Quantitative Reasoning Key Terms

Parabola

A "bowl-shaped" curve in the plane that focuses lines from infinity, such as the graph of a quadratic function ax^2 + bx + c. (The surface of rotation is a paraboloid, a shape used in reflecting telescopes or headlights for its focusing properties.) A parabola can be defined as the set of points equidistant from a chosen focus and a given line (directrix). For the curve y = ax^2 + bx + c, first define the discriminant △ = b^2 - 4ac; then the focus lies at (-b/2a, 1-△/4a), and the directrix is the line y = - 1+△/4a. A parabola can be formed by slicing a cone with any plane parallel to one of the rays forming the cone, or as the trajectory of a particle subject to a constant gravitational force.

Normal distribution

A bell-curve-shaped function describing the probability of measuring a quantity, characterized by the mean and standard deviation. The central limit theorem says that large batch averages of a quantity are described by a normal distribution. In formulas, the normal distribution of a quantity (or "random variable") x with mean x̄ and standard deviation ó is (1/√2πó^2)e^-((x-x̄)^2)/2ó^2).

Outlier

A data point that doesn't follow the pattern of the rest of the data, such was one that is more than three standard deviations away from the mean or from a linear repression

Polynomial

A function of a specific form: for functions of one variable, a polynomial is a sum of (nonnegative) powers of the variable multiplied by a number (coefficient), such as 17x^7 + 4x^3 + 2x + 6. The highest power appearing is called the "degree." For polynomials in more than one variable, replace "powers of the variable" by "products of powers of the variables" in the definition: for example, 13xy^2z - 3yz^3 +xyz is a polynomial in three variables.

Molecule

A group of atoms linked by chemical bonds. For example, a molecule of water (H2O) consists of two hydrogen atoms and an oxygen atom.

Proportionality

A linear relationship between quantities so that one is a constant multiple of the other. For example, sales receipts are proportional to the number of theatergoers: if each ticket cost $13, then T theatergoers will generate sales of S = 13T.

Volume

A measure of the "size" of a three-dimensional object or region. The volume of a box is the product of its length, width, and height. The units of volume are therefore length-cubed, e.g. cm^3.

Area

A measure of the "size" of a two-dimensional shape; the area of a rectangle is the product of its length times its width. The units of area are therefore length-squared.

Slope

A measure of the rate of increase of a linear function; the steepness of a line. A function has slope m if, for each unit increase of the input, the output increases by m. For example, the line y = mx + b has slope m since m(x + 1) + b is precisely m units larger than mx + b, for any value of x. The line, therefore, has constant slope or rate of change. The "instantaneous rate of change" of a function f(x) at a value x0 is represented as the slope of the line tangent to the curve y = f(x) at the point (x0, f(x0)).

Scientific notation

A method of expressing a number, particularly a very large or small one, which highlights its order of magnitude. For example, scientific notation for 0.000000465 would be 4.65 x 10^-6. The number that multiplies the power of 10 must be at least 1 and less than 10.

Mole

A number of molecules ("Avogadro's number"), about equal to the number of hydrogen atoms in 1 gram, approximately 6.022 x 10^23; more precisely, the number of carbon-12 atoms in 12 grams. The mass, measured in grams, of 1 mole of a molecule is approximately equal to the sum of the atomic masses of the constituents, measured in atomic mass units. For example, 1 mole of water (H20) has a mass of about 2x1+8 = 10 grams.

Irrational number

A number that cannot be written as a fraction, i.e. cannot be written an m/n, where m and n are integers. The decimal expansion of an irrational number goes on forever without a repeating pattern.

Conversion factor

A number that you multiply to convert a measurement from one unit to another. For example, since 1in = 2.54cm, dividing both sides by "1in" gives 1 = 2.54 cm/in. this is a conversion factor for going from inches to centimeters, since, for example, 12in = 12in x 1 = 12in x 2.54cm/in = 30.48cm. Note that the "inches" cancel.

Quadratic function

A polynomial of degree two in one variable, such was ax^2 + bx + c.

Linear function

A polynomial with no term involving a power greater than 1. A function f(x) of the form f(x) = ax + b for some numbers are a and b.

Order of magnitude

A power of ten that approximates a number; 890 has the order of magnitude 3, since it is close to 10^3. Two numbers can be said to have the same order of magnitude if they differ by a factor roughly less than 5, such as 13 and 46; otherwise, they differ in order of magnitude by the order of magnitude of their ratio. the number 1,623 is about two orders of magnitude higher than the number 13, as their ratio is about 125, which has order magnitude of 2.

Linear

A relationship between input and output of some process is linear if the output is a linear function of the input, meaning any plot of input values would lie on a straight line. I t also means that if you increase the input by a fixed amount, the output will always increase by a fixed multiple of that amount. For example, the number of tires is a linear function of the number of bikes (i.e. twice).

Function

A rule for assigning a number to an element from an input set. Typically, inputs are also numbers. We denote the value a function f assigns to a number x as f(x). This notation can be used to define the function as well, for example, f(x) = x^3 defines the function that takes a numerical input and outputs its cube. For this case, e.g., f(4) = 64. The input/output process is sometimes denoted x --> f(x), here 4 --> 64.

Cost-benefit

A scheme for analyzing the economic value of a decision. One tallies the benefits (measured or anticipated) and subtracts the cost of implementation and operation. For example, if a company wants to expand its sales force, the cost-benefit analysis will weigh the anticipated gain in revenues from sales against the costs of hiring, paying, and managing the new employees.

Graph

A visual representation of a function or data set. The graph of a function f(x) is a set of points (x,y) satisfying y = f(x).

Integer

A whole number or the negative of a whole number: ... -2, -1, 0, 1, 2... An integer-valued variable is often (though not exclusively) denoted by n or m, as distinct from x or y, which may indicate real numbers. The set of all integers is usually denoted as Z.

Root

A zero of a function, i.e., an input value for which the function returns zero. For example, the roots of the function f(x) = x^3 - x are x = -1, 0, 1, as can be seen by factoring x^3 - x = x(x^2 - 1) = x(x + 1)(x + 1).

Annual Percentage Rate (APR)

An annual rate on which loan interest calculations are based. Because accrued interest will generate further interest in later compounding, the APR does not actually represent the interest accumulated in one year, unless interest is only compounded once

Regression

An approximation of a data set of pairs of values by a specified form of curve. A linear regression is an approximate linear relationship between the values, often formed by the line of "least squares fir." For example, if the data are a set of points (x1, y1), (x2, y2), ..., (xn, yn), then the line of least squares fit is the line y = mx + b determined by choosing m and b so as to minimize the sum (y1 - (mx1+b))^2 + ... + (yn - (mxn +b))^2.

Risk-reward

An assessment of the likely outcome of a decision by evaluating the potential risks/costs involved against the possible rewards/benefits/gains. The assessment may involve a probabilistic model for potential outcomes.

Pythagorean Theorem

An equation expressing the relationship between the lengths of sides of a right triangle. If a and b are the leg lengths and c is the length of the hypotenuse, then c^2 = a^2 + b^2. For instance, if a point (x,y) is a distance 13 from the origin (0,0) and you know y = 12, then since 13^2 = x^2 + 12^2, we conclude x = √169-144 = 5.

Independent event

An event or occurrence whose probability does not depend on prior events. For example, if you roll one die and then roll it again, the second roll is an independent event from the first one.

Quadratic formula

An expression for the roots of a quadratic function: ax^2+bx+c = 0 is solved by x = 1/2a (-b±√b^2-4ac). For example, the roots of x^2-4x+3 are 1/2(4±√6/12), or 1 and 3.

Permutation

An ordered arrangement of a set of distinct objects. For example, ADCB is a permutation of the letters (A, B, C, D). If there are N objects, then there are N! ("N factorial" ) permutations. Sometimes "permutation" refers to ordered arrangements including just k of the N objects, in which case, there are N!/(N-k)! such permutations. For example, when k=2 in our example, there are 4!/2! = 4x3 such arrangements: AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC.

Variable

An unknown or unprescribed quantity denoted with a letter or symbol, as in "let the variable x represent the number of acres of corn grown in Iowa." Variables can also represent input values for functions, as in "the operation of addition can be represented as a function of two variables: f(x,y) = x + y."

Exponential, exponential growth, exponential decay

Exponential growth or decay refers to the behavior of members of a population (such as a population of cancer cells, mold spores, dollars of a stock price, immigrants, fruit flies, electric charge, radioactive isotopes) over time. Exponential growth means that in a fixed period of time, the population increases by a fixed factor greater than 1. Decay means that the factor is less than 1. If we call time t and the population P, then if the function P(t) is exponential, it can be written as P(t) = ab^t. Exponential growth mean b > 1, decay mean b < 1. We can also write P(t) = Poe^kt, where Po = a and e^k = b or k = ln(b), in which case, growth or decay occurs according to whether k > 0 or k < 0, respectively.

Z-score

For a measured value x, its Z-score is the number of standard deviations above the mean, so Z = (x - (x)) / ó. For example, if the mean is 12 and the standard deviation is 2, a measured value of 9 has a Z-score of (9-12)/2 = -1.5.

Scatterplot

Given a data set of pairs (x1, y1), ..., (xN, yN), the scatterplot of them is the set of all points (x1, y1), ..., (xN, yN) drawn in the plane. It is used as a visual aid in representing a correlation, or lack thereof.

Average

Given a set of N numbers (data set), the average - also called the mean - is the sum of them divided by N. If we write the numbers as x1, x2,..., xN, then the average is denoted x = 1/N(x1 + ... + XN)

Variance

Given a set of N numbers x1, ..., xN such as a data set, the variance is a measure of the spread from the mean. It is the average of the squared distance from the mean, defined by var(x) = ((x - x̄)^2). For example, the data set 2, 4, 5, 6, 8 has mean 5, so the variance is the average of 9, 1, 0, 1, 9, which is 4.

Standard deviation

Given a set of N numbers x1,..., xN such as a data set, the standard deviation óx is a measure of the spread from the mean. It is defined by óx = √((x - x̄)^2), in other words, the square root of the variance. For example, the standard deviation of the data set 2, 4, 5, 6, 8 is 2.

Median

Given a set of numbers (data set), the median is the value for which half the data are above and half are below. If there is no single such value - as for the data set {1, 3, 5, 7, 8, 8} - the median is the average of the two closest values. Here 5 and 7 are the closest, so the median is 6.

Correlation

Given a set of pairs of numbers (x1, y1), ... , (xN, YN), the correlation between them measures the degree to which the y values depend on the x values through a linear relationship (or vice versa). We denote the correlation px, y and define it by px, y = ((x-x̄)(y-ȳ)) / óxóy, where the overlies indicate mean value and ó indicates the standard deviation.

Expected value

Given a set of values x1, ..., xN occurring with probabilities p1, ..., PN, the expected value (x) is the weighted sum p1x1 + ... pNxN. Note that when all values are equally likely, the expected value agrees with the mean, justifying the use of the same notation.

Factor

In multiplication, one of a product of terms - so x is a factor of xy and a + b is a factor of (a+b)(a-c).

Confounder

In statistics, a variable that correlates (either positively or negatively) with a variable being examined. For example, suppose you measure the effect of sunglasses on how random people's beauty is perceived. If it turns out that beautiful people are more likely to wear sunglasses in the first place, then the person's beauty will be a confounding factor.

Qualitative

Involving concepts but not quantities. A qualitative understanding of gravity holds that massive objects attract each other, explaining why the Earth stays in orbit around the Sun.

Quantitative

Involving numbers. A quantitative understanding of gravity is that the attractive force between two massive objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

Percent

Literally, "out of a hundred," so, for example, "twelve percent" means "twelve out of a hundred," i.e. 12/100 or 0.12, denoted 12%. For example, 3% of 250 is 0.03 x 250 = 7.5. Changes and errors are often measured in percentages, was in "the Dow Jones Industrial Average fell 2% today" or "your guess of 8,670 jelly beans was off by just two percent: there were 8,500.

Mean

The arithmetic average of a distribution, obtained by adding the scores and then dividing by the number of scores.

Distributivity

The arithmetic property that a(b+c) = b + ac for all numbers a, b, c. This property can be used to prove FOIL: (p+q)(r+s) = (p+q)r + (p+q)s, then use distributivity twice more (after writing the last two terms in the opposite order using commutativity) to write as pr + qr + ps + qs, which can be rearranged using commutativity to obtain the FOIL ordering, pr + ps + qr + qs.

Null hypothesis

The assumption that the treatment is a trial had no effect. If, for a known distribution, this is found to happen with a probability (p-value) less than some chosen threshold or tolerance, then the null hypothesis will be rejected in favor of the alternative hypothesis: that the treatment had an effect.

Molarity

The concentration of a compound in solution, expressed in units of molars, or moles per liter, and denoted M. For example, consider 56 grams of table salt (sodium chloride: NaCl) dissolved in 1 liber of water. Since the atomic mass of Na is 11 and that of Cl is 17, 56 grams represents two moles (2 x 28 = 56). This solution would have molarity of about 2 mol/L = 2M. We write [NaCl] = 2M.

Absolute value

The distance a number is from zero on a number line

Selection bias

The failure to ensure a representative sample for data collection. For example, a survey about e-commerce that is conducted online will be biased by selecting only Internet users.

Interest

The fee paid to a lender for the privilege of borrowing, typically a percentage of the value borrowed. (A bank pays interest to account holders.)

Cumulative distribution function

The function, Φ(x), describing the probability of measuring a value less than x for a random variable obeying a normal distribution with given mean and standard deviation. In this book, we only consider a mean of zero and standard deviation of 1, so we take x to be measured in "number of standard deviations above the mean," or Z-score.

Logarithm

The inverse function to the exponential function. That is, log 10^x = x; so, for example, log 1,000 = 3. The natural logarithm is defined by ln e^x = x.

Circumference

The length around a circle. If the radius is R, the circumference is 2πR.

Significant digits, significant figures

The measure of precision for values/quantities/measurements used in analysis. Counting significant digits is simplest in scientific notation: the number 4.65 x 10^-6 is given to three significant digits, namely, the 4, 6, and 5 in 4.65. The number 0.000032 has two significant digits, since it is 3.2 x 10^-5. To indicate the same quantity with one more significant digits, you might write 0.0000320 or 3.20 x 10^-5.

Coefficient

The number in front of one of several expressions, e.g., as in a polynomial. For example, if f(x) = 3x^7 + 6x^3 + 2, the coefficient of the x^3 term is 6.

p-value

The probability that collected data support the "null hypothesis," which is that the data occur just by chance. For example, the data may show aa decline in measured fish population in the world's fisheries. It could be a coincidence of sudden deaths (the null hypothesis), or there may be something causing the fish to die (the alternative hypothesis). The statistician will want to set a significance level, such as 1% or 5%. If the p-value is below the significance level, the null hypothesis will be rejected in favor of the alternative hypothesis.

Dimensional analysis

The process of estimating a quantity's magnitude by determining what units it must have, then forming a combination of other quantities in the problem that have the same units.

Associativity

The property that expresses the notion that it doesn't matter how you group pairs of things to combine more than two. In arithmetic, it is the property of addition that (a+b)+c = a+(b+c) for all numbers a,b,c. This allows you to write a+b+c unambiguously. In multiplication, it is the property (ab)c=a(bc). We take this for granted, but cooking, for example, is not associative! When making bread, you need to combine water and yeast and flour. If you do ((water+yeast)+flour), you'll be fine, but (water+(yeast+flour)) won't work.

Commutativity

The property that expresses the notion that it doesn't matter which order you combine things. For example, addition and multiplication are commutative because a+b = b+a and ab = ba, but subtraction is not commutative because a-b ≠ b-a.

Half-life

The time it takes fora an exponential decay function, such as the measure of radioactivity over time, to decay to half its value. If the function is f(t) = Ce^-at, the half-life T satisfies f(t+T) = 1/2f(t). This yields e^-aT = 1/2, or T = ln(2)/a. The bigger the decay rate a, the shorter the half-life.

Opportunity cost

When you choose to spend your time and money doing one thing, you simultaneously choose not to spend your time and money on other things. The opportunity cost is the measure of the most valuable thing you have not done with your time and money. If you had to forgo viewing a video of a cute kitten to read this definition, then I hope that the value of understanding opportunity cost is greater than the value in pleasure you would have felt watching Kitty snuggle with a ball of yarn.

Factorial

n!=n(n-1)(n-2)...(3)(2)(1) For example, 5! = 120. The quantity N! counts the number of ways of ordering N objects. For example, 3! = 6 is the number of "words" you can form from 3 letters: ABC, ACB, BCA, BAC, CAB, CBA. Note: 0! = 1.