Data interpretation, oriented

Before reading any value, check what the axis measures and in what units. A bar reaching 4.2 on an axis marked "millions of riders" means 4,200,000 riders, and a column headed $k is in thousands of dollars. With grouped bars or several lines, match each series to the legend first; swapped series give answers that look reasonable and are wrong.

Read the last words of the question to see what it wants. "Which month was wettest?" wants a month: if April to July had 58, 91, 74, and 66 mm of rain, the answer is May, and 91 is only the evidence. A question about combined totals needs the bars in each group added first. If Product A sold 50 and Product B sold 10 in one quarter, and each sold 35 in the next, the second quarter's combined 70 beats the first's 60, even though the tallest single bar is in the first.

For trends, check every step: a series of 12, 15, 14, and 19 ends higher than it starts but did not rise every quarter. Where two lines are farthest apart is the largest vertical gap between them, not the highest point of either line. A scatter plot whose points rise from left to right shows a positive association, but an association alone does not prove that one variable causes the other.

Rule: check the scale, the units, and the legend first, then answer with exactly what the question asks for.

Add a list by pairing values that make round numbers: for sales of 37, 19, 23, and 41, the pairs 37 + 23 and 19 + 41 each make 60, so the total is 120. When a total and all but one part are given, subtract: a yearly total of 731 with three quarters at 215, 188, and 149 leaves 731 − 552 = 179 for the fourth. To find when a cumulative total first passes a target, keep a running sum: daily sales of 23, 16, 31, and 12 give 23, 39, 70, and 82, so a target of 75 is first passed on the fourth day.

For a difference, name both values before subtracting. The gap between the best and second-best of 52, 38, 47, and 29 is 52 − 47 = 5, and the gap between the wettest and driest of 34, 82, 57, and 26 mm is 82 − 26 = 56 mm. Subtracting by counting up through a round number avoids borrowing: from 26 to 30 is 4, and from 30 to 82 is 52, so the gap is 56.

To find where a series fell the most, give each step its sign: 75, 48, 63, 21 changes by −27, +15, and −42, so the biggest fall is from the third value to the fourth. A large rise is not a fall.

Rule: pair values to add, subtract the known parts to find a missing one, and name the two values and the sign of every change before comparing.

The mean is the sum of the values divided by how many there are: scores of 14, 9, 17, and 12 total 52, so the mean is 13. A mean must lie between the smallest and largest values, which is a quick check on the arithmetic.

When one value joins a group, add it to the group's total and divide by the new count. A team of 8 averaging 20 km, joined by a member who logs 47 km, has a total of 160 + 47 = 207 km over 9 people, an average of 23 km. Averaging the old average with the new value, to get 33.5, treats the newcomer as half the team.

Groups of different sizes combine through their totals too. A class of 15 averaging 70 and a class of 45 averaging 82 total 1,050 and 3,690, so together they average 4,740 ÷ 60 = 79, not the midpoint of 76, because the larger class pulls the average toward its own.

Rule: turn every average into a total, combine the totals, and divide by the combined count.

A pie slice is its percent of the whole: a 28% slice of a $2,500 budget is $700. Going the other way, if the remaining 12% of sales was 54 units, total sales were 54 ÷ 0.12 = 450 units. The slices of a pie add to 100%, so an unlabeled slice is 100% minus the others.

A percent change divides the change by the starting value. From 50 to 65 is a rise of 15 on 50, or 30%. From 65 back to 50 is a fall of 15 on 65, about 23%, because the starting value is now larger.

A share of a share multiplies the rates. If 35% of 1,200 people rent and 20% of those renters own a car, then 0.35 × 0.2 = 0.07 of all 1,200, or 84 people, do both. For "what percent of the commuters are women", divide by the commuters: if 24 of 80 women and 36 of 120 men commute by train, women are 24 of the 60 commuters, or 40%. Dividing by the 80 women gives 30%, the answer to a different question. Growth over two periods multiplies as well: 15% and then 40% gives 1.15 × 1.4 = 1.61, a 61% rise, not 55%.

Rule: divide a change by its starting value, multiply the rates for a share of a share or for growth in sequence, and divide by the group the question names.

Raw counts favor bigger groups, so compare rates. A town of 48,000 with 120 accidents has 2.5 per 1,000 people, while a town of 15,000 with 45 accidents has 3 per 1,000, so the town with fewer accidents has the higher rate.

A market share is a company's sales divided by the whole market's. If a company's sales hold at 90 while the market grows from 450 to 600, its share falls from 20% to 15%. Unchanged sales can still mean a shrinking share.

A price index sets its base year to 100, so the index divided by 100 is the price multiplier. An index of 112 means prices are 1.12 times the base year's, so a basket that cost $350 then costs $392 now. Reading the index as a 112% increase is the classic error.

Rule: divide by the size of each group before comparing, and turn an index into a multiplier by dividing it by 100.

Many wrong choices are right answers to a neighboring question: the value instead of the label, a part instead of the total, the change instead of the new amount, or the gap between the extremes instead of the gap between the top two. Reread the last sentence before choosing.

The data also sets bounds. A mean lies between the smallest and largest values, a part cannot exceed its total, a pie chart's slices add to 100%, and a combined average lies between the group averages. An answer outside those bounds is wrong, whatever its arithmetic.

Finally, estimate from rounded values. Sales of 37, 19, 23, and 41 are about 40 + 20 + 20 + 40 = 120. A computed total far from the estimate signals a skipped value or a carrying slip, and the estimate alone often rules out most choices.

Rule: before choosing, confirm that the answer matches the question, fits the data's bounds, and lands near a rough estimate.