Probability and samples

1. Probability and samples Chapter 7

2. Who we care about • In research, we don’t make conclusions about individual people • We make conclusions about what is true for most people •  we don’t care about individual scores • We care about the mean score, across a group of people

3. Thinking back to distributions of individual scores • Scores go on the X-axis • Prevalence goes on the Y-axis • May be skewed • May be normally distributed • Can use to determine how likely it would be to pull out a score in a certain region from that group

4. Thinking in terms of sample means • Can select many different samples from a population, just as you can select many different individuals from a group • From a population of 1000, you could select many different samples of 10 people, and you could calculate a mean for each of those samples • (clearly, you never would do this, but you could)

5. Distribution of sample means • Similar to distribution of individual scores • On Y axis = prevalence • On X axis = possible means of samples of a given size

6. A crucial difference • Distributions of individual scores can take on any shape • Distributions of sample means will always be normally distributed • Theoretical distribution – never actually created, but the idea of it is a crucial basis for inferential statistics • Different distribution for each sample size

7. Comparing distribution of individual scores and sampling distribution of the mean • Central limit theorem: the mean of all individual scores will equal the mean of the means of all the samples of a given size that can be selected from that group of individual scores • Like distribution of individual scores, there is variability • More variability among individual scores than in the distribution of sample means

8. Quantifying variability of the distribution of sample means • Just like standard deviation, captures average distance from the mean •  average distance between means of samples of a given size and the mean of the population • Standard error of the mean: AKA standard error = standard deviation of the distribution of sample means • sm = s/sqrt(n) • Usually referred to in journals as SE or SEM

9. Extending the comparison between types of distributions • We can determine the probability of selecting a sample with different levels of oddness (defined as distance from the mean of the population that the sample is from), just as we did for individual people • Convert mean of sample to z-score: • Z = (X-m)/s • Z of sample = (M-m)/sm • Number of standard errors that the mean of the sample is away from the mean of the population • Use unit-normal table to determine likelihood of selecting such a sample, given the population

10. To keep in mind • Just as there is variability from person to person, there is variability from sample to sample • The more variability, the odder the person or sample, you may obtain – just by chance • Using z-scores, you can figure out how likely it is to select individuals or samples that are different distances from what you think the mean of the population is • Remember to think of all the new terms and equations in English