# Data Visualisation

Mathematics I · Data Analysis and Probability · https://tryals.app/learn/mathematics-i/data-visualisation

## Seeing the Data

A well-chosen graph reveals patterns that a table of numbers hides. **Data visualisation** matches the data type to the right display so shape, trend, relationship, and outliers become visible at a glance.

Here are the core chart types and their purposes:

| Chart Type | Purpose & Structure |
| --- | --- |
| **Histogram** | **Distribution** of a single numerical variable using touching bars. |
| **Bar chart** | Compares values across **categories** using separated bars. |
| **Scatter plot** | **Relationship between two numerical variables** via plotted pairs. |
| **Line chart** | Trends of a value over an ordered sequence, usually **time**. |
| **Box plot** | Summarises a distribution via five numbers to **compare groups**. |

## Chart Distinctions & Pitfalls

The crucial distinction is between a **histogram** and a **bar chart**, which look similar but display different data types.

| Feature | Histogram | Bar Chart |
| --- | --- | --- |
| **Data type** | Continuous numerical | Categorical groups |
| **X-axis** | Number line (binned intervals) | Distinct labels |
| **Bar spacing** | Bars touch | Bars have gaps |

> **Common pitfall:** Using a bar chart for continuous data or a histogram for categories. They answer different questions; mixing them up misrepresents the data.

Good visualisation requires honest design: a truthful axis starting at zero for bar heights, no distorting 3D effects, and a scale that neither exaggerates nor flattens real differences.

## Practice questions

5 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. A bar chart is the wrong display for a continuous variable because its separated bars imply the categories are distinct.

**Answer:** True

**Why:** True. Bar charts encode *categories*, so the gaps assert there is nothing between them, false for a continuous measurement. A histogram's bars touch to show the axis is continuous, and its bar *widths* carry the bin size, which is information a bar chart has no way to express.

Page: https://tryals.app/practice/mathematics-i/data-visualisation/a-bar-chart-is-the-wrong-display-for-a-continuous-variable-because

### 2. Select every design choice that makes a chart misleading.

A. Using unequal bin widths without saying so
B. Truncating the vertical axis so small differences look dramatic
C. Starting a bar chart's value axis at zero
D. Adding 3D effects that distort the apparent size of bars
E. Labelling both axes with their units

**Answer:** A. Using unequal bin widths without saying so; B. Truncating the vertical axis so small differences look dramatic; D. Adding 3D effects that distort the apparent size of bars

**Why:** Truncated axes exaggerate differences, 3D distorts areas, and unequal bins misrepresent frequencies because the eye compares bar _areas_. Clear labelling and a zero baseline are the honest choices, not the deceptive ones.

Page: https://tryals.app/practice/mathematics-i/data-visualisation/select-every-design-choice-that-makes-a-chart-misleading

### 3. Order the steps for constructing a histogram from raw numerical data.

**Answer:**

1. Find the smallest and largest values to fix the range covered
2. Choose a number of bins and give them equal width
3. Assign each data value to the bin whose interval contains it
4. Count how many values fell into each bin
5. Draw a bar of that height over each bin, with the bars touching

**Why:** Bin width is the one free choice, and it matters: too wide hides the shape, too narrow turns the histogram into noise. Everything after that choice is mechanical counting.

Page: https://tryals.app/practice/mathematics-i/data-visualisation/order-the-steps-for-constructing-a-histogram-from-raw-numerical-data

### 4. Sort each feature by whether it describes a histogram or a bar chart.

**Answer:**

- Histogram: Bars touch one another, The horizontal axis is a number line, Shows the distribution of one numerical variable
- Bar chart: Bars are separated by gaps, The horizontal axis carries distinct category labels, Compares one value across unordered groups

**Why:** A histogram bins a continuous variable, so its bins are adjacent intervals and the bars touch. A bar chart compares separate categories with no space between them on any number line, so the bars are drawn apart. The visual difference records a real difference in the data.

Page: https://tryals.app/practice/mathematics-i/data-visualisation/sort-each-feature-by-whether-it-describes-a-histogram-or-a-bar-chart

### 5. The same dataset can produce a histogram with one peak or two, depending only on the bin width. What does that imply about reading a histogram?

A. That bin width is an analytic choice which can create or hide structure
B. That histograms should never be used for continuous data
C. That bimodal data cannot be reliably distinguished from unimodal data
D. That the correct bin width is determined uniquely by the sample size

**Answer:** A. That bin width is an analytic choice which can create or hide structure

**Why:** Wide bins smooth away a genuine second peak and narrow ones manufacture peaks from noise, so the same data supports both pictures. Rules of thumb exist and none is canonical, which is why an honest histogram comes with its bin width stated, and why a suspicious one never does.

Page: https://tryals.app/practice/mathematics-i/data-visualisation/the-same-dataset-can-produce-a-histogram-with-one-peak-or-two