Underlying Principles

for MTE 181

Types of Data Displays

• There are different kinds of graphs for representing data:  stem and leaf plots, dot plots, histograms, bar graph, stacked bar graph, line graphs, scatterplots, pie charts, etc.  Be familiar enough with technology that you can draw these by hand or by computer.

Data Displays that Show Relationships

• For linear regression (approximating data with a straight line), positive correlation results in a line that slopes up (from left to right), negative correlation in a line that slopes down, and zero correlation means the data is too scattered to be represented well by a straight line.
• Correlation just means that as one event happens, then another event happens.  It does NOT mean that one event causes the other.

Describing the Average and Spread of Data

• There are four main types of "Average":  the mean, the median, the mode, and the midrange.
• To calculate the (arithmetic) mean, add up the total score, then divide by the total number of participants.  (This is what math teachers usually mean when they say "average".)  The mean is also the "balance point":  The sum of the distance of the data points from the mean on one side of the mean equals the sum of the distances of the data points from the mean on the other side of the mean.
• The median is the middle score when they are ranked from lowest to highest, or the mean of the middle two, if there are an even number of scores.
• The mode is the most frequently occuring data point.  A distribution with two modes is called "bimodal" (like a camel with two humps instead of just one).  If no one data point occurs most frequently, the distribution is said to have "no mode."
• The midrange is the mean of the highest and lowest scores.
• The range is the distance from the lowest to the highest score.

• The first quartile (Q₁) is the median of the scores to the left of the overall median.  The third quartile (Q₃) is the median of the scores to the right of the overall median.  The overall median itself, of course, is Q₂.  And the interquartile range is the distance from Q₁ to Q₃.
• A box-and-whisker plot shows the lowest score, Q₁, the median, Q₃, and the highest score.
• Variance and standard deviation are a way to measure the spread of the data.  The bigger the variance (or the larger the standard deviation), the more spread out the data is.
• To calculate the variance, first calculate the arithmetic mean of the data.  Then determine the distance from each data point to the mean.  Then square each of those distances.  Then add up all of those squares.  Then divide by the number of data points.  The result is called the population variance.  (If you ever take a statistics class, you calculate a sample variance, instead of a population variance.  The difference is that instead of dividing by the number of data points n, you divide by one less (n - 1).  The reason is that you assume you are only measuring a sample of what could be, instead of measuring the entire population.  And the n - 1 is a kind of "fudge factor", since you don't know what the actual entire population distribution looks like.)
• To calculate the standard deviation, you just compute the square root of the variance.  (In our class where variance is computed by dividing by n, this is called the population standard deviation.  (In a statistics class where variance is computed by dividing by n - 1, this standard deviation is called the sample standard deviation.)
  

Decision Making with Data

• Be wary in accepting conclusions made from data:  Was the data collection method reliable and sound?  Are valid and appropriate conclusions drawn from the data?
• In drawing or interpreting graphs, watch for misleading graphs.

Understanding Probability

• Probability is a fraction:  (part / whole)  or (number of a particular kind of outcome / total number of outcomes possible).  Probability may also be given as a decimal or as a percent.
• An event is the outcome you are watching to see whether or not happens.
• A sample space is the list of all possible outcomes.  You may have different sample spaces for the same experiment (situation) depending on what event (outcome) you are looking for.
• Probability can be between 0 (0%) and 1 (100%).
• If two events are exact opposites (complementary) and one or the other of them always happens, then their probabilities add up to 100%.  For instance, if the probability of rain today is 5%, then the probability of no rain today is 95% (1 - 5%), since 5% + 95% = 100%.
• Venn diagrams are tools to examine probability.  Remember the same rules about intersection and union that you learned about Venn diagrams in MTE 180.
• If A and B are mutually exclusive, then the probability of both A and B happening ( P(A and B) = = zero, since they can't both happen at the same time.
• P(A or B) = P(A) + P(B) - P(A and B).   (to subtract out the intersection or overlap which has been counted twice).
• You need to be familiar with the outcomes when you roll 2 dice.  You should also be familiar with what cards are in a standard deck of 52 playing cards.
• A HELPFUL PROCESS for dealing with the multiplication principle:  (1) To determine the number of outcomes, ask "How many spaces are being filled?"  Then physically write down each space itself.  Label the spaces if they contain different things, for instance, if three contain letters and one contains a digit.  (2) Then in each space, write how many possibilities you have for that space.  If necessary, decide whether you have AND (multiplication) or OR (addition). (If this AND this occur, then multiply. If this OR this occurs but not both, then add.
• Watch out for "with replacement" or "without replacement" in choosing.  These affect the probability (number of successful choices / number of total choices) of the second, third, etc., choices.
• Watch for limitations on the total, the denominator of the probability fraction.  For instance, let's suppose you are choosing a digit from 1 to 9.  What is the probability that the number is even if you know that the number is prime?  In this case the phrase "if you know that" limits your total choices.  Thus, instead of choosing 2, 4, 6, or 8 out of 1, 2, 3, 4, 5, 6, 7, 8, and 9 (which would have been a probability of 4/9), you are instead choosing from among prime numbers.  The primes from 1 to 9 are 2, 3, 5, and 7.  Of these, only 2 is even.  Thus, if you know that the number is prime, the probability of choosing an even digit from 1 to 9 is 1/4.
• If you are given tables of data, it is helpful to construct a row and/or column of totals, if not provided.
• Tree diagrams (or logic diagrams) and Venn diagrams are useful tools to analyze probability.  If two events overlap, use a Venn diagram and the formula to analyze them.
• If needed for simulations, I will provide coins, cards, or dice.  Since I didn't teach you how to use the random number generator on your calculator, you are not responsible for knowing that for this test.

• Odds for = successes / failures, whereas probability = successes / total = successes / (successes + failures).  Odds against = failures / successes.
• Expected value = probability * payoff + probability * payoff + probability * payoff + ..., for as many distinct (non-overlaping) outcomes as there are.
• If expected value of a game = 0, the game is called "fair".  Otherwise, the game is called "unfair" (It is advantageous to one person but disadvantageous to another.)

Geometry

• Understand what the following geometric terms mean:  intersecting lines, parallel lines, perpendicular lines, skew lines, parallel planes, intersecting planes, concurrent lines, coplanar points, collinear points, segment, ray, angle, right angle, acute angle, obtuse angle, complementary angles, supplementary angles, adjacent angles, vertical angles, perpendicular, bisect, perpendicular bisector, closed curve or figure, circle, center of a circle, chord, diameter, radius, circumference, polygon, simple convex polygon, interior angle, regular polygon, triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, dodecagon, median of a triangle, altitude of a triangle, equilateral triangle, equiangular triangle, isosceles triangle, scalene triangle, right triangle, acute triangle, obtuse triangle, trapezoid, kite, parallelogram, rhombus, rectangle, square, similar triangles.  Note:  I am Not going to ask you to define these for me.  I just want you to understand what they are so that when some of them show up in a problem you can draw them and start to solve the problem.  Also, the altitude of a triangle is sometimes called its height, as in the formula for the area of a triangle:  (1/2) * base * height.
• Know sum of two supplementary angles.  Know the sum of the interior angles of a triangle.
  
• Understand when a network is traversable, e.g., understand Theorem:  Network Traversability on p. 526 in our textbook.  Be able to determine whether a vertex is odd or even.