Data Visualization — Histograms, Shape, and Skewness

🎯 Learning Objectives

• Construct and interpret histograms for continuous data
• Create frequency polygons and stem-and-leaf plots
• Identify distribution shape (symmetric, skewed, bimodal, uniform)
• Recognize skewness and understand its implications
• Choose appropriate visualizations based on data type and purpose

📋 Prerequisites

• Data Types & Scales (01-data-types-scales) — histograms are for continuous data
• Central Tendency (03-central-tendency) — understanding how shape affects mean vs median
• Dispersion & Percentiles (04-dispersion-percentiles) — box plots and spread
• Basic algebra and graph reading

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📖 Core Content

6.1 Intuition: A Picture Is Worth a Thousand Numbers

• Where the data is concentrated
• How spread out it is
• Whether it's symmetric or skewed
• Whether there are gaps or clusters
• Whether there are outliers Visualization is the bridge between raw data and insight. Before doing any formal analysis, you should always visualize your data.

Everyday analogy: Imagine describing a city by listing the height of every building (raw data). Now imagine seeing a skyline silhouette (visualization). The silhouette instantly tells you: which part of town has skyscrapers, where it's mostly low-rise, whether there's a clear downtown center. That's what visualization does for data. 🔑 Key Insight: Anscombe's Quartet (four datasets with identical mean, variance, correlation, and regression line) proved that visualizing data is essential — completely different patterns can have identical summary statistics.

6.2 Histograms

6.2.1 Intuition

6.2.2 Construction: Step-by-Step

52, 55, 61, 63, 67, 68, 70, 72, 73, 75, 76, 78, 79, 81, 83, 85, 87, 90, 93, 98 Step 1: Find the range. Min = 52, Max = 98, Range = 46 Step 2: Choose number of bins. A common rule: $\text{Number of bins} \approx \sqrt{n}$. For n = 20, that's about 4-5 bins. For larger datasets, use 5-15 bins. Too few bins hide detail; too many bins create noise. Step 3: Determine bin width. Range / bins ≈ 46 / 5 ≈ 9.2. Let's use width = 10 for simplicity. Step 4: Define bin boundaries.

6.2.3 Key Rules

6.3 Frequency Polygons

1. Create a histogram
2. Mark the midpoint of each bin at the correct height
3. Connect the points with straight lines
4. Close the polygon to the X-axis at both ends Advantages over histograms:

• Easier to compare multiple distributions (overlay lines)
• Shows the shape more cleanly
• Useful for smooth distributions

6.4 Stem-and-Leaf Plots

pseudo
Stem | Leaf
5    | 2 5
6    | 1 3 7 8
7    | 0 2 3 5 6 8 9
8    | 1 3 5 7
9    | 0 3 8

6.5 Distribution Shape

6.5.1 Symmetric Distributions

6.5.2 Skewed Distributions

6.5.3 Other Shapes

6.6 Box Plots and Shape

• Right-skewed: Right whisker longer, median closer to Q1 than Q3
• Left-skewed: Left whisker longer, median closer to Q3 than Q1
• Symmetric: Whiskers roughly equal, median near center of box

6.7 Common Visualization Mistakes

Mistake 1: Misleading Y-Axis

Mistake 2: Inappropriate Bin Width

Mistake 3: Using a Bar Chart for Continuous Data

6.8 Visualizing Categorical Data (Review)

6.9 Worked Examples

Example 1: Creating a Histogram (Easy)

18, 19, 21, 22, 22, 23, 25, 25, 27, 28, 30, 31, 32, 34, 35, 38, 39, 40, 42, 45, 48, 50, 52, 55, 60 Create a histogram with 5 bins. Solution: Step 1: Range = 60 - 18 = 42 Step 2: Bin width ≈ 42/5 ≈ 8.4. Let's use 9 for clean boundaries.

pseudo
18-27: ████████ (8)
27-36: ██████   (6)
36-45: █████    (5)
45-54: ████     (4)
54-63: ██       (2)

Example 2: Comparing Distributions (Medium)

• Class A performed slightly better on average (median 72 vs 70)
• Class A was more consistent (IQR 13 vs 22) — most students scored in a narrow range
• Class B had more variability — some students scored very high (up to 98) and some very low (down to 40)
• The teaching approach might differ: Class A seems uniform, Class B has a mix of strong and weak students

Example 3: Stem-and-Leaf with Skewness (Harder)

pseudo
1  | 5 8
2  | 2 5 8
3  | 0 2 5 8
4  | 0 5
5  | 0 5
6  | 0
7  | 0
8  | 5
9  |
10 |
11 |
12 | 0
...
20 | 0
...
35 | 0

• Right-skewed (positive skew) — clearly visible
• Most values cluster in the lower stems (15-70)
• A long tail stretches to 350
• This is typical income data: most households earn modest amounts, a few earn very high amounts Impact on central tendency: Mean > Median (mean pulled right by the high earners)

6.10 Edge Cases & Gotchas

All Values in One Bin

Gaps in the Data

Outliers in Histograms

6.11 Why This Matters

• EDA (Exploratory Data Analysis): Visualize first, model second
• Communication: A well-designed chart communicates more effectively than a table of numbers
• Quality control: Histograms reveal anomalies in manufacturing processes
• Scientific discovery: Many breakthroughs came from pattern recognition in visualizations (e.g., John Snow's cholera map) This connects to:
• Week 4 (Contingency Tables): Visualizing relationships between variables
• Week 12 (Normal Distribution): The bell curve shape
• BSMA1004 (Stats 2): Residual plots for regression diagnostics

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📐 Key Formulas / Concepts

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⚠️ Common Pitfalls

Pitfall 1: Confusing Histograms and Bar Charts

• Bar chart: Gaps between bars → categorical data
• Histogram: Bars touch → continuous data

Pitfall 2: Misinterpreting Skewness Direction

Pitfall 3: Ignoring the Sample Size

Pitfall 4: Over-interpreting Small Bumps

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📝 Practice Questions

Q1: Histogram Construction

Create a histogram (6 bins) for: 2, 3, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 25

<details> <strong>Solution</strong>

Range = 25 - 2 = 23 Bin width ≈ 23/6 ≈ 3.83 ≈ 4

Wait, let me recount more carefully:

Values sorted: 2,3,5,6,7,8,8,9,10,10,11,12,13,14,15,16,18,20,22,25

Shape: Right-skewed (long tail toward higher values).

</details> > **Q2: Shape Identification** > > Identify the shape of each distribution: > > a) Mean = 50, Median = 50, Mode = 50 b) Mean = 35, Median = 30, Mode = 25 c) Mean = 80, Median = 85, Mode = 90 > > <details> <strong>Solution</strong> > > a) **Symmetric** — all three measures equal → symmetric distribution > > b) **Right-skewed** (positive skew) — Mean > Median > Mode → tail on the right > > c) **Left-skewed** (negative skew) — Mean < Median < Mode → tail on the left > > **Memory check:** The mean is pulled toward the tail. In (b), mean is larger (pulled right), so tail is right. </details> > **Q3: Stem-and-Leaf** > > Create a stem-and-leaf plot for: 12, 14, 18, 21, 23, 25, 28, 30, 33, 35, 37, 40, 42, 48, 51 > > <details> <strong>Solution</strong> > > ```pseudo > Stem | Leaf > 1 | 2 4 8 > 2 | 1 3 5 8 > 3 | 0 3 5 7 > 4 | 0 2 8 > 5 | 1 > ``` > > **Interpretation:** The distribution is roughly symmetric, with a slight concentration in the 20s and 30s. There's one value in the 50s (51) as a mild outlier. </details> > **Q4: Bin Width** > > A dataset has 100 values, range = 50. How many bins would you recommend for a histogram? > > <details> <strong>Solution</strong> > > Using $\sqrt{n}$: $\sqrt{100} = 10$ bins > > Bin width ≈ 50/10 = 5 > > So: 10 bins of width 5 each. > > This is a starting point — you might adjust based on the data's natural groupings. </details> > **Q5: Box Plot from Histogram** > > A histogram shows: left tail short, right tail long, peak on the left. Sketch the approximate box plot. > > <details> <strong>Solution</strong> > > This describes a **right-skewed** distribution. > > Box plot characteristics: > > - Left whisker: short > - Box: median closer to Q1 (left side of box) > - Right whisker: long > > ```pseudo > |----|--|------| > Min Q1 Med Q3 Max > ``` > > The right whisker (Q3 to Max) is longer than the left whisker (Min to Q1). The median is on the left side of the box. </details> > **Q6: Real-World Shape** > > For each variable, predict the shape and explain your reasoning: > > a) Time taken by students to complete an exam b) Number of goals scored per match in football c) Height of all adult humans > > <details> <strong>Solution</strong> > > **a) Exam completion time — Right-skewed** > > - Most students finish within a moderate time (say 45-60 min) > - A few students take much longer (the tail extends right) > - Very few finish extremely fast (hard to finish faster than a minimum time) > > **b) Goals per match — Right-skewed** > > - Most matches have 0-3 goals > - A few high-scoring matches (5+ goals) create a right tail > - Cannot have negative goals > > **c) Adult height — Roughly symmetric (bell-shaped)** > > - Heights cluster around the mean for each gender > - Equal number of people slightly above and slightly below average > - Extreme heights are rare in both directions > - Note: This is actually bimodal if we don't separate genders (men and women have different means) </details> > **Q7: Histogram Comparison** > > Two datasets have the same mean and standard deviation. Dataset A's histogram is bimodal. Dataset B's histogram is bell-shaped. What does this tell you? > > <details> <strong>Solution</strong> > > The identical mean and standard deviation don't reveal that these datasets are fundamentally different: > > **Dataset A (Bimodal):** > > - Likely contains two distinct subgroups (e.g., men and women, or two different treatment groups) > - The mean falls between the two peaks and may not represent either group well > - The standard deviation is inflated because of the gap between groups > > **Dataset B (Bell-shaped):** > > - Likely from a single homogeneous population > - The mean represents a "typical" value well > - The standard deviation reflects natural variation within one group > > **Lesson:** Always visualize! Summary statistics alone can be very misleading. </details> > **Q8: Application** > > A factory monitors the weight of cereal boxes (labeled 500g). The filling machine produces a symmetric distribution with mean 505g and standard deviation 3g. > > a) Sketch the distribution shape. b) What percentage of boxes are underweight (< 500g)? c) If the machine shifts to mean 500g, what changes about the distribution? > > <details> <strong>Solution</strong> > > **a) Shape:** Bell-shaped (normal distribution), centered at 505g, spread of about ±3g. > > **b) Underweight boxes:** > > - 500g is (500 - 505)/3 = -1.67 standard deviations below the mean > - Using the empirical rule: roughly 5% of data falls beyond ±2 s.d. > - More precisely: about 4.75% of boxes weigh less than 500g (this uses the Z-table which we'll learn later) > > **c) If mean shifts to 500g:** > > - The distribution centers exactly at 500g > - Now 50% of boxes are underweight (below 500g) > - This is why factories set the mean slightly above the label weight — to minimize underweight packages </details> * * * ## 🔗 Cross-References - **Next topic:** [Contingency Tables](/notes/01-foundation-bsma1002-stats-1-week04-06-contingency-tables) — visualizing relationships between two categorical variables - **Previous:** [Dispersion & Percentiles](/notes/01-foundation-bsma1002-stats-1-week03-04-dispersion-percentiles) — box plots as visualization of spread - **Week 12 (Normal Distribution):** The bell curve and the empirical rule - **BSMA1004 (Stats 2):** Residual plots for model diagnostics - **BSCS2004 (ML Foundations):** EDA is the first step in any ML project [Join Discord](https://discord.gg/gE2m4Qrdqv) [Previous**Dispersion & Percentiles**](/notes/01-foundation-bsma1002-stats-1-week03-04-dispersion-percentiles)[Next**Contingency Tables**](/notes/01-foundation-bsma1002-stats-1-week04-06-contingency-tables)