Spread and grouped data
| English | Chinese | Pinyin |
|---|---|---|
| grouped data | 分组数据 | fēn zǔ shù jù |
| modal class | 众数组 | zhòng shù zǔ |
| quartiles | 四分位数 | sì fēn wèi shù |
| box plot | 箱线图 | xiāng xiàn tú |
When data comes in groups
- A survey records the heights of 200 students. Instead of listing all 200 values, the data is grouped: $140$–$150$ cm ($15$ students), $150$–$160$ cm ($45$ students), and so on.
- Grouped data 分组数据 loses the individual values — so you can only estimate the mean.
Spread measure lab
Classify data summaries by what they tell you about spread.
Estimating the grouped mean
- Use the midpoint of each class as the representative value:
The class $10$ to $20$ has midpoint $\dfrac{10 + 20}{2} = 15$.
- The modal class 众数组 is the group with the highest frequency.
A bar chart shows the frequency of each group
A grouped class runs from 10 to 20. What midpoint do you use as its value?
(10 + 20)/2 = 15.
The modal class in grouped data is:
The modal class is simply the group with the most data points.
A class runs from 30 to 50. What midpoint do you use?
(30 + 50)/2 = 40.
Quartiles 四分位数 and the IQR
- Put the data in order. The quartiles cut it into four equal parts:
- Lower quartile (LQ) at the $\dfrac{1}{4}$ position.
- Upper quartile (UQ) at the $\dfrac{3}{4}$ position.
IQR, not range. The IQR measures the spread of the middle half of the data, ignoring the extremes. It's more reliable than the range when outliers are present.
Box-and-whisker plot
Build a box plot from the five-number summary: the box is the middle 50% (the interquartile range) and the median's position shows the skew.
A data set has upper quartile 15 and lower quartile 7. Find the interquartile range.
IQR = UQ − LQ = 15 − 7 = 8.
The interquartile range ignores extreme values, unlike the range.
The IQR uses only the middle half, so outliers do not affect it.
The IQR measures the spread of the middle ______ of the data.
The IQR is the range of the middle 50% of values (between LQ and UQ).
Worked example
- UQ $= 15$, LQ $= 7$: $\text{IQR} = 15 - 7 = 8$.
- This means the middle $50\%$ of the data spans a range of $8$ units.
Box plots 箱线图
- A box plot shows the five-number summary: minimum, LQ, median, UQ, maximum.
- The box spans the IQR; the whiskers reach the extremes.
Quartiles cut ordered data into quarters — like marks on a number line dividing the range into four equal-frequency parts.
You've got it
- estimate a grouped mean using each class midpoint; modal class $=$ highest frequency
- IQR $=$ UQ $-$ LQ (spread of the middle half)
- the IQR ignores extreme values, unlike the range