When to use t-distribution instead of normal distribution?

The t-distribution and the normal distribution are essential tools in statistics, but they serve different purposes and are applied under distinct circumstances. Understanding when to use the t-distribution instead of the normal distribution can greatly enhance the accuracy of our statistical analyses.

Here are some of the things that determine which distribution should be used:

Sample size

• Normal distribution: If working with a large sample size (typically, n > 30), the central limit theorem tells us that the sample mean will be approximately normally distributed, regardless of the population distribution. In such cases, it’s safe to use the normal distribution.

• t-distribution: For smaller sample sizes (usually, n < 30), the t-distribution is the better choice. It accounts for the increased uncertainty associated with estimating population parameters from limited data.

Population variance

• Normal distribution: We can use the normal distribution when we know the population variance. This is often the case in theoretical or idealized scenarios.

• t-distribution: When we don’t know the population variance or when we’re estimating the population standard deviation from the sample, we use the t-distribution. It incorporates the additional variability introduced by estimating the population standard deviation from your sample.

Hypothesis testing and confidence intervals

• t-distribution: If we’re conducting hypothesis tests or constructing confidence intervals for a population mean with a small sample size or an unknown population standard deviation, we use the t-distribution. In these cases, the t-statistic replaces the z-statistic.

• Normal distribution: For large sample sizes or when we know the population parameters, the normal distribution is suitable for hypothesis testing and constructing confidence intervals.

Handling skewness and outliers

The t-distribution is more robust than the normal distribution when data deviates from normality, exhibits skewness, or contains outliers. If our dataset has these characteristics, the t-distribution can yield more accurate results.

Conclusion

In conclusion, choosing between the t-distribution and the normal distribution depends on the characteristics of our data and the goals of our statistical analysis. Use the t-distribution for small sample sizes, unknown population variances, and non-normal data. In contrast, the normal distribution is appropriate for large samples with known population parameters. Making the right choice ensures that our statistical inferences are both valid and reliable.

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