# Zach’s facts

Zach’s facts have been extracted from the book to remind you of the key concepts you and Zach have learned in each chapter.

Zach's Facts 9.1 Sources of bias

• Extreme scores and non-normal distributions bias the mean and the variance, the standard error of the mean and the confidence interval for the mean.

• In general, extreme scores and non-normal distributions bias estimates of any parameter (not just the mean) and the standard error and the confidence interval around the estimate.

• Some distributions that look normal are not. One example is the mixed normal distribution, which is where two or more distributions mixed together look normal but in fact the combined distribution has too many scores at the extremes (the distribution is said to have heavy tails). These mixed distributions can severely bias parameter estimates such as the mean and their associated confidence intervals and standard errors.

Zach's Facts 9.2 Reducing bias

• A robust estimate is one that is reliable even when the normal assumptions of the statistic are not met.

• Transforming the data, for example by taking the square root or log of scores, can reduce the impact of extreme high scores, or positive skew.

• The trimmed mean is the mean based on scores that have had a percentage of extreme scores removed. For example, removing the highest and lowest 20% of scores and then computing the mean of the remaining scores would give us the 20% trimmed mean. The 20% trimmed mean is a robust estimate.

• Winsorizing data is where a percentage of the highest scores are replaced with the next highest score in the data and the same percentage of the lowest scores are replaced with the next lowest score in the data. A mean based on a winsorized sample is usually robust.

• The bootstrap is a technique for estimating robust standard errors and confidence intervals. The standard error and confidence interval are estimated empirically by sampling with replacement from the observed data.