Read the axes first
Before looking at any bar or point, answer three questions about the picture: what does one gridline step represent, what are the units, and does the vertical axis start at zero?
A vertical axis labelled “Revenue (millions of dollars)” with gridlines every $5$ means a bar reaching two gridlines above $20$ is $\$30$ million — not $30$, not $\$30$. Answers on the SAT are given in the units named on the axis or in the question, and those two are not always the same.
Lỗi mất điểm rẻ nhất của cả nhóm này không nằm ở phép tính mà ở đơn vị. Trục ghi “thousands of units” mà câu hỏi hỏi “how many units” — đáp án phải nhân thêm $1000$, và phương án chưa nhân bao giờ cũng có mặt. Trước khi đọc bất kỳ cột nào, hãy đọc to nhãn trục trong đầu kèm đơn vị.
Bar charts
The chart below shows monthly rainfall recorded at a weather station.
Between which two consecutive months was the increase in rainfall greatest?
Month-to-month changes: $+20$, $+20$, $+40$, $-30$, $-30$. The greatest increase is from March to April, an increase of $\boxed{40\text{ mm}}$.
Note that April is also the tallest bar, but those are two different facts: the tallest bar answers “which month had the most rain”, not “where did rainfall rise the most”.
April's rainfall is what percent greater than February's?
$\dfrac{120-60}{60}=\dfrac{60}{60}=1=\boxed{100\%}$ greater.
April's rainfall is $200\%$ of February's, which is the same fact stated with a different base. Only one of those two numbers answers the question as asked.
“$A$ is what percent greater than $B$” dùng mẫu số $B$ và tử số là hiệu $A-B$. “$A$ is what percent of $B$” dùng mẫu số $B$ và tử số là $A$. Hai con số luôn lệch nhau đúng $100$, và cả hai luôn có mặt trong bốn phương án. Gạch chân chữ “greater than” hay “of” ngay khi đọc đề.
Line graphs
The graph below shows annual revenue for two companies.
- Value: read the height at a single $x$.
- Change: subtract two heights. Units are the $y$-units.
- Average rate of change: divide the change by the change in $x$. Units are $y$-units per $x$-unit, and this is the slope of the straight line joining the two points — not the slope of any one segment between them.
In which year did Company P's revenue first exceed Company Q's?
Compare year by year: $2018$ ($12<30$), $2019$ ($18<27$), $2020$ ($22<25$), $2021$ ($28>26$). The first year in which P is above Q is $\boxed{2021}$.
What was Company Q's average rate of change in revenue from $2018$ to $2023$?
$\dfrac{39-30}{2023-2018}=\dfrac{9}{5}=\boxed{1.8}$ million dollars per year.
Q's revenue fell for three years and then rose sharply; the average rate of change ignores all of that and reports only the net effect over the five-year span.
Chữ “average rate of change” làm nhiều học sinh đi tìm trung bình cộng của các giá trị. Nó không phải vậy: nó là $\dfrac{\text{giá trị cuối} - \text{giá trị đầu}}{\text{số bước theo trục ngang}}$ — đúng bằng hệ số góc của đoạn thẳng nối hai điểm đầu và cuối. Đường đi ở giữa gấp khúc thế nào cũng không ảnh hưởng.
Histograms
A histogram groups values into intervals (bins) and shows how many values fall in each. The histogram below records the number of books read last month by each of $60$ students.
Recoverable: the total count; the count in any bin or union of bins; the bin that contains the median; upper and lower bounds for the mean, the minimum and the maximum.
Lost: every individual value. You cannot say how many students read exactly $7$ books, nor compute the exact mean, nor name the maximum.
A second school ran the same survey on $50$ students and obtained the bin counts $4$, $8$, $10$, $18$, $10$ for the same five intervals. Which interval contains the median?
Locating a median in a histogram is a cumulative-count exercise, never a tallest-bar exercise. With $50$ values the median is the average of the $25$th and $26$th values in order. Cumulative counts: $4$, then $4+8=12$, then $12+10=22$, then $22+18=40$. Positions $25$ and $26$ are both past $22$ and not past $40$, so they land in the fourth bin and the median lies in $\boxed{9\text{--}11}$.
The tallest bar is also the fourth here, but that agreement is a coincidence: the tallest bar and the median bin are different questions and often give different answers.
What is the smallest possible value of the mean number of books read?
Push every value to the bottom of its bin: \[ \frac{8(0)+14(3)+20(6)+12(9)+6(12)}{60}=\frac{0+42+120+108+72}{60}=\frac{342}{60}=\boxed{5.7}. \] No arrangement consistent with the histogram can produce a smaller mean.
Histogram là chỗ đề hỏi “cái gì không xác định được” nhiều nhất. Nguyên tắc: cột chỉ nói bao nhiêu giá trị nằm trong khoảng, không nói chúng là số nào. Vì vậy đếm được, chặn được, xác định được khoảng chứa trung vị — nhưng không tính được trung bình chính xác, không biết giá trị lớn nhất, không biết có bao nhiêu người ứng với một con số cụ thể bên trong khoảng.
Exact statistics from an ungrouped frequency table
When the table lists each value individually rather than in intervals, nothing is lost and the mean and median are exact.
The table records the number of pets owned by each of $25$ people.
| Number of pets | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Number of people | 7 | 9 | 5 | 3 | 1 |
Mean: multiply each value by its frequency and divide by the total count: \[ \frac{0(7)+1(9)+2(5)+3(3)+4(1)}{25}=\frac{0+9+10+9+4}{25}=\frac{32}{25}=\boxed{1.28}. \] Median: with $25$ values the median is the $13$th. Cumulative counts are $7$, then $7+9=16$, so the $13$th value is a $1$: the median is $\boxed{1}$.
Dividing $32$ by $5$ — the number of columns — gives $6.4$, a value larger than every entry in the table, which is the signal that the wrong denominator was used.
A chart of rates is not a chart of counts
When the vertical axis carries percentages rather than counts, no question about how many individuals are involved can be answered without the size of each group.
A bar chart shows the percentage of students who passed an examination at each of four schools. School A's bar reads $80\%$ and School B's reads $60\%$. Does it follow that more students passed at School A than at School B?
No. The chart reports rates, not counts. If School A enrols $50$ students and School B enrols $500$, then $40$ students passed at School A and $300$ at School B. Without the enrolment figures the comparison of counts cannot be made.
Cột cao hơn không đồng nghĩa với “nhiều người hơn” nếu trục đứng là phần trăm. Đọc nhãn trục: “Number of students” là đếm, “Percent of students” là tỉ lệ. Với biểu đồ tỉ lệ, mọi câu hỏi về số lượng đều cần thêm cỡ của từng nhóm — không có cỡ nhóm thì đáp án đúng là “không xác định được”.
Practice
Questions 1, 2, 4 and 18 refer to the rainfall bar chart in this unit; questions 3, 5, 6, 14 and 15 refer to the revenue line graph; questions 7, 8, 11, 12 and 13 refer to the books histogram.
Questions 9, 10 and 17 refer to the table below, which records the number of siblings of each of $40$ students.
| Number of siblings | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Number of students | 9 | 14 | 11 | 4 | 2 |