Percentages, data, probability and inference break into 18 question types, each with a standard first move. Learn them with worked examples and graphs you can drag, then name the type of fresh practice questions and solve them.
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18
Types
Percentages Which amount is the 100%?
Part, whole & percent
Bands 1 to 2
What is 35% of 240?
Write the percent as a decimal: 35%=0.35
Multiply: 0.35×240=84
Answer84
The gray bar is the whole. The blue bar is the part. Change the percent and the whole.
35% of 240=0.35×240=84. The part is 35% of the whole.
Spot it“what percent of”“% of a number”“how many did NOT”
First movePart ÷ whole = percent. The whole is the number after the word “of.”
Watch outTurn the percent into a decimal before you multiply.
Percent increase & decrease
Band 4
A price changed from $80 to $100. What was the percent change?
Change: 100−80=20
Divide by the original: 8020=0.25
Answer25% increase
Change the percent. The percent change always compares to the original (gray) bar.
80100−80=0.25, a 25% increase. Dividing by the new value instead gives 20%, the common mistake.
Spot it“percent change”rose from … to …“greater percent increase”
First move(new − old) ÷ old × 100.
Watch outDivide by the original value, not the new one.
Successive percent changes
Band 6
A price rises 20%, then the new price falls 25%. What is the net change?
Multipliers: 1.20 and 0.75
Multiply them: 1.20×0.75=0.90
Answer10% decrease
Start at 100. Apply the increase, then the decrease to the new amount.
1.20×0.75=0.9, a net change of −10%. Adding and subtracting the percents would wrongly say −5%.
Spot it“increased … then decreased”“an extra 10% off”net change
First moveMultiply the multipliers (1.20 × 0.75). Don’t add the percents.
Watch outThe second percent applies to the new amount, not the original.
Find the original
Band 6
After a 20% discount a jacket costs $60. What was the original price?
You pay 80% of the original: 0.8x=60
Divide: x=0.860=75
Answer$75
The sale price (blue) is a fraction of the original (gray). Divide to work backward.
0.8×original=60, so original =0.860=75. Adding 20% back to 60 would give 72, which is wrong.
Spot it“after a 20% discount, it costs…”“before the increase”asks for the original
First moveWrite final = factor × original, then divide by the factor.
Watch outDon’t add the percent back onto the final amount.
Percent as the unknown
Bands 5 to 7
If p% of 250 is 40, what is the value of p?
Write the equation: 100p⋅250=40
Solve: p=25040⋅100=16
Answer16
Spot it“what is the value of p”“p% of … is …”the same percent applied twice“interest rate each year”
First moveWrite the situation as an equation with p/100 as a multiplier, then solve for p.
Watch outIf the same percent is applied twice you get p squared, so take the square root at the end.
Describing data Center, spread, tables
Weighted & missing averages
Bands 5 to 7
175 compounds have an average boiling point of 184. 50 of them are type A with an average of 339. What is the average of the type B compounds?
Total of all: 175×184=32,200. Type A: 50×339=16,950
Type B total: 32,200−16,950=15,250, shared among 125 compounds
Answer12515250=122
Spot it“average of the rest”two groups, different averages“what score is needed for a mean of…”“how many are in the group”
First moveTurn every average into a total (average × count). Then add or subtract totals and divide by the right count.
Watch outNever average two averages unless the groups are the same size.
Mean, median & outliers
Bands 3 to 7
The mean of 5 numbers is 12. A sixth number, 18, is added. What is the new mean?
Sum of the five: 5×12=60
New sum: 60+18=78, shared among 6 numbers
Answer678=13
Slide the extra value up. The mean chases it. The median barely moves.
Data: 8,10,11,12,12,13,14,18. Mean =12.3, median =12. An unusually large value pulls the mean toward it; the median only cares about the middle.
Spot it“the mean of n numbers”“median”“which statistic will not change?”
First moveSum = mean × count. Sort the data to find the median.
Watch outAn outlier moves the mean much more than the median.
Spread & standard deviation
Bands 4 to 6Not on PSAT 8/9
Every value in a data set is increased by 10. What happens to the mean and the standard deviation?
Every value moves 10 to the right, so the center moves 10
The values are just as spread out as before
AnswerMean +10, standard deviation unchanged
Slide the spread. The mean stays at 20, but the standard deviation grows as values move away from it.
Mean =20. Standard deviation ≈2.31. Doubling the spread doubles the standard deviation; moving everything together changes only the mean.
Spot it“greater standard deviation”“every value is increased by…”“multiplied by…”
First moveFarther from the mean means larger SD. Shifting moves the center. Scaling changes both.
Watch outAdding the same amount to every value never changes the spread.
Frequency tables & histograms
Bands 1 to 2
The values 1,2,3 appear 2,5,3 times. What is the mean?
Total: 1(2)+2(5)+3(3)=21
Count: 2+5+3=10
Answer1021=2.1
Change how often 2 and 5 appear. The mean moves toward whichever is more frequent.
Total =1(3)+2(3)+3(4)+4(3)+5(3)=48. Count =16. Mean =1648=3.
Spot ita table of values and frequenciesa bar graph of counts“mean / median of the data set”
First moveMean: add value × frequency, divide by the count. Median: add up frequencies to the middle.
Watch outAveraging the listed values ignores how often each appears.
Two-variable data Lines of fit, residuals, tables
Line of best fit
Band 3
The line of best fit is y^=50+6x. Predict y when x=4, and say what 6 means.
y^=50+6(4)=74
6 is the slope: predicted y rises by 6 for each increase of 1 in x
Answer74
Drag the slope and intercept to make your line (blue) match the best-fit line (dashed).
your line: ŷ = 14 + 1xbest fit
Your line predicts 23 at x=9. The slope 1 means predicted y rises 1 per 1 in x. Total squared error: yours 214, best fit 10.
Spot it“line of best fit”“predicted value”“interpret the slope”
First moveSubstitute x into the line. The slope is the change in predicted y per 1 in x.
Watch outThe line predicts. It is not exact for every dot.
Residuals
Band 4
A model predicts y^=20+3x. At x=5 the actual value is 41. Find the residual.
Predicted: 20+3(5)=35
Residual = actual − predicted =41−35
Answer6 (the point is above the line)
Pick a point. The red segment is its residual: actual minus predicted.
At x=3: actual =12, predicted =11. Residual =12−11=1. Above the line, so positive.
Spot it“residual”one highlighted point and a lineactual vs predicted
First moveResidual = actual − predicted.
Watch outA point above the line has a positive residual.
Choose a model from a scatterplot
Band 4
The points rise slowly at first, then faster and faster. Which type of model fits?
A straight line rises at one steady rate, so it doesn't fit
A curve that bends upward and speeds up is exponential growth
AnswerExponential growth
Step through the five shapes. Name the pattern you see before you read the label.
Shape 4: exponential growth. Speeding up (bending upward) is growth.
Spot it“best models the relationship”points that rise, peak, then fallpoints that bend upward
First moveDescribe the shape first, then match it to a function type.
Watch outExponential curves speed up or flatten out. A straight line never does.
Two-way tables
Band 3
Group A: 30 said Yes and 20 said No. What percent of Group A said Yes?
"Of Group A" makes Group A the whole: 30+20=50
5030=0.6
Answer60%
Change the counts. The percent of Group A that said Yes uses only the Group A total.
Group A (blue): 30 Yes, 20 No. Group B (red): 10 Yes, 40 No. Percent of Group A who said Yes =5030=60%. Dividing by the grand total (100) would wrongly give 30%.
Spot ita table with row and column totals“what percent of the ___ group”“in all, how many”
First moveFind the group after “of.” Its total is the denominator.
Watch outDividing by the grand total answers a different question.
Probability Count the right outcomes
Basic probability
Bands 1 to 2
A bag has 3 red and 7 blue marbles. What is the probability of picking a marble that is NOT red?
Total marbles: 3+7=10
Not red means blue: 7 of the 10
Answer107
Change the marble counts. P(not red) is blue over the total.
Total =10. P(not red)=107=0.7. P(red)=0.3, and the two always add to 1.
Spot it“selected at random”“independent events”“NOT”
First moveFavorable ÷ total. “Not” means 1 − P. Independent “and” means multiply.
Watch outAdding probabilities is not “and.”
Conditional probability
Band 6
40 students play a sport, and 15 of them are in Grade 11. One athlete is chosen at random. What is P(Grade 11)?
"Of the athletes" makes the pool 40 students
Grade 11 athletes: 15
Answer4015=83
Change the counts. The condition decides the denominator, so the two conditional probabilities differ.
P(Yes∣Group A)=5030=0.6 (the pool is Group A). P(Group A∣Yes)=4030=0.75 (the pool is everyone who said Yes).
Spot it“given that”“from the students who…”a two-way table
First moveThe condition is the new total. Count the favorable ones inside that group.
Watch outP(A given B) is not the same as P(B given A).
Inference & claims What can you conclude?
Margin of error
Band 5Not on PSAT 8/9
A random sample finds 54% favor a plan, with a margin of error of 4 points. What values are plausible for all voters?
Interval: 54±4
Low: 54−4=50. High: 54+4=58
Answer50% to 58%
Move the estimate and margin. A claim is plausible only if it lands inside the interval.
Interval: 54±4, so 50% to 58%. A claim of 60% is outside the interval, so it is not supported.
Spot it“margin of error”“a random sample”“plausible value”
First moveInterval = estimate ± margin. A bigger random sample shrinks the margin, by the square root.
Watch outThe interval describes the whole population, not the sample.
Sampling & generalizing
Band 7SAT only
A random sample of 100 students is chosen from one high school. The results can be generalized to which group?
Generalize only to where the random sample came from
That is the one school, not all schools
AnswerAll students at that high school
A bigger random sample shrinks the random error. It cannot fix a biased sample.
random error (about 100 ÷ √n)
At n=100 the random error is about 10 points. Quadrupling the sample halves it. A bias of 4 points stays 4 no matter how large the sample gets.
Spot it“can be generalized to”“chosen at random from…”a volunteer sample
First moveGeneralize only to the group the random sample was drawn from.
Watch outA volunteer or convenience sample cannot be generalized.
Studies & causation
No matching band
Volunteers are randomly assigned to two groups, and one group improves more. What can be concluded?
Random assignment allows a cause-and-effect conclusion for the participants
Volunteers were not randomly selected, so the result can't be generalized
AnswerCause and effect for participants only
Spot it“randomly assigned”“volunteers”“which conclusion is supported?”
First moveRandom selection lets you generalize. Random assignment lets you claim cause and effect.
Watch outEach needs its own random step. One does not give you the other.