AP Statistics Curriculum 2007 IntroTools
Contents
General Advance-Placement (AP) Statistics Curriculum - Statistics with Tools
Statistics with Tools (Calculators and Computers)
A critical component in any data analysis or process understanding protocol is that one needs to develop a model that has a compact analytical representation (e.g., formulas, symbolic equations, etc.) The model is used to study the process theoretically. Emperical validation of the model is carried by pluggin in data and actually testing the model. This validation stop may be done manually by computing the model prediction or model inference from recorded measurements. This typically may be done by hand only for small number of observations (<10). In practice, most of the time, we use or write algorithms and computer programs that automate these calculations for better efficiency, accuracy and consistency in applying the model to larger datasets.
There are a number of statistical software tools (programs) that one can employ for data analysis and statistical processing. Some of these are: SAS, SYSTAT, SPSS, R, SOCR.
Approach & Model Validation
Before any statistical analysis tool is employed to analyze a dataset, one needs to carefully review the prerequisites and assumptions that this model demands about the data and study design.
For example, if we measure the weight and height of students and want to study gender, age or race differences or association between weight and height, we need to make sure our sample size is large enough, these weight and height measurements are random (i.e., we do not have repeated measurements of the same student or twin-measurements) and that the students we can measure are a representative sample of the population that we are making inference about (e.g., 8th-grade students).
In this example, suppose we record the following 6 pairs of {weight (kg), height (cm)}:
| Student Index | 1 | 2 | 3 | 4 | 5 | 6 |
| Weight | 60 | 75 | 58 | 67 | 56 | 80 |
| Height | 167 | 175 | 152 | 172 | 166 | 175 |
We can easily compute the average weight (66 kg) and height (167 cm) using the sample mean-formula. We can also compute these averages using the SOCR Charts, or any other statistical package, as shown in the image below.
Computational Resources: Internet-based SOCR Tools
Several of the SOCR tools and resources will be shown later to be useful in a variety of sitiations. Here is just a list of these with one example of each:
- SOCR Charts and Histogram Charts Activity
- SOCR Analyses and Simple Linear Regression Activity
- SOCR Modeler and Random Number Generation Activity
Hands-on Examples & Activities
- As part of a brain imaging study of Alzheimer's disease *, the investigators collected the following data. We will now demonstrate how computer programs, software tools and resources, loke SOCR, can help in statistically analyzing larger datasets (certainly data size over 10 are difficult to calculate by hand correctly). In this case we'll work with 240 measurements derived from data acquired by this study.
- Let's first try to plot some of these data.
- Now we can demonstrate the use of SOCR Analyses to look for Left-Right hemispheric (HEMISPHERE) effects of the average MRI intensities (MEAN) in one Region of Interest (Occipital lobe, ROI=2). For this we can apply simple Paired T-test. This analysis is justified as the average intensities will follow Normal Distribution by the Central Limit Theorem and because the left and right hemispheric observations are naturally paired.
- Copy in your mouse buffer the 6th (MEAN), 8th (HEMISPHERE) and 9th (ROI) columns of the following data table. You can paste these three columns in Excel, or any other spreadsheet program, and reorder the rows first by ROI and then by HEMISPHERE. This will give you a exerp of 240 rows of measurements (MEAN) for ROI=2 (Occipital lobe) for each of the two hemisperes. The break down of this number of observations is as follows 240 = 2(hemispheres) * 3 (3D spatial locations, blobs) * 40 (Patients).
- Copy these 240 Rows and paste them in the Paired T-test Analysis under SOCR Analyses. Map the MEAN and HEMISPHERE columns to Dependent and Independent variables and then click Calculate. The results indicate that there are significant differences between the Left and Right Occipital mean intensities for these 40 subjects.
References
- Mega MS, Dinov, ID, Thompson, P, Manese, M, Lindshield, C, Moussai, J, Tran, N, Olsen, K, Felix, J, Zoumalan, C, Woods, RP, Toga, AW, Mazziotta, JC. Automated Brain Tissue Assessment in the Elderly and Demented Population: Construction and Validation of a Sub-Volume Probabilistic Brain Atlas, NeuroImage, 26(4), 1009-1018, 2005.
- SOCR Home page: http://www.socr.ucla.edu
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