Glossary

absolute value

The absolute value of a number is its distance from 0 on the number line. 


association

In statistics we say that there is an association between two variables if the two variables are statistically related to each other; if the value of one of the variables can be used to estimate the value of the other.


average rate of change

The average rate of change of a function \(f\) between inputs \(a\) and \(b\) is the change in the outputs divided by the change in the inputs: \(\frac{f(b)-f(a)}{b-a}\). It is the slope of the line joining \((a,f(a))\) and \((b, f(b))\) on the graph.


bell-shaped distribution

A distribution whose dot plot or histogram takes the form of a bell with most of the data clustered near the center and fewer points farther from the center. 


bimodal distribution

A distribution with two very common data values seen in a dot plot or histogram as distinct peaks. In the dot plot shown, the two common data values are 2 and 7,


categorical data

Categorical data are data where the values are categories. For example, the breeds of 10 different dogs are categorical data. Another example is the colors of 100 different flowers.


categorical variable

A variable that takes on values which can be divided into groups or categories. For example, color is a categorical variable which can take on the values, red, blue, green, etc.


causal relationship

A relationship is one in which a change in one of the variables causes a change in the other variable.


coefficient

In an algebraic expression, the coefficient of a variable is the constant the variable is multiplied by. If the variable appears by itself then it is regarded as being multiplied by 1 and the coefficient is 1.

The coefficient of \(x\) in the expression \(3x + 2\) is \(3\). The coefficient of \(p\) in the expression \(5 + p\) is 1.


completing the square

Completing the square in a quadratic expression means transforming it into the form \(a(x+p)^2-q\), where \(a\), \(p\), and \(q\) are constants.

Completing the square in a quadratic equation means transforming into the form \(a(x+p)^2=q\).


constant term

In an expression like \(5x + 2\) the number 2 is called the constant term because it doesn't change when \(x\) changes. 

In the expression \(5x-8\) the constant term is -8, because we think of the expression as \(5x + (-8)\). In the expression \(12x-4\) the constant term is -4.


constraint

A limitation on the possible values of variables in a model, often expressed by an equation or inequality or by specifying that the value must be an integer. For example, distance above the ground \(d\), in meters, might be constrained to be non-negative, expressed by \(d \ge 0\).


correlation coefficient

A number between -1 and 1 that describes the strength and direction of a linear association between two numerical variables. The sign of the correlation coefficient is the same as the sign of the slope of the best fit line. The closer the correlation coefficient is to 0, the weaker the linear relationship. When the correlation coefficient is closer to 1 or -1, the linear model fits the data better.

The first figure shows a correlation coefficient which is close to 1, the second a correlation coefficient which is positive but closer to 0, and the third a correlation coefficient which is close to -1.


decreasing (function)

A function is decreasing if its outputs get smaller as the inputs get larger, resulting in a downward sloping graph as you move from left to right.

A function can also be decreasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is decreasing for \(x \ge 0\) because the graph slopes downward to the right of the vertical axis.


dependent variable

A variable representing the output of a function.

The equation \(y = 6-x\) defines \(y\) as a function of \(x\). The variable \(x\) is the independent variable, because you can choose any value for it. The variable \(y\) is called the dependent variable, because it depends on \(x\). Once you have chosen a value for \(x\), the value of \(y\) is determined. 


distribution

For a numerical or categorical data set, the distribution tells you how many of each value or each category there are in the data set.


domain

The domain of a function is the set of all of its possible input values.


elimination

A method of solving a system of two equations in two variables where you add or subtract a multiple of one equation to another in order to get an equation with only one of the variables (thus eliminating the other variable).


equivalent equations

Equations that have the exact same solutions are equivalent equations.


equivalent systems

Two systems are equivalent if they share the exact same solution set.


exponential function

An exponential function is a function that has a constant growth factor. Another way to say this is that it grows by equal factors over equal intervals. For example, \(f(x)=2 \boldcdot 3^x\) defines an exponential function. Any time \(x\) increases by 1, \(f(x)\) increases by a factor of 3. 


factored form (of a quadratic expression)

A quadratic expression that is written as the product of a constant times two linear factors is said to be in factored form. For example, \(2(x-1)(x+3)\) and \((5x + 2)(3x-1)\) are both in factored form.


five-number summary

The five-number summary of a data set consists of the minimum, the three quartiles, and the maximum. It is often indicated by a box plot like the one shown, where the minimum is 2, the three quartiles are 4, 4.5, and 6.5, and the maximum is 9.


function

A function takes inputs from one set and assigns them to outputs from another set, assigning exactly one output to each input.


function notation

Function notation is a way of writing the outputs of a function that you have given a name to. If the function is named \(f\) and \(x\) is an input, then \(f(x)\) denotes the corresponding output.


growth factor

In an exponential function, the output is multiplied by the same factor every time the input increases by one. The multiplier is called the growth factor.


growth rate

In an exponential function, the growth rate is the fraction or percentage of the output that gets added every time the input is increased by one. If the growth rate is 20% or 0.2, then the growth factor is 1.2.


horizontal intercept

The horizontal intercept of a graph is the point where the graph crosses the horizontal axis. If the axis is labeled with the variable \(x\), the horizontal intercept is also called the \(x\)-intercept. The horizontal intercept of the graph of \(2x + 4y = 12\) is \((6,0)\).

The term is sometimes used to refer only to the \(x\)-coordinate of the point where the graph crosses the horizontal axis.


increasing (function)

A function is increasing if its outputs get larger as the inputs get larger, resulting in an upward sloping graph as you move from left to right.

A function can also be increasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is increasing for \(x \le 0\) because the graph slopes upward to the left of the vertical axis.


independent variable

A variable representing the input of a function.

The equation \(y = 6-x\) defines \(y\) as a function of \(x\). The variable \(x\) is the independent variable, because you can choose any value for it. The variable \(y\) is called the dependent variable, because it depends on \(x\). Once you have chosen a value for \(x\), the value of \(y\) is determined. 


inverse (function)

Two functions are inverses to each other if their input-output pairs are reversed, so that if one functions takes \(a\) as input and gives \(b\) as an output, then the other function takes \(b\) as an input and gives \(a\) as an output.

You can sometimes find an inverse function by reversing the processes that define the first function in order to define the second function.


irrational number

An irrational number is a number that is not rational. That is, it cannot be expressed as a positive or negative fraction, or zero.


linear function

A linear function is a function that has a constant rate of change. Another way to say this is that it grows by equal differences over equal intervals. For example, \(f(x)=4x-3\) defines a linear function. Any time \(x\) increases by 1, \(f(x)\) increases by 4.


linear term

The linear term in a quadratic expression (In standard form) \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, is the term \(bx\). (If the expression is not in standard form, it may need to be rewritten in standard form first.)


maximum

A maximum of a function is a value of the function that is greater than or equal to all the other values. The maximum of the graph of the function is the corresponding highest point on the graph.


minimum

A minimum of a function is a value of the function that is less than or equal to all the other values. The minimum of the graph of the function is the corresponding lowest point on the graph.


model

A mathematical or statistical representation of a problem from science, technology, engineering, work, or everyday life, used to solve problems and make decisions.


negative relationship

A relationship between two numerical variables is negative if an increase in the data for one variable tends to be paired with a decrease in the data for the other variable.


non-statistical question

A non-statistical question is a question which can be answered by a specific measurement or procedure where no variability is anticipated, for example:

  • How high is that building?
  • If I run at 2 meters per second, how long will it take me to run 100 meters?

numerical data

Numerical data, also called measurement or quantitative data, are data where the values are numbers, measurements, or quantities. For example, the weights of 10 different dogs are numerical data.


outlier

A data value that is unusual in that it differs quite a bit from the other values in the data set. In the box plot shown, the minimum, 0, and the maximum, 44, are both outliers.


perfect square

A perfect square is an expression that is something times itself. Usually we are interested in situations where the something is a rational number or an expression with rational coefficients.


piecewise function

A piecewise function is a function defined using different expressions for different intervals in its domain.


positive relationship

A relationship between two numerical variables is positive if an increase in the data for one variable tends to be paired with an increase in the data for the other variable.


quadratic equation

An equation that is equivalent to one of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).


quadratic expression

A quadratic expression in \(x\) is one that is equivalent to an expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).


quadratic formula

The formula \(x = {\text-b \pm \sqrt{b^2-4ac} \over 2a}\) that gives the solutions of the quadratic equation \(ax^2 + bx + c = 0\), where \(a\) is not 0.


quadratic function

A function where the output is given by a quadratic expression in the input.


range

The range of a function is the set of all of its possible output values.


rational number

A rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point on the number line that you get by dividing the unit interval into \(b\) equal parts and finding the point that is \(a\) of them from 0. We can always write a fraction in the form \(\frac{a}{b}\) where \(a\) and \(b\) are whole numbers, with \(b\) not equal to 0, but there are other ways to write them. For example, 0.7 is a fraction because it is the point on the number line you get by dividing the unit interval into 10 equal parts and finding the point that is 7 of those parts away from 0. We can also write this number as \(\frac{7}{10}\).

The numbers \(3\), \(\text-\frac34\), and \(6.7\) are all rational numbers. The numbers \(\pi\) and \(\text-\sqrt{2}\) are not rational numbers, because they cannot be written as fractions or their opposites.


relative frequency table

A version of a two-way table in which the value in each cell is divided by the total number of responses in the entire table or by the total number of responses in a row or a column.

The table illustrates the first type for the relationship between the condition of a textbook and its price for 120 of the books at a college bookstore. 

  $10 or less more than \$10 but less than $30 $30 or more
new 0.025 0.075 0.225
used 0.275 0.300 0.100

residual

The difference between the \(y\)-value for a point in a scatter plot and the value predicted by a linear model. The lengths of the dashed lines in the figure are the residuals for each data point.


skewed distribution

A distribution where one side of the distribution has more values farther from the bulk of the data than the other side, so that the mean is not equal to the median. In the dot plot shown, the data values on the left, such as 1, 2, and 3, are further from the bulk of the data than the data values on the right.


solutions to a system of inequalities

All pairs of values that make the inequalities in a system true are solutions to the system. The solutions to a system of inequalities can be represented by the points in the region where the graphs of the two inequalities overlap.


solution to a system of equations

A coordinate pair that makes both equations in the system true.

On the graph shown of the equations in a system, the solution is the point where the graphs intersect.


standard deviation

A measure of the variability, or spread, of a distribution, calculated by a method similar to the method for calculating the MAD (mean absolute deviation). The exact method is studied in more advanced courses.


standard form (of a quadratic expression)

The standard form of a quadratic expression in \(x\) is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not 0.


statistic

A quantity that is calculated from sample data, such as mean, median, or MAD (mean absolute deviation).


statistical question

A statistical question is a question that can only be answered by using data and where we expect the data to have variability, for example:

  • Who is the most popular musical artist at your school?
  • When do students in your class typically eat dinner?
  • Which classroom in your school has the most books?

strong relationship

A relationship between two numerical variables is strong if the data is tightly clustered around the best fit line.


substitution

Substitution is replacing a variable with an expression it is equal to.


symmetric distribution

A distribution with a vertical line of symmetry in the center of the graphical representation, so that the mean is equal to the median. In the dot plot shown, the distribution is symmetric about the data value 5.


system of equations

Two or more equations that represent the constraints in the same situation form a system of equations. 


system of inequalities

Two or more inequalities that represent the constraints in the same situation form a system of inequalities.


two-way table

A way of organizing data from two categorical variables in order to investigate the association between them. 

  has a cell phone does not have a cell phone
10–12 years old 25 35
13–15 years old 38 12
16–18 years old 52 8

uniform distribution

A distribution which has the data values evenly distributed throughout the range of the data.


variable (statistics)

A characteristic of individuals in a population that can take on different values


vertex form (of a quadratic expression)

The vertex form of a quadratic expression in \(x\) is \(a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants, and \(a\) is not 0.


vertex (of a graph)

The vertex of the graph of a quadratic function or of an absolute value function is the point where the graph changes from increasing to decreasing or vice versa. It is the highest or lowest point on the graph.

 


vertical intercept

The vertical intercept of a graph is the point where the graph crosses the vertical axis. If the axis is labeled with the variable \(y\), the vertical intercept is also called the \(y\)-intercept.

Also, the term is sometimes used to mean just the \(y\)-coordinate of the point where the graph crosses the vertical axis. The vertical intercept of the graph of \(y = 3x - 5\) is \((0,\text-5)\), or just -5.


weak relationship

A relationship between two numerical variables is weak if the data is loosely spread around the best fit line.


zero (of a function)

A zero of a function is an input that yields an output of zero. If other words, if \(f(a) = 0\) then \(a\) is a zero of \(f\).


zero product property

The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.