Spread and grouped data · Datos agrupados y dispersión
| English | Español |
|---|---|
| grouped data/ɡruːpt ˈdeɪtə/ | datos agrupados |
| modal class/ˈməʊdl klæs/ | clase modal |
| quartiles/ˈkwɔːtaɪlz/ | cuartiles |
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 · Laboratorio de medida de dispersión
Classify data summaries by what they tell you about spread. · Clasificar resúmenes de datos según lo que indican sobre la dispersión.
Estimating the grouped mean
- Use the midpoint of each class as the representative value:
The class $10$ to · hasta $20$ has midpoint $\dfrac{10 + 20}{2} = 15$.
- The · El 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? · Un grupo de clase va de 10 a 20. ¿Qué punto medio usas como su valor?
(10 + 20)/2 = 15.
The modal class in grouped data is: · La clase modal en datos agrupados es:
The modal class is simply the group with the most data points. · La clase modal es simplemente el grupo con la mayor cantidad de datos.
A class runs from 30 to 50. What midpoint do you use? · Una clase va de 30 a 50. ¿Qué punto medio usas?
(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 · Diagrama de caja y bigotes
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. · Construir un diagrama de caja a partir del resumen de cinco números: la caja representa el 50% central (rango intercuartílico) y la posición de la mediana indica la asimetría.
A data set has upper quartile 15 and lower quartile 7. Find the interquartile range. · Un conjunto de datos tiene cuartil superior 15 y cuartil inferior 7. Encuentra el rango intercuartílico.
IQR = UQ − LQ = 15 − 7 = 8. · RIC = CU − CI = 15 − 7 = 8.
The interquartile range ignores extreme values, unlike the range. · El rango intercuartílico ignora valores extremos, a diferencia del rango.
The IQR uses only the middle half, so outliers do not affect it. · El RIC usa solo la mitad central, por lo que los valores atípicos no lo afectan.
The IQR measures the spread of the middle ______ of the data. · El RIC mide la dispersión del ______ central de los datos.
The IQR is the range of the middle 50% of values (between LQ and UQ). · El RIC es el rango del 50% central de valores (entre el CI y el CU).
Worked example · Ejemplo resuelto
- 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. · Los puntos medios de clase 5,15,30 tienen frecuencias 4,6,5. Encuentra la media estimada con 1 decimales.
Weighted total = 260 and frequency total = 15, so mean ≈ 17.3. · La suma ponderada es 260 y la suma de frecuencias es 15, por lo tanto media ≈ 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 · y $Q_3=(10+12)/2=11$.
- $\text{IQR}=Q_3-Q_1=11-4.5=6.5$. 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. · Los cuartiles son 4.5 y 11. Encuentra el IQR.
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