Reading the Numbers: Be a Data Detective
The Lincoln Middle School survey just came back, and the numbers are a mess. Your job is to turn a pile of raw data into a clear story — using center, spread, and the shape of the data. Tú eres detective de datos: conviertes números en una historia clara.
1 Engage
Hook & essential question
A reporter for the school news asked two questions in the hallway. Read both carefully:
Question A
"How tall is the tallest 6th grader?"
You get one single answer.
Question B
"How tall are the 6th graders at our school?"
You get many different answers.
Only one of these is a statistical question. A statistical question expects variability — the answers differ from person to person, so you need a whole data set to answer it. Una pregunta estadística espera variabilidad: las respuestas son diferentes.
Quick think (no calculator): Which one is the statistical question — A or B? In your notebook, write your choice and one sentence explaining why. You will check your thinking in the Apply section.
Essential question: When I have a whole set of numbers, how do I describe the "typical" value, how spread out the data is, and what the data is shaped like?
2 Explore
Investigate before the lesson
Explore each tool below for a few minutes. As you go, hunt for one big idea: a single data set has a center (what's typical), a spread (how much it varies), and a shape (how it piles up).
The survey data: minutes of homework last night
Ten 6th graders answered "How many minutes did you spend on homework last night?" Here is the raw data set you will use all the way through this HyperDoc:
20 · 25 · 30 · 30 · 35 · 40 · 40 · 45 · 50 · 85
Below is the dot plot of that same data. Each dot is one student. Drag your eyes across it — where do the dots cluster, and is there a dot sitting all alone?
Most dots pile up between 20 and 50. One dot sits far away at 85.
3 Explain
The math, in plain language
Every data set can be described three ways: its center (the typical value), its spread (how far apart the values are), and its shape (how the values pile up). Here are the tools for this unit.
- statistical question
- a question whose answers vary · pregunta estadística
- mean
- the balance point: add all, divide by how many · la media / el promedio
- median
- the middle value when sorted · la mediana
- mode
- the value that appears most · la moda
- range
- biggest − smallest · el rango
- outlier
- a value far from the rest · valor atípico
- MAD
- mean absolute deviation — typical distance from the mean · desviación media absoluta
Center: three ways to say "typical"
Using our sorted data: 20 25 30 30 35 40 40 45 50 85
Mean (average)
mean = sum ÷ count
20+25+30+30+35+40+40+45+50+85 = 400.
400 ÷ 10 = 40 minutes.
Median (middle)
middle of sorted list
With 10 values, average the 5th and 6th: (35 + 40) ÷ 2 = 37.5 minutes.
Mode (most common)
value seen most
Both 30 and 40 appear twice, so this set has two modes.
Mean vs. median
The 85 (outlier) pulls the mean up to 40, but the median stays at 37.5. With an outlier, the median describes "typical" better.
Spread: range and MAD
Range = 85 − 20 = 65 minutes. Range is quick, but one outlier makes it huge. MAD is a fairer measure of spread: it asks "on average, how far is each value from the mean?"
Worked example — MAD of a smaller set
To see MAD clearly, use five quiz scores: 6, 8, 10, 12, 14. The mean is (6+8+10+12+14) ÷ 5 = 10. Now measure each distance from 10:
| Value | 6 | 8 | 10 | 12 | 14 |
|---|---|---|---|---|---|
| Distance from mean (10) | 4 | 2 | 0 | 2 | 4 |
| MAD = (4+2+0+2+4) ÷ 5 | = 12 ÷ 5 = 2.4 | ||||
A MAD of 2.4 means scores are typically about 2.4 points away from the average. MAD = (sum of distances) ÷ count
Shape: histograms & box plots
A histogram groups data into equal intervals (bins) and shows how many fall in each bin. A box plot shows the five-number summary: minimum, Q1, median, Q3, and maximum — and the box holds the middle half of the data.
Histogram of the homework data
Tall on the left, one short bar far right = skewed right.
Box plot of the homework data
The long right whisker is the outlier (85) stretching the data.
Reading shape: when one tail stretches far to the right (like ours), we say the data is skewed right. When the data is roughly even on both sides of the center, it is symmetric.
Level 1 · support Step-by-step helper / Ayuda paso a paso
- Sort first. Always put numbers smallest → largest before finding median, range, Q1, or Q3.
- Mean: add every value, then divide by how many there are (count the values!).
- Median: cross off one from each end until you reach the middle. If two are left, average them.
- MAD: (1) find the mean, (2) find each distance from the mean (always positive), (3) average those distances. La distancia siempre es positiva.
Level 2 · enrichment Push your thinking
Our data is skewed right because of the 85. Predict: if that student had answered 40 instead of 85, which would change more — the mean or the median? Recalculate both to check, then write a rule about which measure of center is more resistant to outliers and why.
4 Apply
Show what you can do — auto-checked
Level 1 · support Sentence starters for explaining
"I sorted the data first. To find the ____ I ____. The typical value is about ____ minutes. The data is shaped ____ because ____."
"Ordené los datos primero. Para hallar el/la ____ yo ____. El valor típico es de unos ____ minutos. Los datos tienen forma ____ porque ____."
Level 2 · enrichment Build a five-number summary
For the homework data, list all five numbers of the box plot: minimum, Q1, median, Q3, and maximum. Then explain why the IQR (the width of the box) is a "fairer" spread than the range for this data set, which contains the outlier 85.
Teacher Notes & Answer Key (not printed)
Stage-by-Stage Notes (Engage → Explore → Explain → Apply → Reflect)
- Engage: sort questions into statistical vs not (do answers vary?).
- Explore: compute center and spread on a small data set.
- Explain: mean, median, mode, range, MAD, and reading box plots/histograms.
- Apply: answer key below.
- Reflect: students explain how an outlier pulls the mean.
Apply — Answer Key
- Q1 — Statistical question (MC): answer b.
- Q2 — Mean: 40.
- Q3 — Median: 37.5.
- Q4 — Range: 65.
- Q5 — MAD: 2.
- Q6 — Outlier effect (MC): answer c.
- Q7 — Shape (MC): answer b.
- Q8 — IQR (Q3 − Q1 = 45 − 30): 15.
Standard
CCSS 6.SP.A.1–B.5.
5 Reflect
Think about your thinking
A. In this data set the mean (40) and the median (37.5) were different. Which one would you report to describe a "typical" student, and why?
B. Mean, MAD, and shape each tell you something different about a data set. In your own words, what does each one tell you that the others do not?
C. One thing about reading data that is now clear to me / Una cosa sobre los datos que ahora entiendo bien:
Your reflections save with this HyperDoc when you use Save as PDF or Save as DOC above.
6 Extend
Optional challenge project
Run your own data investigation
Pick a statistical question you actually care about (for example, "How many hours of screen time do students get on a weekend?"). Then:
- Survey at least 10 people and record the raw data.
- Find the mean, median, mode, range, and MAD. Mark any outliers.
- Draw a dot plot and a histogram of your data, and describe its shape (symmetric or skewed).
- Write one paragraph: what is the "typical" answer, how spread out is it, and which measure of center is the most honest for your data?
Haz tu propia encuesta de 10 personas, calcula la media, la mediana y la MAD, dibuja un histograma y describe la forma de los datos.