1. Inferring Population Parameters

2. z-Scores Standardizing scores by reporting them as distance from the mean in units of standard deviation
Mean
1 Std. Deviation
Test Score
z = (score-mean)/sd

3. Gaussian distribution Continuous probability distribution Normal curve
Normal distribution
Gaussian distribution
Continuous probability distribution
Normal curve
Characteristic curve
Bell curve
0.13%
0.13%
2.14%
13.59%
34.13%
34.13%
13.59%
2.14%
Standard Deviations (z scores) -3 -2 -1 +1 +2 +3
Cumulative Percentages
.01%
2.3%
15.9%
50.0%
84.1%
97.7%
99.9%
Percentile Equivalents 1 5 10 20 40 60 80 90 95 99 30 50 70
IQ Scores 55 70 85
100
115
130
145
SAT Scores (sd 209)
400
608
817
1026
1235
1444
1600

4. Percentile Rank (z-Scores) When the distribution is normal, identification of percent responses lower than a given score
Mean
1 Std. Deviation
Percentage of scores lower than target score
Test Score
z = (score-mean)/sd

5. Sample Mean When a sample group is being compared to the population, use a sampling distribution
1 Std. Deviation
Group of 47 Mean Score
z = (score-mean)/sd

6. Apply the central limit theorem
Sample Mean When a sample group is being compared to the population, use a sampling distribution
Mean
Group of 47 Mean Score
Apply the central limit theorem

7. Sample Mean When a sample group is being compared to the population, use a sampling distribution
Standard Error (SD adjusted for group size)
Percentage of group means lower than target group score
Group of 47 Mean Score
Compute a z-score

8. Probability that the group mean could have appeared randomly
Sampling Distribution of the Mean represents distribution of randomly selected groups
Mean
Standard Error (SD adjusted for group size)
Percentage of group means lower than target group score
Group of 47 Mean Score
Probability that the group mean could have appeared randomly

9. Probability that the group mean could have appeared randomly
Sampling Distribution of the Mean Reporting the probability a group mean might appear randomly based on something that looks like a z-score
Mean
Standard Error (SD adjusted for group size)
Group of 47 Mean Score
Probability that the group mean could have appeared randomly

10. What if you do not know the SD of the population? A one sample t-test
Mean
Standard Error (sample SD adjusted for group size)
Estimate the SD of the population using the sample SD
Group of 47 Mean Score
Probability that the group mean could have appeared randomly

11. One Sample t-Test To do this you need the population mean and the sample statistics
You do not need to know anything about the population to which you are inferring except the mean score. EZAnalyze calls this the numerical test value (NTV)
Group of 47 Mean Score

12. An example in Excel excel file: One Sample

13. OK, this was fun but you will almost never encounter a situation where you would do one sample tests. What we need to figure out is how to compare the means from two groups.

14. Central Limit Theorem
When the means of randomly selected groups of a given size are plotted, a histogram approaching normal will appear.
The almost normal distribution is called a sampling distribution of the mean.
The population distribution does not have to be normal for this to work.
The larger the group size the more normal the resulting distribution will be.
This does not work very well for group sizes under 30.

15. Sampling Distribution of the Mean
Remember what this is—a distribution of the mean of randomly selected groups of a given size.
So, any give mean value on the distribution will represent how frequently that value was likely to appear randomly.

16. t-Test for Independent Samples
Now the computer figures out the Sampling Distribution of the Mean for samples of x?
Use the sample mean to estimate the population mean.
Use the sample SD to estimate the population SD.
Start with the mean and SD for sample of x?
This starts the same way.

17. t-Test for Independent Samples
Now the computer figures out the Sampling Distribution of the Mean for samples of x?
Use the sample mean to estimate the population mean.
Use the sample SD to estimate the population SD.
Plot the mean of the comparison group on the sampling distribution
What is the probability that the second mean could have appeared by chance in the sampling distribution.

18. Probability Testing
Using the central limit theorem, a sampling distribution of the mean for groups the size of one of the samples is produced using estimates of the population parameters.
The second group mean is plotted on the distribution to determine the probability it could have appeared randomly in the population the first sample represents.
If the probability it could have appeared randomly is low, then the assumption is that the two samples came from different populations. Something made these two groups different.

19. An example in Excel Practice Problems: Performance Groups