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Quantitative Reasoning: Data Analysis

Descriptive statistics, every graph ETS names, counting, probability and the normal curve — and the hard line where inferential statistics starts.

26 topics · 109 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Mean, Median and Mode

  1. How it worksHow does the mean act as the balance point of a dot plot while the median just splits the count?
  2. Worked exampleFind the mean, the median and the mode of 6, 3, 8, 6, 4, 6 and 9.
  3. Common mistakeWhy isn't the median always sitting somewhere between the mean and the mode?
  4. Going furtherWhat happens to the mean and to the median when every value in a list goes up by 7?
Your turneasy1 of 3
Find the mean of the five values 7, 16, 34, 46 and 63.

Weighted Means and Averages of Averages

  1. How it worksHow does weighting each value by how often it occurs rebuild the true overall mean?
  2. Common mistakeWhy isn't an average of averages the same as the average of everything pooled together?
  3. Worked exampleFind the combined mean when 12 students average 78 and 18 students average 88.
  4. Going furtherWhat happens to a weighted mean as one group's share of the total grows toward everything?
Your turnmedium1 of 3
One class of 16 students averages 79 on a test and a second class of 30 students averages 89. Find the combined mean for all the students together.

Quartiles and the Interquartile Range

  1. How it worksHow do Q1, the median and Q3 cut sorted data into four groups of roughly equal count?
  2. Worked exampleFind Q1, Q3 and the interquartile range of 2, 4, 4, 5, 6, 7, 7, 8, 9, 10, 11 and 14.
  3. Going furtherWhat happens to the IQR when the middle half of the data bunches tightly around the median?
  4. Common mistakeWhy doesn't the interquartile range move when the largest value jumps from 14 to 140?
Your turneasy1 of 3
In a sorted data set, Q1 is 20 and Q3 is 63. Find the interquartile range.

Percentiles

  1. How it worksHow do 99 percentile cut points slice a large ordered data set into a hundred equal groups?
  2. Worked exampleFind roughly what percentile a score sits in when 240 of 300 students scored below it.
  3. Common mistakeWhy isn't the 50th percentile the midpoint between the smallest and largest value?
  4. Ties togetherHow do percentiles line up with quartiles once a data set gets large?
Your turneasy1 of 3
Of 365 students who took a test, 231 scored below Jordan. Find Jordan's approximate percentile.

Range and Standard Deviation

  1. How it worksWalk me through squaring each distance from the mean, averaging those, then taking a square root.
  2. Worked exampleFind the standard deviation of 0, 7, 8, 10 and 10.
  3. Common mistakeWhy doesn't adding 10 to every value in a list change its standard deviation at all?
  4. Going furtherWhat happens to the standard deviation when every value in a list is multiplied by 3?
  5. Common mistakeWhy isn't a larger range automatically a larger standard deviation?
Your turnhard1 of 3
Find the standard deviation of the five values 24, 24, 24, 24 and 24 plus 28.

Standardization and Standard Units

  1. How it worksHow does subtracting the mean and dividing by the deviation move a value onto a shared ruler?
  2. Worked exampleFind the standard units for a value of 48 when the mean is 32.5 and the deviation is 7.1.
  3. Common mistakeWhy doesn't a standardized value of 1.5 mean the raw value was 1.5 times the mean?
  4. Going furtherWhat happens to every standardized value when the whole data set is shifted up by 20?
Your turneasy1 of 3
A data set has mean 34 and standard deviation 9. Find the standard units of the value 54.

Boxplots and the Five-Number Summary

  1. How it worksWalk me through reading a five-number summary off a sorted list and drawing the box and whiskers.
  2. Common mistakeWhy doesn't a wide box mean more data points landed in that part of the range?
  3. Worked exampleDraw the boxplot for 2, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 12, 13, 14, 15 and 18.
  4. Going furtherWhat happens to two boxplots side by side when one of the data sets is shifted 100 higher?
Your turnhard1 of 3
A boxplot has minimum 7, Q1 21, median 39, Q3 49 and maximum 72. Find the width of the box as a fraction of the full range.

Frequency and Relative Frequency Distributions

  1. How it worksHow does dividing each frequency by the total turn raw counts into percents that add to 100?
  2. Common mistakeWhy isn't the mean of a frequency table the average of the values in its left column?
  3. Worked exampleFind the relative frequency of 2-child households when 7 of 25 households have exactly 2.
  4. Going furtherWhat changes when a table groups values into intervals instead of listing each value separately?
Your turneasy1 of 3
Of 33 households surveyed, 10 have exactly 2 children. Find the relative frequency of 2-child households as a percent.

Histograms

  1. How it worksHow does bar height carry the frequency while bar position carries the value or the interval?
  2. Common mistakeWhy isn't a histogram the same thing as a bar graph of separate named categories?
  3. Worked exampleFind the median of a histogram with bars of height 3, 7, 9, 5 and 1 over the values 0 through 4.
  4. Going furtherWhat happens to a histogram's shape when the bin width is doubled?
Your turnhard1 of 3
A histogram over the values 0, 1, 2, 3 and 4 has bars of height 5, 10, 20, 5 and 1. Find the mean of the data.

Reading Tables and Percents of a Total

  1. How it worksHow does a percent column plus a total row let you rebuild every raw count in a table?
  2. Worked exampleFind the number of credit complaints when they are 1.0 percent of 22,998 total complaints.
  3. Worked exampleFind the percent decrease when total complaints fall from 22,998 in one year to 13,278 the next.
  4. Common mistakeWhy isn't a move from 20.0 percent to 22.1 percent an increase of more than 2 percent?
Your turnhard1 of 3
A category accounts for 16 percent of a year-one total of 24661 complaints and 24 percent of a year-two total of 11836 complaints. Find the change in the actual number of complaints in that category, year two minus year one.

Bar Graphs and Segmented Bar Graphs

  1. How it worksHow does a segmented bar stack several categories into one column that totals the whole?
  2. Worked exampleFind the third segment's value when a bar totals 240 and the first two segments read 85 and 62.
  3. Going furtherWhat changes when a segmented bar graph is redrawn with every bar stretched to equal height?
  4. Common mistakeWhy doesn't the tallest bar always stand for the biggest share of the total?
Your turneasy1 of 3
A segmented bar totals 284, and its first two segments read 86 and 87. Find the value of the third and final segment.

Circle Graphs

  1. How it worksHow does each sector's central angle come from its percent multiplied by 360 degrees?
  2. Worked exampleFind the central angle of a sector standing for 18 percent of a circle graph.
  3. Worked exampleFind the dollar value of a 25 percent sector when the whole graph covers 3,980 million dollars.
  4. Common mistakeWhy isn't a slice that looks twice as wide as another necessarily worth twice as much?
Your turneasy1 of 3
Find the central angle, in degrees, of the sector standing for 34 percent of a circle graph.

Line Graphs

  1. How it worksHow does reading a line graph's slope between two points give you the change per year?
  2. Worked exampleFind the year of greatest increase when enrollment reads 500, 560, 590, 700 and 720.
  3. Common mistakeWhy doesn't the highest point on a line graph mark the year of the biggest increase?
Your turnmedium1 of 3
A line graph shows enrollment of 539 in year 2011 and 933 in year 2020. Find the average increase in enrollment per year over that span.

Scatterplots and Trend Lines

  1. How it worksHow does a line of best fit get placed through a scatterplot, and what does its slope mean?
  2. Worked exampleFind the predicted value at x equal to 20 when a trend line passes through 5,12 and 15,32.
  3. Common mistakeWhy isn't a strong upward trend in a scatterplot evidence that one thing causes the other?
  4. Going furtherWhat happens to a trend line when one far-out point is dropped into the corner of the plot?
Your turnhard1 of 3
A trend line passes through the points (5, 13) and (18, 31). Find the value it predicts at x equal to 21.

Data Interpretation Sets

  1. How it worksWalk me through working a data interpretation set: title first, then units, then the question.
  2. Worked exampleFind 23 percent of 4,860 using only the basic on-screen calculator and a few seconds of estimating.
  3. Common mistakeWhy doesn't a data interpretation set ever use the quantitative comparison format?
  4. Going furtherWhat changes about your approach when a data set question says select all that apply?
Your turneasy1 of 3
A data interpretation table reports a total of 4947. Find 36 percent of that total.

Sets, Lists and the Multiplication Principle

  1. How it worksHow does a branching tree of choices turn into a single product of counts?
  2. Worked exampleFind how many 4-character codes use one of 5 letters followed by three digits from 0 to 9.
  3. Going furtherWhat changes in the count when a choice is not allowed to repeat an earlier one?
  4. Common mistakeWhy isn't the number of outcomes the sum of the choices available at each stage?
Your turneasy1 of 3
A wardrobe holds 7 shirts, 6 ties and 6 jackets. Find how many different shirt-tie-jacket outfits are possible.

Permutations and Factorials

  1. How it worksHow does filling seats one at a time with one fewer candidate each time build a factorial?
  2. Worked exampleFind how many ways 3 of 6 runners can take first, second and third place.
  3. Worked exampleFind how many distinct arrangements the letters of LEVEL can be put into.
  4. Common mistakeWhy doesn't the order-matters count apply when the prizes handed out are identical?
Your turneasy1 of 3
A race has 12 runners. Find how many different ways first, second and third place can be awarded.

Combinations

  1. How it worksHow does dividing by the arrangements of the chosen group collapse the duplicate orderings?
  2. Worked exampleFind how many different 3-person committees can be formed from a group of 5 people.
  3. Worked exampleFind how many ways to choose 2 books from 9 when the order of choosing does not matter.
  4. Common mistakeWhy isn't choosing 4 from 10 a different count than choosing 6 from 10?
  5. Going furtherWhat happens to the committee count when one particular person has to be included?
Your turnmedium1 of 3
Find how many different 3-person committees can be formed from a group of 12 people.

Venn Diagrams and the Overlap

  1. How it worksHow does subtracting the overlap exactly once repair the double count in a two-circle diagram?
  2. Worked exampleFind how many of 50 students take neither when 28 take French, 25 take Spanish and 12 take both.
  3. Common mistakeWhy doesn't only-French mean the same thing as French in a two-circle Venn diagram?
  4. Worked exampleFind the overlap when 40 of 58 people own a dog, 30 own a cat and 12 own neither animal.
  5. Going furtherWhat changes in the counting rule once a third circle joins the diagram?
Your turnmedium1 of 3
Of 74 students, 30 take French, 27 take Spanish and 14 take both. Find how many take neither language.

Elementary Probability and Sample Spaces

  1. How it worksHow does listing every equally likely outcome turn a probability into a plain fraction?
  2. How it worksHow does the sample space for two dice become a 6 by 6 grid of 36 equally likely outcomes?
  3. Worked exampleFind the probability of rolling a sum of 7 with two fair six-sided dice.
  4. Common mistakeWhy isn't every sum from 2 through 12 equally likely when you roll two dice?
Your turneasy1 of 3
A bag holds 10 red, 11 green and 13 blue marbles. Find the probability that one marble drawn at random is green.

Mutually Exclusive Events and the Addition Rule

  1. How it worksHow does the chance of A or B equal the two chances added, minus the chance of both?
  2. Worked exampleFind the probability of drawing a heart or a face card from a standard 52-card deck.
  3. Common mistakeWhy doesn't rolling a 4 and rolling an even number count as mutually exclusive?
  4. Going furtherWhat happens to the addition rule when the two events genuinely cannot both occur?
Your turnhard1 of 3
A deck of 55 cards contains 14 hearts and 14 face cards, and 3 cards are both. Find the probability that one card drawn at random is a heart or a face card.

Independent Events and the Multiplication Rule

  1. How it worksHow does multiplying probabilities work only when one event leaves the other's odds untouched?
  2. Worked exampleFind the probability of two heads in a row followed by a 5 on a fair six-sided die.
  3. Common mistakeWhy isn't mutually exclusive just another way of saying independent?
  4. Worked exampleFind the chance of drawing two red cards in a row from 52 cards without replacing the first.
  5. Going furtherWhat changes about independence when the sample is drawn from an enormous population?
Your turnhard1 of 3
A spinner lands on red with probability 65 percent on each spin, independently of every other spin. Find the probability of red on all 4 of the next 4 spins.

Conditional Probability

  1. How it worksHow does conditioning on an event shrink the sample space down to one row of a two-way table?
  2. How it worksWalk me through turning a two-way table of 200 people into any conditional probability you want.
  3. Worked exampleFind the chance a card from a 52-card deck is a king, given it is already known to be a face card.
  4. Common mistakeWhy isn't the chance of rain given clouds the same as the chance of clouds given rain?
  5. Going furtherWhat happens to a conditional probability when the two events turn out to be independent?
Your turneasy1 of 3
A survey of 220 people includes 79 who commute by train, and 48 of those train commuters also carry a season ticket. Find the probability that a randomly chosen train commuter carries a season ticket.

Random Variables and Expected Value

  1. How it worksHow does a probability distribution hand a weight to each value a random variable can take?
  2. Common mistakeWhy isn't the expected value just the middle of the range of possible values?
  3. Worked exampleFind the expected value of X when X is 0, 1, 2 or 3 with probabilities 0.2, 0.3, 0.4 and 0.1.
  4. Going furtherWhat happens to the histogram of a random variable as the number of trials grows very large?
Your turnmedium1 of 3
A game pays 48 dollars with probability 66 percent and 7 dollars otherwise. Find the expected payout in dollars.

The Normal Distribution

  1. How it worksHow do the bands at one, two and three deviations fill in about 68, 95 and 99.7 percent?
  2. How it worksHow does the coarser ETS wording, two-thirds within one deviation, still answer the question?
  3. Worked exampleFind the chance that W lands between 3 and 7 when W is normal with mean 5 and deviation 2.
  4. Common mistakeWhy doesn't a smaller standard deviation make a normal curve shorter instead of taller?
  5. Ties togetherHow does standardizing a normal value connect any bell curve to one standard normal curve?
Your turneasy1 of 3
A normal distribution has mean 75 and standard deviation 11. Almost all of its values fall within 2 standard deviations of the mean. Find the largest value in that interval.

Why Inferential Statistics Is Not Tested

  1. How it worksHow does the line fall between descriptive statistics, which are tested, and inference, which is not?
  2. Common mistakeWhy isn't a p-value, a confidence interval or a regression slope ever the answer on the GRE?
  3. Going furtherWhat still counts as fair game about a normal distribution once inference is ruled out?
  4. Ties togetherHow does the no-inference rule line up with the bans on trigonometry, calculus and proofs?
Your turnmedium1 of 3
A GRE data interpretation set presents a table of 151 observations. Which task could the set legitimately ask for?

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