Notes on Qualitative & Quantitative Data, Frequency Distributions, and Graphical Summaries

Qualitative and Quantitative Data, Frequency Distributions, and Graphical Summaries

Overview and readiness

Course intro focus: sampling, bias, and evaluating sample quality; distinguishing good vs bad samples and how to alleviate bias or randomness.
Data types preview: qualitative (categorical) vs quantitative (numerical) data; graphical summaries used for each in chapter two.
Preparations for activities: have calculators, pen/paper ready; from chapter three, have the formula sheet available.
Qualitative data definition elaborated:
- Qualitative data are categories or labels that describe characteristics of individuals.
- Usually words, but can be numbers if the numbers are categorical (i.e., cannot be meaningfully added or averaged).
- Example given: jersey numbers (numeric but not meaningful to add/subtract for categorization).

Qualitative data example setup

Dataset: 50 customers of a retailer; analyzing credit cards used (Discover, Visa, American Express, Mastercard).
Data structure: 50 values (one per customer) summarized by category.
Purpose of summarization: to understand the distribution of categories, not to compute totals across numeric values.

Frequency distribution (qualitative data)

Definition: frequency of a category = number of times that category occurs in the dataset.
Example (credit cards): counts for each category; Mastercard frequency = 11 (corrected from 10 after counting all cases).
Completed counts (example): Mastercard = 11; Visa = 23; American Express = 9; Discover = 7.
Important check: sum of frequencies should equal the total sample size N (here, N = 50). If you miss a category, the sum will be off.
Frequency distribution table structure:
- Columns: Category | Frequency
- Example snippet: Mastercard | 11; Visa | 23; American Express | 9; Discover | 7

Relative frequency (qualitative data)

Definition: relative frequency = frequency of a category divided by the total number of observations.
Calculation: for each category, $f_{rel} = rac{f}{N}$
In the example: total N = 50; relative frequencies are
- Mastercard: $rac{11}{50} = 0.22$
- Visa: $rac{23}{50} = 0.46$
- American Express: $rac{9}{50} = 0.18$
- Discover: $rac{7}{50} = 0.14$
Properties: the sum of all relative frequencies equals 1 (or 100% when expressed as percentages).
- Sum check: $0.22 + 0.46 + 0.18 + 0.14 = 1.00$
Rounding: when converting to percentages or rounding decimals, you may introduce rounding error; the numbers should still sum close to 1 (or 100%).
Percentage form:
- Mastercard: 22%; Visa: 46%; American Express: 18%; Discover: 14% (based on the same fractions).

Graphs for qualitative data

Bar graphs (qualitative):
- Y-axis represents frequency (counts) or relative frequency (proportions).
- Bars have equal width, are separated, and aligned with each category on the x-axis.
- Primary use: visualize the frequency distribution.
Pareto chart (a special bar graph):
- Bar order is descending by frequency (highest to lowest).
- Useful for identifying the most important categories quickly.
- Example interpretation: shows which credit card types occur most often (Visa highest, followed by Mastercard, American Express, Discover).
Horizontal bar graph: useful when category names are long; bars extend horizontally to accommodate long labels.
Side-by-side bar graph: compare two datasets side by side (e.g., year-over-year data, e.g., 2022 vs 2023).
- Key use: visually assess changes between two time points or groups.
Qualitative data recap on graphs:
- Bar graphs visualize frequencies or relative frequencies.
- Pie charts provide a proportional visualization of qualitative categories.

Pie charts (qualitative data)

Definition: circle divided into slices; each slice represents a category.
Slice size corresponds to the category’s proportion (relative frequency).
Example setup (credit cards): slices for Discover, Visa, American Express, Mastercard with given relative shares (e.g., 0.22, 0.46, 0.18, 0.14).
Necessary components for usefulness:
- Percentages or proportions for each slice.
- Labels identifying the category.
- A legend or direct labeling that matches the category to the color/slice.
Important note: slices should be proportional to the percentages; mislabeling or mismatched values reduces interpretability.

Qualitative data vs quantitative data (summary)

Qualitative data focus on categories and proportions; often summarized with frequencies, relative frequencies, and visualizations such as bar charts and pie charts.
Quantitative data focus on numerical values and summaries such as mean, median, standard deviation, and graphical summaries like histograms.
Qualitative charts emphasize percentages; quantitative charts provide more numerical summaries and properties.

Bar graphs, horizontal bars, and side-by-side examples (qualitative) – a practical set

Bars illustrate category frequencies side by side with other bars when ready for comparison (e.g., platform usage across years).
Side-by-side examples show a direct comparison across two datasets (e.g., 2022 vs 2023) to see where increases or decreases occur (e.g., Facebook, YouTube, WhatsApp, Instagram, WeChat).

Qualitative data examples (bar, Pareto, and pie)

Example: credit card data summary (categories: Discover, Visa, American Express, Mastercard) with frequencies and relative frequencies as above.
Pie chart example: share of responses in a poll showing Better, Same, Worse, No opinion; percentages given: Better 39%, Same 19%, Worse 38%, No opinion 14%.
Pie chart interpretation rules:
- The largest slice indicates the most common category (Better in this example).
- If two categories share a category with No opinion, sum corresponding percentages for combined categories (e.g., Same + Worse).
- The pie chart should include both category labels and their percentage values for clarity.

Bar charts for quantitative data (intro to quantitative graphs)

For quantitative data, a closely related graph is the histogram, where bars touch (no gaps) to represent continuous data in intervals (classes).
When using bar graphs for quantitative data, the y-axis can represent frequencies or relative frequencies, similar to qualitative charts, but the data are numerical.

Quantitative data: frequency distributions and classes

Purpose: to summarize numerical data by grouping values into classes (intervals) rather than individual values.
Why classes: raw quantitative data can span large ranges; using classes reduces rows and organizes data for interpretation.
Key terms:
- Lower class limit: the smallest value that can appear in a class.
- Upper class limit: the largest value that can appear in a class.
- Class width: the difference between consecutive lower class limits.
Example of class width: For classes 0–4 and 5–9, the width is 5 (consecutive lower limits 0 and 5).
Class width determination methods:
1) Fixed number of classes method: decide the number of classes k, determine the starting lower limit, then compute a uniform class width.
2) Width-determined method: choose a class width w, determine the first lower limit, and then add w to form subsequent classes until the largest data value is included.
Important guidance: with smaller data sets, use fewer classes to avoid over-fragmentation; too many classes can dilute information.
General formula (to ensure coverage):
- Class width: $w = \left\lceil \frac{\max - \min}{k} \right\rceil$
- Where k is the desired number of classes and (\lceil \cdot \rceil) is the ceiling function (round up).
Step-by-step approach when building classes (two equivalent methods):
- Method A (choose k):
- Select the smallest value as a starting lower limit (or something just below it).
- Decide the number of classes k.
- Compute width w using the formula above.
- Create classes by adding w to the previous lower limit until all data are covered.
- Method B (choose width):
- Choose a class width w.
- Select a starting lower limit for the first class.
- Create subsequent classes by adding w to the previous lower limit.
- Stop when the largest data value is included in the last class.
After classes are formed, count the frequency of observations in each class, then compute relative frequencies if desired:
- Total number of data points: $N = \sum f_i$
- Relative frequency for a class: $f{rel,i} = \frac{fi}{N}$
Quantitative example (decimal data): emissions data with 65 observations; choose class width of 1; smallest value = 0.25; largest value = 6.64; first class lower limit chosen as 0 to keep width simple; class boundaries extend to include 6.64 (e.g., 0–0.99, 1–1.99, …, 6–6.99).
- Number of classes with width 1 that cover 0.25 to 6.64 is 7.
- Using the width-based calculation to verify:
- Largest value − smallest value = 6.64 − 0.25 = 6.39
- If k = 7, then $w = \frac{6.39}{7} \approx 0.913$ , ceiling to 1, confirming width 1 works.
Frequencies and relative frequencies example (65 data points):
- Frequencies per class are counted from the data.
- Relative frequencies: $f{rel,i} = \frac{fi}{65}$
- Note: with decimals, relative frequencies may have several decimal places; rounding to 3 decimals is common in practice.
Open-ended class: example in histograms where the last class is open-ended (e.g., 85 and older) to accommodate potentially very large values while acknowledging the tail of the distribution.
Discrete vs continuous data in histograms:
- Histograms are used for quantitative data and are typically continuous (bars touch).
- Discrete data (e.g., number of children) may be shown with bars but often do not require class intervals if the data are counts of distinct integers.

Histograms and distribution shapes (quantitative data)

Histogram basics:
- Bars are drawn for each class with width equal to the class width.
- The number of bars equals the number of classes.
- For continuous data, bars touch to indicate continuity between classes.
Class width and number of bars trade-off:
- Too few classes = less detail; too many classes = overly granular and hard to read.
Open-ended last class example: age or other data with an upper tail extended into an open interval (e.g., 85 and older).
Discrete vs continuous distinction reiterated:
- Discrete example: number of children is a discrete count; may not be grouped into overlapping continuous classes.
- Continuous example: emissions or ages, where class intervals are natural and meaningful.

Distribution shapes and modal properties

Skewness concepts (two main directions):
- Left-skewed (tail to the left): most data on the right, with a tail extending to smaller values on the left.
- Right-skewed (tail to the right): most data on the left, with a tail extending to larger values on the right.
- Visualization aid: imagine a sled going down a hill to the right (right-skew) or up a hill to the left (left-skew) to help remember tail direction.
Symmetry and shapes:
- Bell-shaped (normal, symmetric): peak around the center; often used for many natural phenomena.
- Uniform distribution: relatively equal heights across the range; all outcomes have similar probability.
Unimodal vs bimodal distributions:
- Unimodal: a single peak.
- Bimodal: two distinct peaks; there may be two different subgroups contributing to the data distribution.
Key takeaways:
- The shape of a distribution is determined by data; skewness, symmetry, and modality are descriptive characteristics.
- In this course, bell-shaped (normal) and uniform distributions are highlighted; other shapes like skewed distributions are also important.

Interpreting distribution shapes from examples

Graph interpretation exercise (four graphs): identify skewness, symmetry, and modality.
- Example A: skewed to the right (tail on the right); one peak near the left side may indicate right skew with a tail to the right.
- Example B: bell-shaped (approximately symmetric, unimodal).
- Example C: approximately uniform (bars of roughly similar height).
- Example D: left-skewed (tail on the left).
Unimodal vs bimodal practice:
- A graph with one clear peak is unimodal.
- A graph with two distinct peaks is bimodal.

Concepts in practice: applying graph interpretation to everyday scenarios

Practice questions (interpretation from graphs): several real-world scenarios were discussed to practice identifying distribution shapes:
- Coffee shop: two rush-hour peaks with a dip in between → bimodal.
- Ice cream stall: highest sales at opening, then gradually decrease → right-skewed (tail to the right).
- Ticket sales for a festival: slow at first, then rise toward the event → skewed left (tail to the left).
- Local music shop pricing: most purchases in a moderately priced range with fewer very cheap or very expensive purchases → bell-shaped (roughly normal) distribution.
- Park visitors throughout the day: roughly even distribution → uniform distribution.
Note on interpretation: a bell-shaped curve is a special case of unimodal and symmetric; uniform implies equal probability across values.

Practical workflow and summary takeaways

Start with understanding data type and the goal of the summary (describe categories vs summarize numerical values).
For qualitative data:
- Use frequency and relative frequency tables.
- Use bar charts, Pareto charts, horizontal bars, and pie charts to visualize the distribution.
For quantitative data:
- Use frequency distributions with classes (intervals) to summarize data.
- Decide class width and the number of classes, ensuring the entire data range is covered.
- Compute frequencies and relative frequencies for each class.
- Use histograms (bars touching) to visualize the distribution.
- Interpret shapes (skewness, symmetry, uniformity) and modality (unimodal vs bimodal).
Core formulas and relationships to memorize:
- Frequency of a category: f_i = #{\text{observations in category } i}
- Total observations: $N = \sumi fi$
- Relative frequency: $f{rel,i} = \frac{fi}{N}$
- Sum of relative frequencies: $\sumi f{rel,i} = 1$
- Percentage form: $\%i = 100 \times f{rel,i}$
- Class width (for frequency distributions of quantitative data): $w = \left\lceil \frac{\max - \min}{k} \right\rceil$ where k is the desired number of classes.
- If you fold the data into classes by a chosen width or by choosing k, you’ll still ensure the largest value is included in the last class.
Real-world connections: these methods help summarize large data sets, identify dominant categories, compare groups (e.g., year-over-year changes), and assess distribution characteristics that influence further statistical analyses (e.g., choosing appropriate models or tests based on symmetry or skewness).

Key terms to remember

Qualitative data
Quantitative data
Frequency distribution
Relative frequency
Pie charts
Bar graphs
Pareto chart
Histogram (for quantitative data)
Class (interval) in a histogram
Lower class limit, Upper class limit
Class width
Open-ended class
Skewness (left vs right)
Symmetric distribution
Uniform distribution
Unimodal vs bimodal

Examples recap and quick answers

Qualitative sample: credit cards (Discover, Visa, American Express, Mastercard)
- Mastercard frequency: 11; Visa: 23; American Express: 9; Discover: 7
- Relative frequencies: Mastercard 0.22, Visa 0.46, American Express 0.18, Discover 0.14
Quantitative example: aircraft types for landings (single type)
- Single frequency: 10; total N = 30; Relative frequency: $\frac{10}{30} = 0.333… \approx 0.333$ (≈ 33.3%; reported 0.3 or 30% in the narration)
Quantitative class construction example (decimal data 0.25 to 6.64, target 7 classes):
- If you fix width w = 1 and start at 0, the classes are approximately [0,0.99], [1,1.99], …, [6,6.99].
- With the width-based method: max − min = 6.64 − 0.25 = 6.39; k = 7; w ≈ 0.913 → rounded up to 1; yields 7 classes of width 1 as in the fixed-width approach.
Open-ended last class example: 85 and older in histograms.
Distribution shapes recap:
- Skew right: majority on the left, tail to the right; examples discussed include some data sets with a right tail.
- Skew left: majority on the right, tail to the left.
- Bell-shaped: symmetric, unimodal.
- Uniform: relatively equal heights across categories or class intervals.

If you want, I can tailor these notes further to a specific chapter or provide a condensed checklist for quick revision before the exam.