Representation of data

The diagram that changed history

• In 1854, John Snow mapped cholera deaths in London and spotted a cluster around one water pump. That diagram saved lives.
• Choosing the right statistical diagram is a skill — it reveals patterns that raw numbers hide.

Statistical diagrams

• Choose a diagram to suit the data:
  • Stem-and-leaf 茎叶图 (keeps the values, shows shape),
  • Box-and-whisker 箱线图 (lowest, three quartiles 四分位数, highest),
  • Histogram 直方图 (grouped data — bar area = frequency),
  • Cumulative frequency 累积频数 (running totals → median 中位数 and quartiles).

A cumulative frequency curve: read the median across from half the total, and the quartiles from a quarter and three-quarters

A box plot shows the five-number summary; the box spans the interquartile range 四分位距, IQR = Q₃ − Q₁.

Averages and spread

• Central tendency: mean 平均数 $\bar{x} = \dfrac{\sum x}{n}$, median (middle), mode 众数 (most common).
• Spread: range, interquartile range, and standard deviation 标准差 $\sigma = \sqrt{\dfrac{\sum x^2}{n} - \bar{x}^2}$.
• Variance 方差 = $\sigma^2$.

A Venn diagram: the overlap is the intersection of two events

Worked example — standard deviation

• Data: 2, 4, 4, 4, 5, 5, 7, 9. $n = 8$, $\sum x = 40$, $\sum x^2 = 232$.
• Mean $\bar{x} = \dfrac{40}{8} = 5$.
• Variance $= \dfrac{232}{8} - 5^2 = 29 - 25 = 4$.
• Standard deviation $\sigma = \sqrt{4} = 2$.

Choosing the right average

• Mean: uses all data, but affected by outliers 离群值.
• Median: not affected by outliers — good for skewed data (like salaries).
• Mode: useful for categorical data (most popular colour, most common shoe size).
• Display data with stem-and-leaf diagrams, box-and-whisker plots and cumulative frequency graphs, and measure its variation (spread).

You've got it

Write from a mean of 7

• The values are 4, 4, 8 and 12. A class of width 4 has frequency 12.
• Find the mean, the median and the frequency density.