ch. 5, z-scores

define z-score

combining the mean and standard deviation into one score, transforming x (the raw score) into z-scores

2 purposes of z scores

1. see the position of a score within a distribution

   • this is helpful to identify outliers (participants who are extreme compared to the rest of the sample)

2. to standardize the entire distribution

   • this is helpful when scores are not marked on the same task (for example one test is marked 0-10 and another is 0-16). z-scores allow us to see them as they fall on the same distribution

formula for z score

to change z-scores back into x scores

X = μ + zσ

distribution of sample means

the collection of sample means from all possible random samples (of a particular size) that can be obtained from a population

central limit theorem

allows us to estimate if the sample mean is normal without testing the entire population mean.

2 conditions for central limit theorem (only 1 needs to be met)

1. the population from which the samples are selected is a normal distribution (used in specific scenarios like IQ scores)

2. the sample size (n) is at least 30

   this tends to lead to a normal distribution regardless the population distribution shape

standard error of M

the average distance between a sample mean and population mean

tells you how well your sample represents the population

formula for standard error

small standard error means …

• sample means are closer together

  • the sample mean is a more reliable estimate of the population

larger standard error means …

• sample means are more spread out

  • sample mean is a less reliable estimate of the population

law of large numbers

• as a sample size increases, standard error decreases

  • a larger sample is better able to represent the population

calculating standard error from the variance