Calculate confidence intervals using either standard deviation or standard error and input confidence level, Z-score, or p-value.
Understanding Confidence Intervals
A confidence interval (CI) is a range of values used to estimate the true value of a population parameter. For example, a 95% confidence interval for the mean suggests that if we take repeated samples and compute a confidence interval for each sample, approximately 95% of those intervals will contain the population mean.
Margin of Error, Lower Bound, and Upper Bound
The Margin of Error represents the range within which we expect the true population parameter to fall. It is calculated as the product of the Z-score and the standard error or standard deviation. The Lower Bound is the lowest value within the confidence interval, calculated by subtracting the Margin of Error from the sample mean. The Upper Bound is the highest value within the confidence interval, calculated by adding the Margin of Error to the sample mean. Together, the lower and upper bounds define the confidence interval range.
Formula for Confidence Interval
The confidence interval for a sample mean \( \bar{x} \) is calculated as:
$ CI = \bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{N}} $
Where:
- \( \bar{x} \): Sample mean
- \( Z \): Z-score corresponding to the confidence level (e.g., 1.96 for 95%)
- \( \sigma \): Population standard deviation (or sample standard deviation)
- \( N \): Sample size
Real-Life Example: Average Weight of Apples
Imagine you’re a researcher interested in determining the average weight of apples in an orchard. You randomly sample 100 apples and find an average (mean) weight of 150 grams with a standard deviation of 20 grams. If you want to calculate a 95% confidence interval for the mean weight of all apples in the orchard, follow these steps:
- Since the confidence level is 95%, use a Z-score of 1.96 (common for 95% confidence).
- Calculate the standard error: \( SE = \frac{20}{\sqrt{100}} = 2 \) grams.
- Calculate the margin of error: \( ME = 1.96 \times 2 = 3.92 \) grams.
- Determine the confidence interval by adding and subtracting the margin of error from the sample mean:
$$ CI = 150 \pm 3.92 $$
Therefore, the 95% confidence interval is approximately [146.08, 153.92] grams. This means we are 95% confident that the average weight of all apples in the orchard falls between 146.08 and 153.92 grams.
Z-values for Common Confidence Levels
| Confidence Level (%) | Z-value | p-value |
|---|---|---|
| 70% | 1.04 | 0.30 |
| 75% | 1.15 | 0.25 |
| 80% | 1.28 | 0.20 |
| 85% | 1.44 | 0.15 |
| 90% | 1.645 | 0.10 |
| 95% | 1.96 | 0.05 |
| 98% | 2.33 | 0.02 |
| 99% | 2.576 | 0.01 |
| 99.5% | 2.807 | 0.005 |
| 99.9% | 3.291 | 0.001 |
| 99.99% | 3.891 | 0.0001 |
| 99.999% | 4.417 | 0.00001 |
Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.