Summary Statistics for a Quantitative Variable
Introduced| English | Meaning |
|---|---|
| mean/miːn/ | The sum of a set of values divided by the number of values. |
| median/ˈmiːdiːən/ | The middle value when data are arranged in order, or the mean of the two middle values for an even number of observations. |
| Range/reɪndʒ/ | Maximum − minimum (simple, but sensitive to extremes). |
| Standard deviation/ˈstændəd ˌdiːvɪˈeɪʃn/ | A measure of spread equal to the square root of the variance. |
| resistant/rɪˈzɪstənt/ | resistant |
Numbers that summarize the data
- A graph shows the shape; summary statistics boil a distribution down to a few key numbers.
- Two jobs: measure the center (a typical value) and the spread (how variable).
- Which numbers you use depends on the shape — especially whether there are outliers.
- These summaries are the backbone of every later inference.
Measures of center
- The mean 均值 $\bar{x}$ is the arithmetic average: add all values, divide by how many.
- The median 中位数 is the middle value when the data are sorted (average the two middle if even).
- For a symmetric distribution they're close; for a skewed one they differ.
- The mean gets pulled toward a long tail; the median stays put.
The five-number summary and the IQR on a box plot
Center and spread of data
The mean and median mark the center; the range, IQR, and standard deviation measure the spread.
Find the median of $4, 5, 6, 7, 100$.
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The middle value of the sorted data is $6$.
In $4,5,6,7,100$, the mean ($24.4$) is dragged upward by the value $100$.
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The mean is sensitive to extreme values.
Measures of spread
- Range 极差 = maximum − minimum (simple, but sensitive to extremes).
- IQR (interquartile range) $=Q_3-Q_1$ — the spread of the middle $50\%$.
- Standard deviation 标准差 $s_x$ — the typical distance of values from the mean.
- Bigger spread numbers mean more variability.
For a data set with $Q_1=5$ and $Q_3=7$, find the IQR.
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$\text{IQR}=Q_3-Q_1=7-5=2$.
Resistant vs. sensitive
- Median and IQR are resistant 稳健 — outliers barely move them.
- Mean and standard deviation are sensitive — one extreme value can shift them a lot.
- So for skewed data or data with outliers, report the median and IQR.
- Outlier rule: a value is an outlier if it's below $Q_1-1.5\times\text{IQR}$ or above $Q_3+1.5\times\text{IQR}$.
Which measures are resistant to outliers?
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Median and IQR resist outliers; mean and SD do not.
By the $1.5\times\text{IQR}$ rule, a high outlier is any value above...
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Measure from the quartile: $Q_3+1.5\,\text{IQR}$.
For strongly skewed data, the best center/spread summary is...
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Resistant measures suit skewed data.
Match the summary to the shape: for skewed data or outliers, use the resistant median and IQR — the mean and standard deviation get dragged by extremes. And the $1.5\times\text{IQR}$ rule measures from the quartiles ($Q_1,Q_3$), not the mean: flag values below $Q_1-1.5\,\text{IQR}$ or above $Q_3+1.5\,\text{IQR}$.
Data: $4, 5, 6, 7, 100$.
- Mean $=\tfrac{4+5+6+7+100}{5}=24.4$ (dragged up by $100$). Median $=6$ (unaffected).
- The median ($6$) better represents the typical value here.
- With $Q_1=5$, $Q_3=7$, $\text{IQR}=2$: $100>7+1.5(2)=10$, so $100$ is an outlier.
Summarize center with the mean $\bar{x}$ or median, and spread with the range, IQR $=Q_3-Q_1$, or standard deviation $s_x$. The median and IQR are resistant to outliers; the mean and SD are not — so prefer them for skewed data. Flag outliers with the $1.5\times\text{IQR}$ rule.
Write from a mean of 5
- The values are 4, 4, 4 and 8.
- The lesson summaries include a mean of 6 and a spread of 2.
What is the mean?
The mean is the total divided by the number of values.
What is the mean of 4, 4, 4 and 8?
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20 divided by 4 is 5. Other options — 6 (the lesson mean); 8 (the largest value).