Week 2, Day 2 (Scientific Notation, Roots, Powers, Measurements; Data Analysis)

Scientific Notation p. 59

The brief look we had at a kind of different notation earlier was expanded notation. Scientific notation is the more compact version, used to express very large and very small numbers. Significant digits in scientific notation include all numerals except zero. So if a number has several digits, we want to make sure that there is only one significant digit to the left of the decimal in scientific notation to have the correct answer.

Examples:

Note that small numbers have negative powers of ten, while large numbers have positive powers of ten.

Operations with Scientific Notation

We multiply the numbers and add the powers of ten when multiplying with scientific notation.

We divide the numbers and subtract the powers of ten when dividing with scientific notation.

Squares & Cubes p. 63

Memorizing perfect squares and cubes helps you recognize and calculate faster.

The list of perfect squares below goes from 1 to 25.

The list of perfect cubes below goes from 0 to 7.

Square Roots & Cube Roots

The list of perfect square roots below goes from 0 to 10.

The list of perfect cube roots below goes from 0 to 4.

Approximating Square Roots

The list of approximate square roots below goes from 1 to 10. These can help with estimations.

Simplifying Roots

To simplify roots, look at the factors of the number underneath the radical and try to find a perfect square. A perfect square can be pulled from under the radical, and any other factors will remain beneath the radical sign.

Chapter 3: Statistics and Probability

Statistics p. 185

The basic formulas for statistics problems are all listed on the chart, but each needs some explanation. For measures of central tendency, we will look at three different meanings for average: mean, median, and mode. The mean is also known as the arithmetic mean.

For measures of dispersion or spread, we will look at range, variance, and standard deviation. I will break standard deviation down into simpler steps, because the formula is complex and lengthy.

Graphic Displays p. 185

Frequency distribution table - most detailed - ex. p. 186

Stem-and-Leaf plot - also detailed - ex. p. 186

Bar graph - multiple bars or columns - ex. p. 187

Line Graphs - lines connect dots - ex. p. 187

Central Tendency p. 187

Central tendency measures how the data in a given set clusters or clumps together, or in other words where the middle point of the data is. However, there are three meanings for “middle” or “average” that we must distinguish carefully.

Mean p. 187

The mean or arithmetic mean is the most commonly used measure of where the center of the data lies. This is what people usually mean when they say “find the average” of several values or data points. I remember this one with the mnemonic, “When she said I was average, she was just being mean.”

We take the sum (represented by the capital Greek letter sigma or Σ) and divide it by the number of data points (represented by n) to get the mean (represented by the lowercase Greek letter mu or μ). Thus, we get the following formula:

μ = Σ ¸ n

Median p. 188

The median is the middle number in a list of numbers arranged from least to greatest (ascending order) or from greatest to least (descending order). I remember that the middle number is the median because they both have an “i” in them.

When there is an odd number of data points, there is one middle number, so the median is easy to locate. For example: {1, 2, 3, 4, 5} has five data points, an odd number. The 3 is in the middle, so the 3 is the median.

When there is an even number of data points, you must take the two middle numbers and find their mean. For example: {6, 7, 8, 9} has four data points, an even number. The 7 and 8 are in the middle, so find their mean (7 + 8) ¸ 2 = 7.5, and that is the median for this set.

Most problems you will encounter at higher levels of difficulty will require you to put the numbers in order before you can find the median – and some of these numbers will repeat, so you have to include all repetitions to do the calculation. For example: {1, 2, 1, 0, 5, 5, 3, 4, 4, 4} is not in order, and is a set with ten data points, an even number. {0, 1, 1, 2, 3, 4, 4, 4, 5, 5} is this set arranged in ascending order. The two middle numbers are now 3 and 4, which give us the median 3.5.

Mode p. 189

The mode is the number that occurs with the highest frequency in a set where values occur more than once. In other words, I remember it because the mode is the most frequent number – both words have an “o” in them.

In the set from our previous example: {0, 1, 1, 2, 3, 4, 4, 4, 5, 5}, the one occurs twice, the four occurs three times, and the five occurs twice. The four occurs the most frequently, so the mode is 4.

It is possible that a data set will either have no modes or more than one mode, so be careful when counting frequency.

Dispersion

Dispersion is how spread out the data is. There are several such measures that tell us the distances from each data point to the others.

Range p. 189

Many books treat range as a measure of central tendency because it measures the distance from the greatest value to the least value – but it is precisely because of this measure that I treat the range as a measure of spread or dispersion. To calculate the range, find the smallest value (x₁, pronounced “x sub 1,” the first data point when arranged in ascending order) and subtract it from the largest value (x_n, pronounced “x sub n,” the last data point when arranged in ascending order). Thus the range = x_n – x₁ or (if G = greatest value and L = least value) G – L.

Variance

Variance measures how far each data point varies from the mean, and is required to determine the standard deviation. More detail comes in the next section. I have never seen variance per se on the GRE, but it is essential.

Standard Deviation p. 191

Standard deviation, simply stated, measures the average of the distance from each data point from the mean – though it is a little more complicated than that. In fact, it is the square root of the variance. The formula makes use of the Greek letters and math symbols we have been using so far, and looks very complex:

The lowercase Greek letter sigma (σ) stands for standard deviation. The symbol for each individual data point is x_i, pronounced “x sub i.” I usually write a lowercase n instead of a capital N for the number of the total population included. Before we take the square root of the result, we have the variance, but after finding the square root of the result we have the standard deviation.

Let me break this process down into its component steps:

1. Find the mean of the data set.

            μ = (x₁ + x₂ + … + x_n) ¸ n

2. Find the difference between each data point and the mean.

            (x₁ – μ), (x₂ – μ), … (x_n – μ)

3. Square the differences.

            (x₁ – μ)², (x₂ – μ)², … (x_n – μ)²

4. Take the sum of the squares.

            Σ(x_i – μ)² = (x₁ – μ)² + (x₂ – μ)² +… + (x_n – μ)²

5. Divide the sum by the number of data points. ß The quotient is the variance.

            Σ(x_i – μ)² ¸ n = quotient = variance

6. Take the square root of the quotient. ß The square root is the standard deviation.

            √[Σ(x_i – μ)² ¸ n] = square root = standard deviation = σ

A large standard deviation is when most of the data spreads out far away from the mean. A small standard deviation is when most of the data is very close to the mean.

Below you will see the standard bell curve, called that because it is shaped sort of like a bell. Most averages when graphed will take on this shape, such as average height or weight. Most people will cluster within one standard deviation of the mean, in the center of the curve, while very few people will be beyond two standard deviations of the mean – at the very, very small or the very, very large on either the left or right extreme ends of the curve.

It is important to memorize the percentages inside the curve because the GRE may ask you about the probability that a measure will occur within the first two standard deviations above the mean, for example.

Usually this means that about 68.2% of the population will exist within 1 standard deviation of the mean, while the remaining 31.8% of the population is 1 standard deviation or more away from the mean value. Exactly half (50%) of the population will fall before the mean, and exactly half will fall after. Memorizing these percentages makes probabilities easier to discern when dealing with standard deviation problems.

When the data points increase or decrease all by the same amount, the measures of central tendency (mean, median, and mode) all change, but range, standard deviation, and variance do not. Adding a brand new data point may change the standard deviation and variance, but ONLY when it is outside of the existing standard deviation. Adding a new data point only changes the range when it is greater than the largest data point or less than the smallest data point.

Think of it this way, using the example data set from page 89: the mean for {3, 7, 7, 8, 10} is 7. The standard deviation works out to be about 2.28. If a new data point is discovered, greater or lesser than 7 ± 2.28 – in other words, smaller than 7 – 2.28 OR larger than 7 + 2.28 – this will change the standard deviation. However, if all the data points increase by 2 (so the new set becomes {5, 9, 9, 10, 12}), then the standard deviation does not change at all.

START HERE MONDAY p. 193 Data Quartiles

Data Quartiles p. 193

• Quartiles are four equally divided groups from smallest to largest of any given data set; percentiles are 100 equally divided groups from smallest to largest of any given data set. 
• The least value is the beginning of the first quartile, which goes up to Q1.
• Q1 is the point halfway from the smallest value to the median, or up to the 25th percentile.
• The second quartile is from Q1 to Q2.
• Q2 is the median, or up to the 50th percentile.
• The third quartile is from the median to Q3.
• Q3 is halfway from the median to the greatest value, or up to the 75th percentile, and
• The fourth quartile is from Q3 to the greatest value.
• Q4 is the greatest value, the end of the 99th percentile (there is no 100th percentile, this is why no one has a 100%ile score on the GRE or any other standardized test).

Interquartile range = Q3 - Q1 (from one end of the boxes to the other in the box-and-whisker plot)

Box-and-Whisker plot - diagram p. 193

Bivariate Data p. 195 - two-way frequency and two-way relative frequency.

"Bivariate" means the data varies or changes in two different ways. In other words, it captures data about two different variables. (e.g.: men vs. women and teachers vs. students in a school)

Bivariate Correlation Trends p. 196-7

No correlation - no pattern

Positive correlation - a trend line increasing for both variables

Negative correlation - a trend line decreasing in one variable as the other variable increases

Probability p. 199

The formulas from our memorization sheet are simplified to facilitate space; the ones in the book are a bit more detailed.

For example, P(A) = frequency/outcomes; or in other words, the probability of any event A is frequency of the desired result over all possible outcomes. This is the Classical Probability Formula:

Or P(A) = f ¸ n

1 is the value of absolute certainty, 0 is the value of absolute nonoccurrence. Thus, virtually all probabilities are fractions (or decimals).

Contrast with betting odds:

The “odds” are usually the frequency of A versus the frequency of the other possible events (u = unfavorable events), or

n – f = u                      odds(A) = f:u

Examples:

The odds of landing on heads in a coin flip: 1:1

The odds of drawing a queen of hearts w/o jokers in the deck: 1:51

The odds of rolling a three: 1:5

We make two basic assumptions in Classical Probability (which hold for the GRE):

1. All possible outcomes are taken into account.

            In other words, we do not count unlikely possibilities like a coin landing on edge.

2. Principle of indifference: All possible outcomes are equally probable.

            In other words, we do not think anyone is cheating.

In reality, these assumptions are not true – people can cheat, and unlikely events are still possible. However, on the GRE and other standardized tests, we do not run into these scenarios.

Basic Probabilities

Some of the most basic probabilities are for determined scenarios involving coin flips, dice rolls, and card pulls from a poker deck (usually without jokers).

For coin flips, you have two possible outcomes – heads or tails. Thus, for either desired/favored outcome, you have a ½ probability. E.g.: P(heads) = ½

For dice rolls, each number appears once on a six-sided cube-shaped die, i.e. 1, 2, 3, 4, 5, 6.

Thus, for any given number, you have a 1/6 probability. E.g.: P(5) = 1/6

For even numbers, you have a 3/6 or a ½ probability, and the same applies to odd numbers. E.g.: P(even) = 3/6 = ½

For other situations, such as numbers greater than 2, you have a 4/6 or a 2/3 probability, etc. E.g.: P(x>2) = 4/6 = 2/3

For card pulls from a poker deck (minus jokers), you have several choices out of 52 possible cards. There are two colors: red and black. There are four suits: hearts, diamonds, clubs, and spades (♥♦♣♠). Note that hearts and diamonds are red, and clubs and spades are black.

The probability of a red card is half of the deck, or P(red) = 26/52 or ½.

The same is true for black. P(black) = 26/52 = ½

For any given suit, there are 13 cards: 2, 3, 4, 5, 6, 7, 8, 9, 10, J (Jack), Q (Queen), K (King), A (Ace). Thus, the probability of any suit is 13/52 or ¼. E.g.: P(♥) = 13/52 = ¼

For each face value, there are four such cards; a Queen of hearts, a Queen of diamonds, a Queen of clubs, and a Queen of spades, so the probability of a Queen is 4/52 or 1/13. E.g.: P(Q) = 4/52 = 1/13

For each color and face value, there are two cards, i.e. two red Queens and two black Queens (or 2s or Aces, etc.). E.g.: P(red Q) = 2/52 = 1/26

But for any one single specific face value and suit, there is only a 1/52 probability of drawing that card. E.g.: P(Q♥) = 1/52

Other situations outside of classical probability may require you to calculate both the number of desired occurrences and the number of total possible outcomes. Several questions in our book deal with multiple spinners or jars of marbles or other scenarios where we have to do this calculation.

Complex Probabilities

The other formulas are for situations where you want the probability of more than one event. AND à multiply         OR à add       NOT à subtract

The first is for one event AND another event where the two events do not change each other; this is known as independent conjunction. P(A & B) = P(A) * P(B)            (p. 200)

The second is for one event AND another event where the first event changes the second event; this is known as dependent conjunction. P(A & B) = P(A) * P(B given A)                    (bottom p. 201)

The third is for one event OR another event where the two events cannot happen simultaneously; this is known as exclusive disjunction. P(A or B) = P(A) + P(B)  (p. 201)

The fourth is for one event OR another event where it is possible, but not desired, for the two events to happen simultaneously; this is known as inclusive disjunction. We have to subtract the possibility of both events happening simultaneously to avoid double-counting; this is the inclusion-exclusion principle from set theory. P(A or B) = P(A) + P(B) – P(A & B) or in longer form P(A or B) = P(A) + P(B) – [P(A) * P(B)]                        ()

Some events are even more complex, and require both conjunction and disjunction formulas – see ex. 6 on p. 78.

The fifth is for when you want to know the probability of an event NOT happening; this is known as negation. Because 1 is absolute certainty, we subtract the possibility of the event happening from 1 to get the probability of the event not happening. P (not A) = 1 – P(A)      (modify ex. 7)

Combinations & Permutations p. 204

In the case that we have to make a number of successive choices, we have to do either permutations or combinations of those choices. The simplest successions are just multiplication problems and factorials.

See top of p. 204 for one set of choices that do not affect each other successively. This is just a multiplication problem.

See bottom of p. 204 for a set of successive choices that do affect each other. This is a factorial.

A factorial is when you multiply a given whole number by all the whole numbers that come before it counting down to 1. Ex.: 5! (pronounced “five factorial”) is 120. 5! = 5 * 4 * 3 * 2 * 1 = 120.

I list the first 10 factorials on our memorization chart:

It is easier and less time-consuming to memorize these than to calculate them anew each time you encounter them.

When there are actually very few options (only 2 or 3), it may be easier to list the possible combinations. For example, three people are on the schedule to wash dishes; how many ways can they be ordered? Call those people A, B, and C. The possible orders are: ABC, ACB, BAC, BCA, CAB, and CBA, so there are only 6 possible orders. This list is not very time-consuming, but when there are more than three possible choices, the list gets more complicated, lengthy, and error-prone.

For more these more complicated combinations and permutations, we use the following formulas to simplify the process, make it shorter, and we are less likely to make mistakes:

Permutations are where order matters (for example, when three contestants win a race and one receives a gold medal, the other a silver, and the third a bronze). See p. 205 top example

Combinations are where order does not matter (for example, when just three contestants win, but the problem does not specify placement). See p. 205 bottom example