Cumulative frequency diagrams
| English | Chinese | Pinyin |
|---|---|---|
| cumulative frequency | 累积频数 | lěi jī pín shuò |
| median | 中位数 | zhōng wèi shù |
| quartile | 四分位数 | sì fēn wèi shù |
The running total
- In a marathon, you don't just count runners per age group — you want to know "how many runners have finished by time $t$?"
- That's a cumulative frequency 累积频数: a running total that builds up from the lowest class to the highest.
Cumulative frequency route
Follow raw data into a cumulative frequency curve and percentile reading.
Drawing the curve
- Plot cumulative frequency against the upper end of each class.
- Join the points with a smooth curve (or straight lines).
- The curve always goes up (or stays flat) — it never goes down, because you're adding.
Classes $0$–$10$ (freq $5$), $10$–$20$ (freq $8$), $20$–$30$ (freq $12$): cumulative frequencies are $5, 13, 25$. Plot at $10, 20, 30$.
On a histogram with unequal class widths the bar AREA (not height) is the frequency, so the axis is frequency density
Cumulative frequency is:
Each point adds the next class frequency to the total so far.
Cumulative frequency is plotted against the upper end of each class.
Always plot the running total at the top boundary of the class.
A cumulative frequency curve can go down.
Cumulative frequency is a running total — it always increases or stays flat, never decreases.
Reading the curve
- Median 中位数: go across at half the total cumulative frequency, then down to the $x$-axis.
- Lower quartile 四分位数 (LQ): at $\dfrac{1}{4}$ of the total.
- Upper quartile (UQ): at $\dfrac{3}{4}$ of the total.
- IQR $=$ UQ $-$ LQ.
Across then down. To read a value from a cumulative frequency curve: go across from the $y$-axis to the curve, then down to the $x$-axis. Going the wrong way gives the wrong answer.
Read the median at half the total and the quartiles at one quarter and three quarters: go across to the curve, then down
A cumulative frequency curve has total 80. At what cumulative frequency do you read the median?
The median is at half the total: 80 ÷ 2 = 40.
Total cumulative frequency is 120. At what cumulative frequency do you read the upper quartile?
UQ is at 3/4 of the total: (3/4) × 120 = 90.
To read the median from a cumulative frequency curve, go ______ from the y-axis to the curve, then down to the x-axis.
Go across (horizontally) from the y-value to the curve, then down to read the x-value.
Worked example
- Total cumulative frequency $= 80$. Median at $80 \div 2 = 40$. Read across at $40$ on the $y$-axis, then down — suppose it reads $24$. The median is $24$.
A cumulative frequency curve always rises — the shape is like an elongated S, steep where data is concentrated.
You've got it
- cumulative frequency $=$ running total, plotted at each class's upper end
- read the median at half the total; quartiles at $\dfrac{1}{4}$ and $\dfrac{3}{4}$
- across to the curve, then straight down to the value