SAT II Math: Sizing Up Central Tendency
The SAT II Math test is bound to have at least one or two questions related to probability and statistics. The science of statistics involves organizing, analyzing, and interpreting data in order to reach reasonable conclusions and to help in decision-making. While you will most likely encounter very few questions involving statistics, you no doubt may run across the occasional query asking you to determine probability or some type of statistical average or variation from the average. The questions you'll encounter in this area are not too difficult, but it pays to give this subject a cursory review.
The Level IC test is more likely to include questions about probability, mean, median, and mode. Level IIC is more likely to test you on standard deviation.
To evaluate data correctly, you should know the central tendency of numbers and the dispersion of their values. Mean and mode give you tendency and range provides you with dispersion information.
Much of statistics involves measurement of the central tendency of numbers. A measurement of central tendency is a value that is typical, or representative, of a group of numbers or other information.
Common tools for describing a central tendency follow.
- Mean (the arithmetic mean)
- Weighted mean
- Geometric mean
Analyzing the arithmetic mean
The most common of all of these is the arithmetic mean, or simply the mean. This is what people are talking about when they say the average.
The important concept to remember about mean is the following formula: Mean average = the sum of all numbers divided by the amount of numbers that make up the sum.
While people commonly refer to the mean as the mean average, it does not "mean" the SAT II Math testers are mean by any means.
Take a peek at this example.
Sara tried to compute the mean average of her 8 test scores. She mistakenly divided the correct sum of all her test scores by 7, which yields 96. What is Sara's mean test score?
Because you know Sara's mean test score when divided by 7, you can determine the sum of Sara's scores. This information will then allow you to determine her mean average over 8 tests.
Apply the average formula to Sara's mistaken calculation.
96 = sum of scores ÷ 7
96 × 7 = sum of scores
672 = sum of scores
Now that you know Sara's test score sum, you can figure her true mean average.
Mean average = 672 ÷ 8
Mean average = 84
You know that her average must be less than 96 because you are dividing by a larger number, so you can automatically eliminate C, D, and E. The correct answer is B.
The median is another type of average you may see on the SAT II Math test. The median is a value among a list of several values or numbers that falls exactly in the middle. To find out the median, put the values or numbers in order, usually from low to high. You can think of the median income of a community as, for example, $48,000. This means that half the people's income is below that amount, and half is above. For an odd number of values, just select the middle value. If there is an even number of values, just arrange the numbers as before and determine the value halfway between the two middle values.
The mode is the other common type of average you may encounter on the SAT II test. The mode is the value that occurs most often in a set of values. Questions about mode use a word such as frequency or "how often" something happens. An example of mode could be something like income (again), where you look to the amount of income that occurs most frequently in a given population or sample. Maybe more people in the population or sample have an income of $30,000 than any other amount of income by others. In that case, you'd say the mode is $30,000.
The mean, median, and mode are usually different numbers. If they were all the same, the numbers would fall into a bell curve pattern. The mean average of the bell curve pictured falls right in the middle. If you were to add up all the values and divide by the total number of values, the mean average would be right the in the middle of the curve. This amount is also the mode because the greatest number of any one value falls directly in the middle. It's also the median because there is an equal amount of numbers on either side of the center of the curve. This bell curve is symmetrical, meaning it's the same shape on both sides, and there is an even distribution of numbers throughout the entire curve.
Wondering about weighted means
A weighted mean is where each individual value is multiplied by the number of times it occurs, and then the sum of these products is divided by the total number of times they all occur. In plain English, you may look at an example of grade point average (GPA). Suppose Becky's grades in classes with the amount of credits specified appear in the following Table:
Table Grade Point Average As Weighted Mean
|Table||Grade Point Average As Weighted Mean|
|Class||Number of Credits||Grade||Total Grade Points|
Like any weighted mean, in this example, you multiply the individual values (grades) by the number of times they each occur (credits) to get total grade points. Then you divide the total of these products (total grade points: 41.7) by the total number of times they all occur (total credits: 15). This gives you 41.7 ÷ 15 = 2.78 GPA.
Getting the geometric mean
The geometric mean is simply the average of relative numbers, percents, index numbers, growth rates, and so on. You should know what a geometric mean is, but you won't need to know how to calculate it. If you must know how to calculate it, however, the formula for a geometric mean is:
the nth root of the product of n values