Spread and grouped data
Introduced| English | Meaning |
|---|---|
| grouped data/ɡruːpt ˈdeɪtə/ | Data summarised in groups or class intervals; the grouped table alone does not show each original value. |
| modal class/ˈməʊdl klæs/ | The class interval with the greatest frequency in grouped data. |
| quartiles/ˈkwɔːtaɪlz/ | One of the three boundary values that divide ordered data into four parts of approximately equal size. |
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?
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(10 + 20)/2 = 15.
The modal class in grouped data is:
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The modal class is simply the group with the most data points.
A class runs from 30 to 50. What midpoint do you use?
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(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.
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IQR = UQ − LQ = 15 − 7 = 8.
The interquartile range ignores extreme values, unlike the range.
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The IQR uses only the middle half, so outliers do not affect it.
The IQR measures the spread of the middle ______ of the data.
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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.
Compare spread fairly
- Compare IQRs when you want the spread of the middle half; a smaller IQR means that middle group is more consistent.
- Compare medians as well: equal IQRs do not mean equal typical values.
Calculate a grouped estimate
- Classes $0\le x<10,10\le x<20,20\le x<40$ have frequencies 4,6,5. Midpoints are 5,15,30; weighted total is $T=4(5)+6(15)+5(30)=260$.
- Estimated mean $\bar x\approx260/15=17.3$ to one decimal place. The modal class is $10\le x<20$, with highest frequency 6. The mean is an estimate because the individual values are unknown.
Class midpoints 5,15,30 have frequencies 4,6,5. Find the estimated mean to 1 dp.
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Weighted total = 260 and frequency total = 15, so mean ≈ 17.3.
Quartiles for individual data
- For ordered values $2,4,5,7,8,10,12,14$, median is $(7+8)/2=7.5$. Medians of the lower and upper halves give $Q_1=(4+5)/2=4.5$ and $Q_3=(10+12)/2=11$.
- $\text{IQR}=Q_3-Q_1=11-4.5=6.5$. State the convention used when working from a list; graph questions use the quarter and three-quarter cumulative positions.
Quartiles are 4.5 and 11. Find the IQR.
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IQR = 11 − 4.5 = 6.5.
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