探索单变量数据

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.

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.

Common mistakes. Naming skew by the peak instead of the tail: a long tail to the left is skewed left, even though most values pile up on the right. Writing “the data are normal” from a histogram: say “roughly symmetric and unimodal”, because real data are only ever approximately normal. Leaving out context: “the median is $7$” earns less than “the median commute is $7$ minutes”.

• 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.

Worked example: standard deviation. For the same data, $\bar x=7$. Work in a table: each deviation from the mean, then its square.

$x_i$	$4$	$6$	$7$	$8$	$10$
$x_i-\bar x$	$-3$	$-1$	$0$	$1$	$3$
$(x_i-\bar x)^2$	$9$	$1$	$0$	$1$	$9$

The squares add to $20$. Divide by $n-1=4$, not by $n$, then take the square root:

Interpretation: a typical value is about $2.24$ away from the mean of $7$. The deviations always add to $0$ (here $-3-1+0+1+3=0$), which is why they are squared first.

Common mistakes. Dividing by $n=5$ instead of $n-1=4$ (that gives $\sqrt4=2$, not $2.24$); stopping at the variance, $5$, and reporting it as the standard deviation; and describing $s_x$ as a distance from the median, or as the largest distance from the mean, instead of a typical distance from the mean.

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.

Common mistakes. Building the fences from the range instead of the IQR; adding $1.5\times\text{IQR}$ to the median instead of to the quartiles; and drawing a whisker out to the fence ($50$) instead of to the largest value that is not an outlier.

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 $3\sigma$.

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.

Worked example: between two values, and backwards. With the same $\mu=500$, $\sigma=100$:

• Proportion between $450$ and $650$: $z=\dfrac{450-500}{100}=-0.5$ and $z=\dfrac{650-500}{100}=1.5$. The area is $P(z<1.5)-P(z<-0.5)=0.9332-0.3085=0.6247$, so about $62.5\%$ of scores.
• The score at the $90$th percentile: the table (or invNorm) gives $z=1.28$ for an area of $0.90$ below. Solve $z=\dfrac{x-\mu}{\sigma}$ for $x$: $x=\mu+z\sigma=500+1.28(100)\approx 628$.

On the exam, show the distribution, the boundary and the area: “Normal, $\mu=500$, $\sigma=100$, $P(450<X<650)\approx 0.6247$”. A bare calculator command does not earn full credit.

Common mistakes. Using the empirical rule for a $z$-score that is not a whole number ($z=1.5$ needs the table or technology); giving the area below when the question asks for the proportion above (subtract from $1$); and using a normal model for a distribution the question describes as strongly skewed.

Answer each before reading the answers below.

1. For the data $3,\ 5,\ 7,\ 8,\ 9,\ 12,\ 15$, find the median and the IQR, and check for outliers with the $1.5\times IQR$ rule.
2. Household incomes in a city are strongly right-skewed. Which is larger, the mean or the median? Which better describes a typical household?
3. A test has mean 70 and standard deviation 8. Find the $z$-score of a student who scored 82.
4. If the scores in question 3 are approximately normal, what proportion of students scored below 58?
5. A student says: "The mean is always the best measure of center." Correct the student.

1. Median $8$. $Q_1=5$ and $Q_3=12$ (medians of the lower and upper halves), so $IQR=7$. The fences are $5-10.5=-5.5$ and $12+10.5=22.5$; no value lies outside them, so there are no outliers.
2. The mean is larger, pulled up by the few very high incomes. The median better describes a typical household, because it resists extreme values.
3. $z=\dfrac{82-70}{8}=1.5$: one and a half standard deviations above the mean.
4. $z=\dfrac{58-70}{8}=-1.5$, so the proportion is about $0.067$, or $6.7\%$.
5. The mean is sensitive to outliers and skew. For skewed data or data with outliers, the median is the better summary of a typical value; report the IQR with it.

• 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.