Statistical Questions and Data Display

📊 Statistical Questions and Data Display

Part 1 of 5 — What Makes a Question Statistical?

Topics in This Part

🔑 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.

🔑 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.

📊 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:

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

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:

✅ 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).

📊 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

The Mean

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

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 $2,\ 7,\ 7,\ 4,\ 7,\ 2$

📊 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 $72,\ 95,\ 88,\ 60,\ 91$.

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

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:

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:

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 $= 25$.

📊 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

⚠️ 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.