Confidence Interval for a Population Mean
A confidence interval gives a range of plausible values for a population parameter — in this case, the population mean — based on sample data. This calculator computes the two-sided confidence interval for a mean using the sample mean, sample standard deviation, sample size, and a chosen confidence level (e.g., 90%, 95%).
Understanding confidence intervals is crucial in statistics, as they provide insight into the uncertainty surrounding sample estimates. By calculating a confidence interval, researchers can express the reliability of their findings, helping to make informed decisions based on the data.
How it works
For large samples or when the population standard deviation is unknown but the sample size is reasonably large, the confidence interval for the mean is computed as:
CI = x̄ ± z * (s / √n)Where:
- x̄ = sample mean
- s = sample standard deviation
- n = sample size
- z = z-score corresponding to the desired confidence level (two-tailed)
It is essential to note that the chosen confidence level reflects the degree of certainty you have regarding the estimates. Higher confidence levels (like 99%) will yield wider intervals, indicating a broader range of plausible values for the population mean.
Example
Suppose you have a sample mean of 50, a sample standard deviation of 10, and a sample size of 100. For a 95% confidence level, the calculator computes the standard error s/√n = 10 / √100 = 1, the z-score ≈ 1.96, and a margin of error ≈ 1.96 * 1 = 1.96. The 95% confidence interval is 50 ± 1.96, or approximately (48.04, 51.96). This indicates that you can be 95% confident that the true population mean lies within this range.
When to use this calculator
Use this tool when you want to estimate the plausible range of a population mean based on sample data and when the sample size is large enough for the normal approximation to be reasonable. For small samples (typically n < 30) where the population standard deviation is unknown, consider using a t-distribution based interval instead.
Always ensure that the assumptions behind the confidence interval calculations are satisfied, such as the independence of observations and approximate normality. Violating these assumptions can lead to misleading results.
FAQs about Confidence Intervals
What does the confidence level mean?
The confidence level (e.g., 95%) indicates the proportion of similarly constructed intervals that would contain the true population mean if you repeated the sampling process many times.
Can I use this for proportions?
This calculator is specifically for means. For proportions, use a proportion (p̂) confidence interval calculator which uses a different standard error formula.
What if my sample size is small?
For small samples and unknown population standard deviation, a t-distribution should be used instead of the normal z. This tool uses the z-approximation appropriate for larger samples.
How can I interpret my results effectively?
When interpreting results, consider the context of your data and the implications of your confidence intervals in decision-making. Wider intervals suggest more uncertainty, while narrower intervals offer more precision.
Tips for Using the Confidence Interval Calculator
- Round results appropriately for your application; the calculator provides high precision by default.
- Always check assumptions: independence of observations and approximate normality (or sufficiently large sample size).
- Consult a statistician if you're unsure about your sample data or the appropriate methods to apply.
- Utilize the calculator for multiple configurations to better understand variability and uncertainties inherent in your data.