a confidence interval for each sample, approximately 95% of those intervals would contain the true population parameter. 
2. Range of Values: The interval provides a range of values within which we are reasonably confident that the true population parameter lies. For example, if you have a 95% confidence interval for the population mean of [L, U], it means that you are 95% confident that the true population mean falls between [L] and [U]. L represents the ‘lower endpoint’ of the confidence interval and U represents the ‘upper endpoint.’ 

References 

1. Altman, D., Machin, D., Bryant, T., & Gardner, M. (Eds.). (2013). Statistics with confidence: confidence intervals and statistical guidelines. John Wiley & Sons. 
2. Rohatgi, V. K., & Saleh, A. M. E. (2015). An introduction to probability and statistics. John Wiley & Sons. 
3. Pek, J., Wong, A. C., & Wong, O. C. (2017). Confidence intervals for the mean of non-normal distribution: transform or not to transform. Open Journal of Statistics, 7(3), 405-421. 

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