🎯⭐ INTERACTIVE LESSON

Statistical Questions and Data Display

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Statistical Questions and Data Display - Complete Interactive Lesson

Part 1: What Makes a Question Statistical?

📊 Statistical Questions and Data Display

Part 1 of 5 — What Makes a Question Statistical?

Topics in This Part

Section
Statistical vs. Non-Statistical Questions
Why Variability Is the Key
Spotting the Difference

🔑 Key Concept: A statistical question is one you expect to have a variety of answers. If everyone would give the exact same answer, it is not a statistical question.

Statistical vs. Non-Statistical

A statistical question anticipates variability — the answers are expected to differ. To answer it, you collect data from many people, things, or moments.

A non-statistical question has a single, definite answer. There is nothing to collect — you just look it up or measure once.

Question	Statistical?	Why
"How old am I?"	❌ No	One exact answer
"How old are the students in my class?"	✅ Yes	Ages vary student to student
"How tall is the Eiffel Tower?"	❌ No	A single fixed value
"How tall are the trees in this park?"	✅ Yes	Heights vary tree to tree

🔑 The Test: Ask yourself, "Would I expect different answers?" If yes → statistical. If there is only one answer → not statistical.

Concept Check 🎯

Variability Is the Whole Point

The reason we have a whole subject called statistics is that real-world answers vary. Statistics is the science of collecting, displaying, and making sense of data that has variability.

Example: "How many pets do the families on my street have?"

Some families have $0$ pets, some have $1$ , some have $3$ . The answers spread out — they vary — so we collect them and study the pattern.

💡 A good way to phrase a statistical question is to ask about a group, not one individual: not "How tall am I?" but "How tall are the students in my class?"

See the Variability 🧮

Someone answered the statistical question "How many pets does each family on my street have?" and recorded these $6$ families:

$0,\ 1,\ 3,\ 1,\ 2,\ 0$

1) How many families were surveyed? 2) How many different answers appear in the data? 3) How many families have at least 1 pet?

Sort the Questions 🔽

Label each question as Statistical or Not statistical.

What You Can Do Now

You can tell a statistical question (expects a variety of answers → collect data) from a non-statistical one (single answer).

In Part 2, we start turning collected data into pictures called dot plots, so we can actually see the variability.

Part 2: Dot Plots (Line Plots)

📊 Statistical Questions and Data Display

Part 2 of 5 — Dot Plots (Line Plots)

🔑 The Idea: A dot plot shows every single data value as a dot above a number line. Stacking the dots lets you see which values are common and which are rare.

Building a Dot Plot

Suppose we asked $10$ students, "How many pets do you have?" and got:

$2,\ 0,\ 1,\ 3,\ 2,\ 1,\ 0,\ 2,\ 1,\ 2$

Part 3: Center: Mean, Median, and Mode

📊 Statistical Questions and Data Display

Part 3 of 5 — Center: Mean, Median, and Mode

🔑 The Idea: A whole data set can be summarized by a single measure of center — a typical value. The three most common are the mean, the median, and the mode.

The Three Measures of Center

Measure	What it is	How to find it
Mean	the "balance point" average	add all values, divide by how many
Median	the middle value	sort, then take the middle
Mode	the most frequent value	the value that appears most often

The Mean

$\text{mean} = \frac{\text{sum of all values}}{\text{number of values}}$

Part 4: Spread, Histograms & Box Plots

📊 Statistical Questions and Data Display

Part 4 of 5 — Spread, Histograms & Box Plots

🔑 The Idea: Center tells you the typical value; spread tells you how spread out the data is. We also meet two new displays for larger data sets: the histogram and the box plot.

Range: A Simple Measure of Spread

The range measures how far the data spreads from lowest to highest:

$\text{range} = \text{maximum} - \text{minimum}$

Example: Test scores $72, 95, 88, 60,$ .

Part 5: Mixed Practice & Mastery Check

📊 Statistical Questions and Data Display

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) spot a statistical question, (2) read a dot plot, (3) find the mean, median, and mode, and (4) measure spread with range and read histograms and box plots. Let's put it together.

Quick Reference

Goal	Key move
Statistical question?	Expect a variety of answers → yes
Dot plot	one dot per value, stacked over a number line
Mean	$\dfrac{\text{sum of values}}{\text{number of values}}$

To make a dot plot, draw a number line and place one dot for each value, stacking repeats:

              •
        •     •
  •     •     •
  •     •     •     •
 ─┼─────┼─────┼─────┼─
  0     1     2     3

Now count the dots in each column:

Pets	Number of students (dots)
0	2
1	3
2	4
3	1

✅ Check: $2 + 3 + 4 + 1 = 10$ dots — one for each student. The total number of dots always equals the number of data values.

Read the Dot Plot 🎯

Use the pets dot plot above (0→2 dots, 1→3 dots, 2→4 dots, 3→1 dot).

Count from the Plot 🧮

A class recorded "How many siblings do you have?" The dot plot has:

$0$ siblings → $3$ dots
$1$ sibling → $5$ dots
$2$ siblings → $4$ dots
$3$ siblings → $2$ dots

1) How many students have 1 sibling? 2) How many students have 2 or more siblings? 3) How many students are in the class altogether?

What a Dot Plot Tells You

A dot plot reveals three things at a glance:

Most common value — the tallest stack of dots (this is the mode).
Range of the data — how far the dots spread from lowest to highest.
Gaps and clusters — values where no one landed (a gap) or where dots bunch up (a cluster).

💡 In the pets example, the tallest stack is at $2$ pets, so $2$ is the most common answer. The data runs from $0$ to $3$ .

Interpret the Plot 🔽

Still using the siblings dot plot (0→3, 1→5, 2→4, 3→2).

number of valuessum of all values

Example: Find the mean of $4,\ 8,\ 6,\ 2$ .

$\text{mean} = \frac{4 + 8 + 6 + 2}{4} = \frac{20}{4} = 5$

💡 The mean is like sharing equally: if four people had $4, $8, $6, $2 and pooled it, each would get $5.

Find the Mean 🧮

Add the values, then divide by how many there are.

1) Mean of $5,\ 7,\ 9$ 2) Mean of $10,\ 2,\ 6,\ 6$ 3) Mean of $3,\ 3,\ 4,\ 8,\ 2$

The Median (Middle Value)

To find the median, first put the values in order. Then:

Odd number of values → the median is the single middle value.
Even number of values → the median is the mean of the two middle values.

Example (odd): $7,\ 3,\ 9,\ 1,\ 5$

Sort: $1,\ 3,\ \mathbf{5},\ 7,\ 9$ . The middle value is $\mathbf{5}$ , so the median is $5$ .

Example (even): $2,\ 8,\ 4,\ 6$

Sort: $2,\ \mathbf{4},\ \mathbf{6},\ 8$ . The two middle values are $4$ and $6$ :

$\text{median} = \frac{4 + 6}{2} = 5$

⚠️ Always sort first. Finding the "middle" of an unsorted list gives the wrong answer.

Center Check 🎯

Median and Mode 🧮

1) Median of $12,\ 4,\ 8,\ 10,\ 6$ (sort first!) 2) Median of $1,\ 9,\ 3,\ 7$ (even count — average the two middle) 3) Mode of $2,\ 7,\ 7,\ 4,\ 7,\ 2$

72, 95, 88, 60, 91

$\text{range} = 95 - 60 = 35$

A large range means the values are very spread out; a small range means they are tightly clustered.

💡 Two data sets can have the same mean but very different ranges — spread is its own kind of information.

Histograms (Grouping into Intervals)

When there are many values, we group them into equal intervals (bins) and draw a bar for each. This is a histogram.

Suppose $20$ students' quiz scores are grouped:

Score interval	Number of students
$0$ – $4$	2
$5$ – $9$	7
$10$ – $14$	8
$15$ – $19$	3

The bar height is the count (the frequency) in that interval.

⚠️ A histogram is not a bar graph. In a histogram the bars touch and the horizontal axis shows number intervals (ranges of values), not separate categories.

✅ Check: $2 + 7 + 8 + 3 = 20$ students — the frequencies add up to the total.

Read the Histogram 🎯

Use the quiz-score histogram (0–4 → 2, 5–9 → 7, 10–14 → 8, 15–19 → 3).

Find the Range 🧮

Range $= \text{maximum} - \text{minimum}$ .

1) Range of $14,\ 9,\ 20,\ 11,\ 6$ 2) Range of $30,\ 30,\ 45,\ 22$ 3) A data set has minimum $8$ and maximum $52$ . What is the range?

Box Plots (The Five-Number Summary)

A box plot (box-and-whisker plot) splits ordered data into four equal parts using five numbers:

Marker	Meaning
Minimum	smallest value
Q1 (lower quartile)	median of the lower half
Median (Q2)	middle value
Q3 (upper quartile)	median of the upper half
Maximum	largest value

The box stretches from $Q1$ to $Q3$ , with a line at the median; the whiskers reach out to the min and max.

The width of the box is the interquartile range (IQR):

$\text{IQR} = Q3 - Q1$

💡 A box plot is great for comparing groups, because it shows center (median) and spread (IQR and range) in one compact picture.

Box Plot Parts 🔽

A box plot has: minimum $= 4$ , $Q1 = 7$ , median $= 12$ , $Q3 = 18$ , maximum $= 25$ .

⚠️ Remember: sort before finding the median, and a histogram groups intervals (bars touch) while a bar graph shows separate categories.

Mixed Practice 🎯

Put It Together 🧮

Use the data set $3,\ 7,\ 5,\ 9,\ 1$ .

1) Find the mean. 2) Find the median (sort first!). 3) Find the range.

Exit Quiz ✅

Answer all three to finish the lesson.

Statistical Questions and Data Display

Statistical Questions and Data Display - Complete Interactive Lesson

Part 1: What Makes a Question Statistical?

📊 Statistical Questions and Data Display

Topics in This Part

Statistical vs. Non-Statistical

Variability Is the Whole Point

What You Can Do Now

Part 2: Dot Plots (Line Plots)

📊 Statistical Questions and Data Display

Building a Dot Plot

Part 3: Center: Mean, Median, and Mode

📊 Statistical Questions and Data Display

The Three Measures of Center

The Mean

Part 4: Spread, Histograms & Box Plots

📊 Statistical Questions and Data Display

Range: A Simple Measure of Spread

Part 5: Mixed Practice & Mastery Check

📊 Statistical Questions and Data Display

Quick Reference

What a Dot Plot Tells You

The Median (Middle Value)

Example (odd): 7, 3, 9, 1, 57,\ 3,\ 9,\ 1,\ 57, 3, 9, 1, 5

Example (even): 2, 8, 4, 62,\ 8,\ 4,\ 62, 8, 4, 6

Histograms (Grouping into Intervals)

Box Plots (The Five-Number Summary)

Example (odd): $7,\ 3,\ 9,\ 1,\ 5$

Example (even): $2,\ 8,\ 4,\ 6$