Exploring One-Variable Data

Statistics 统计学 is the science of learning from data 数据 – numbers or labels collected from the real world. Data vary, so we describe patterns and account for the variation 变异 rather than expecting every value to match. A statistical question anticipates an answer based on data that vary.

Two distinctions run through the whole course. A parameter 参数 is a numerical summary of a whole population; a statistic 统计量 is a numerical summary of a sample - we use the statistic to estimate the parameter we cannot measure directly. And descriptive statistics 描述统计 only summarise the data set in hand, while inferential statistics 推断统计 use a sample to make and test claims about the larger population.

A variable 变量 is a characteristic that can differ between individuals. Two kinds:

• Categorical 分类 (qualitative): values are labels/groups (eye colour, brand).
• Quantitative 定量: values are numbers you can do arithmetic on (height, age). Quantitative variables are discrete (countable) or continuous (measured).

Choosing the right graph and summary depends on which kind you have.

A frequency table 频数表 lists each category's count (frequency); a relative frequency 相对频率 table lists each category's proportion 比例 (count ÷ total). Relative frequencies let you compare groups of different sizes fairly.

Bar charts 条形图 show the count or proportion of each category as separated bars; a pie chart shows each category's share of the whole. The bar heights (or slices) let you compare categories at a glance. Bars may be ordered by size or by a natural category order.

For numbers, use a dotplot 点图, stem-and-leaf plot 茎叶图, or histogram 直方图 (bars over value intervals called bins). These show the distribution 分布 – how the values spread out. A histogram's bin width changes the picture, so choose it to reveal the shape.

On a histogram with unequal class widths the bar area is the frequency

Describe four things (remember SOCS):

• Shape 形状: symmetric, or skewed 偏斜 left/right (a long tail on that side), and how many peaks - one main peak is unimodal 单峰, two prominent peaks bimodal 双峰, and roughly equal bars uniform 均匀.
• Outliers 离群值: unusual values far from the rest.
• Center: a typical value (mean or median).
• Spread: how much the values vary (range, IQR, standard deviation).

Always describe shape/center/spread in context, with units.

The shape of a distribution: symmetric, skewed right (long right tail), or skewed left

• Center: the mean 均值 $\bar{x}=\dfrac{\sum x_i}{n}$ (average) and the median 中位数 (middle value). The median resists outliers; the mean is pulled toward a skew.
• Spread: the range, the interquartile range 四分位距 $\text{IQR}=Q_3-Q_1$ (middle 50%), and the standard deviation 标准差 $s_x=\sqrt{\dfrac{\sum(x_i-\bar{x})^2}{n-1}}$ (typical distance from the mean; its square is the variance 方差).
• The five-number summary 五数概括: min, $Q_1$, median, $Q_3$, max.

Use resistant measures (median, IQR) for skewed data; mean and standard deviation for roughly symmetric data.

The percentile 百分位数 of a value is the percent of the data at or below it – so the median is the 50th percentile and $Q_1$ the 25th. A cumulative relative frequency graph 累积相对频率图 makes percentiles easy to read: for each value it plots the proportion of the data at or below it, rising from 0 to 1. Go up from a value to the curve and across to its percentile, or reverse the steps to find the value at a given percentile (the same reading works from a cumulative-frequency table).

Worked example. For the data $4, 8, 6, 10, 7$: the mean is $\bar{x}=\dfrac{4+8+6+10+7}{5}=\dfrac{35}{5}=7$. Sorting to $4,6,7,8,10$, the median is the middle value, $7$. The mean and median agree here because the data are roughly symmetric.

A boxplot 箱线图 draws the five-number summary: a box from $Q_1$ to $Q_3$ with the median inside, and whiskers to the most extreme non-outlier values. A point is an outlier if it lies more than $1.5\times\text{IQR}$ beyond a quartile – a rule you may be asked to apply. Boxplots are ideal for comparing several groups side by side.

Worked example. A dataset has $Q_1=20$ and $Q_3=32$, so $\text{IQR}=12$. The outlier fences are $Q_1-1.5(12)=2$ and $Q_3+1.5(12)=50$. Any value below $2$ or above $50$ is flagged as an outlier.

A box-and-whisker plot shows the quartiles and the range

A boxplot draws the five-number summary; the box spans the IQR

To compare two or more groups, compare shape, center, and spread, and mention outliers – always with comparative words ("Group A has a higher median than Group B") and in context. Do not just describe each group separately; make the comparison explicit.

A normal distribution 正态分布 is a symmetric, bell-shaped model described by its mean $\mu$ and standard deviation $\sigma$. The empirical rule 经验法则 (68–95–99.7): about 68% of values lie within $1\sigma$ of the mean, 95% within $2\sigma$, and 99.7% within $3\sigma$.

The normal curve: a probability is the area under it, centred on the mean

A $z$-score 标准分数 measures how many standard deviations a value is from the mean:

Convert to a $z$-score, then use the normal table or technology to find the proportion (area) below, above, or between values – and reverse the process to find a value from a given percentile.

Worked example. Test scores are normal with $\mu=500$ and $\sigma=100$. A score of $700$ has $z=\dfrac{700-500}{100}=2$. By the empirical rule, $95\%$ of scores lie within $2\sigma$, so $2.5\%$ lie above $700$ – meaning a $700$ is at about the $97.5$th percentile.

The normal curve and the 68-95-99.7 empirical rule

• Describe a distribution by shape, center, spread, and outliers (SOCS) — always in context.
• The mean is pulled by outliers; the median resists them, so prefer the median for skewed data.
• For a normal distribution use the 68–95–99.7 rule and z-scores $z=\tfrac{x-\mu}{\sigma}$.
• Compare distributions with side-by-side boxplots and comment on center, spread, and shape.
• Standard deviation measures a typical distance from the mean; the IQR pairs with the median.