With a new exam comes new topics. The redesigned SAT is going to make its debut come March 2016 armed with a hefty repertoire of new math topics. Among these is going to be statistics, something that you’ve likely seen on the Math II SAT Subject Test. If you’re like most other people, this subject (particularly probability) probably gives you frequent headaches. Even so, it’s not really anything serious to worry about.
Statistics looks confusing at first, but it’s really quite simple if you boil it down enough. Let’s go over some common topics:
1. Basic Fractions/Percentages
2. Counting Principle
3. Factorials + Repeating Occurrences
4. Measures of Central Tendency
5. Standard Deviation
6. Combinations
7. Permutations
Basic Fractions/Percentages
You’ve probably seen this before. It goes like this:
P = # of favorable outcomes / # of total outcomes
where P is the probability of event X happening.
Counting Principle
This is another common topic. The (fundamental) counting principle is way to calculate the total amount of ways events can occur. If Event X has a ways of occurring and Event Y has b ways of occurring, then the total amount of ways both Event X and Y can occur is ab.
n = ab
where a is the # of ways the first event can occur and b is the # of ways the second event can occur.
Factorials + Repeating Occurrences
You can easily tell when a problem is asking you to calculate a factorial if you see an exclamation mark (!) next to a number n. Here is the formula:
n! = n(n - 1)(n - 2) ... 1
Repeating occurrences isn’t actually an official type of question, but they do appear quite often. If a question asks you how many different ways the letters in a certain word or phrase can be arranged, then it’s this kind of question. The way you should go around solving this is to first find the factorial of the total amount of letters. Then for each letter than appears more than once, divide that first factorial you calculated by the factorial of the number of times the specified letter repeats. Keep on doing this until all remaining letters appear only once.
Measures of Central Tendency
This is a fancy way for saying mean, median, and mode. The mean is the sum of all numbers in a set divided by the amount of numbers there are. The median is the middle value of the data set, when the numbers are arranged in order. Finally, the mode is the value(s) that occurs the most frequently.
Source: http://www.ablongman.com/graziano6e/text_site/MATERIAL/statconcepts/mean.png
where x bar represents the mean, sigma X is the sum of all numbers within set X, and N is the total amount of numbers in set X.
Standard Deviation
This is one of the most important concepts in statistics. Standard deviation is, in essence, the average distance the values in the data set are from the mean. Standard deviation can go both ways (both above and below the mean). Usually there is a bar of three standard deviations, but it’s not a steadfast rule. Here’s the formula, which you should never try to calculate by hand while taking the test.
Source: https://s-media-cache-ak0.pinimg.com/originals/24/7d/88/247d88c2a9e349df58c0a4d4d0895676.jpg
Don’t worry if this looks overwhelming and confusing; your graphing calculator can easily do all the math for you if you have a set of numbers. Here’s how you do so:
1. Go to mode and turn “stat wizards” off
2. Go to stat, edit
3. Input the numbers in your data set in any order
4. Go to stat, calc, 1 vars stats
Scroll down and you’ll see the following:
Source: https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3Ad6e8dfdd1d1727a886b8c9c130b98d10bd645f2eed200544827c2c42%2BIMAGE%2BIMAGE.1
Ignore the numbers. Right above n, there are two symbols (Sx and σx). Use Sx if you have a sample and σx if you have a population. You don’t actually have to know the exact details as to which one to use; the problem will tell you if the data set is comprised from either a sample or a population.
Combinations
Combinations, as the name implies, are used when you want to find the total number of ways something can occur when the order does not matter. For instance, if a problem asks you to find the amount of different groupings of two between three people A, B, and C, then this is a combinations problem. This is because a group of AB is the equivalent to BA, as far as this problem is concerned. Here is the formula:
Source: http://mathwithlarry.com/lessons/lessonimages/img65.gif
n stands for the total amount in a set and r represents the amount of choices you have (in the grouping). Note that n is always the larger value and r is the smaller value.
You can either do this the hard way by manually entering in the formula and values into your calculator, or the easy way by letting the machine do it for you. Because people are lazy and you’re taking this test under time constraints, I’ll tell you how to do it the fast way:
1. Punch in n by itself
2. Go to math, prb, nCr
3. Punch in r by itself and press enter
You have your answer. It’s that simple.
Permutations
Permutations are similar to combinations and people often confuse them with good reason. You use a permutation when the order does matter. For instance, if you want to find out the total number of different ways five people A, B, C, D, and E can stand in line, then you will want to use a permutation. ABCDE is a different arrangement from BEDAC. Here’s the formula:
Source: http://ww2.tnstate.edu/ganter/BIO-311-Perm.gif
Similarly, you can have your trust Ti-84 do the hard stuff for you. Here’s how:
1. Punch in n by itself
2. Go to math, prb, nPr (yes, like the radio station)
3. Punch in r by itself and press enter
It’s very easy to do this by calculator. The only thing you have to do is distinguish it from a combinations problem and not accidentally pressing nCr.