Graphs and Data

• Read bar graphs, histograms, dot plots, stem-and-leaf plots, and box plots, and pull a median, range, or count from the display
• Read a scatter plot: describe the association (positive, negative, or none) and use a line of best fit to predict a value
• Interpret the slope and y-intercept of a line of best fit in the context of the data
• Compute an expected value: multiply each outcome by its probability, then add
• Count with two overlapping sets using a Venn diagram or |A or B| = |A| + |B| − |A and B|

Data displays: always read the title, axis labels, and scale before the data. A histogram groups values into intervals and the bar height is the count in that interval, so to find the median, count in from one end to the middle position. The same works for a frequency table: with 16 values the median is the average of the 8th and 9th, found by adding up the counts from the smallest value. A dot plot and a stem-and-leaf plot show every value, so mean, median, and mode can be found exactly (in a stem-and-leaf plot, 4 | 2 7 means 42 and 47). A box plot shows the five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum. The range is maximum − minimum and the interquartile range is Q3 − Q1; each of the four sections holds about a quarter of the data, however wide it looks. Scatter plots show two variables at once. An upward drift is a positive association, a downward drift is negative, and a shapeless cloud is none; association does not prove causation. A line of best fit summarizes the trend: substitute an x-value to predict y. A model of the form y = kx passes through the origin, so k is about y ÷ x for the data points: find that ratio, then multiply. Its slope is the predicted change in y for each one-unit increase in x (for example, 'each extra hour of study predicts 4.5 more points'), and its y-intercept is the predicted value when x = 0. A point above the line is higher than predicted. Expected value is a long-run average: multiply each outcome by its probability and add. If a spinner pays 0 points with probability 0.5, 1 point with probability 0.3, and 2 points with probability 0.2, the expected value is 0(0.5) + 1(0.3) + 2(0.2) = 0.7 points per spin. The probabilities must add to 1. Sets: the union (A or B) is everything in either set, and the intersection (A and B) is what they share. Because the overlap gets counted twice, |A or B| = |A| + |B| − |A and B|. Example: of 30 students, 18 take Spanish, 15 take French, and 8 take both, so 18 + 15 − 8 = 25 take at least one and 30 − 25 = 5 take neither. In a Venn diagram, fill in the overlap first, then subtract it from each circle's total. With three circles the regions are: in only one group, in exactly two (the three two-way overlaps outside the center), in all three (the center), and in none; 'exactly two' leaves out the center, and 'not in C' is the total minus every region inside C. Data collection: results from a sample generalize to a population only if the sample was chosen at random from that population; volunteer and convenience samples are biased, and a bigger biased sample is still biased. Only a randomized experiment, with subjects randomly assigned to treatments, supports a cause-and-effect conclusion; an observational study shows only an association. A larger random sample gives a smaller margin of error.

• In a Venn diagram problem, write the overlap in first. Every other region comes from subtracting it.
• For expected value, make a two-row table of outcomes and probabilities, multiply down each column, and add across.
• For a line of best fit, slope questions are about units: 'y-units per one x-unit.' Say it in words before picking a choice.
• A wide section of a box plot does not hold more data. Each section holds about 25 percent.
• A graphing calculator can verify a prediction from a line of best fit, but plugging in by hand is usually faster.