Frequency Distribution
Create Frequency Tables and Histograms
[Image of histogram frequency distribution]
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Frequency Distribution Table
(Scroll horizontally to see full table)
| Class Interval | Midpoint | Freq | Rel. Freq | Cum. Freq |
|---|
Detailed Construction
👨🏫
By Prof. David Anderson
Statistics Professor | 20+ Years Exp.
“Raw data is like a messy room—it contains everything you need, but you can’t find anything. In my Statistics 101 class, the first thing I teach is ‘Data Organization’. A Frequency Distribution Table is simply a way to tidy up that mess. By grouping data into bins (or classes), we can instantly see patterns, spot outliers, and prepare for the ultimate visualization: the Histogram. Today, I’ll walk you through how to construct one perfectly.”
Frequency Distribution Calculator: Table, Class Width & Histogram
Create Grouped Frequency Tables, Cumulative Frequency, and Midpoints
The Frequency Distribution Calculator transforms raw, ungrouped data into an organized Frequency Distribution Table. It automatically calculates the optimal Number of Classes (using Sturges’ Rule), Class Width, Midpoints, and Cumulative Frequency.
Whether you are calculating Relative Frequency for test scores or preparing a Grouped Frequency Distribution for business analytics, this tool bridges the gap between raw numbers and a Histogram Maker.
1. Anatomy of a Frequency Distribution Table
✨ What do the columns mean?
A professional Frequency Table Calculator generates 5 specific columns. Here is your roadmap:
Class Interval
Lower – Upper
The range (e.g., 10-19). Includes Class Limits.
Frequency
f
The raw count of data points in that class.
Midpoint
x
Center of the class. Essential for Mean of Grouped Data.
Relative Freq.
rf = f/n
Percentage of total data. (Relative Frequency).
Cumulative Freq.
cf
Running total. Used for Ogive charts.
Boundaries
-.5 / +.5
Gapless Class Boundaries for Histograms.
2. How Many Classes? (Sturges’ Rule Calculator)
The hardest part of creating a Grouped Frequency Distribution is deciding how many bins (classes) to use.
• Too few classes = Oversimplified (Everything looks the same).
• Too many classes = Too much noise (Looks like raw data).
This Frequency Distribution Calculator uses Sturges’ Rule to find the “Goldilocks” number of classes ($k$):
Sturges’ Formula
$$ k = 1 + 3.322 \log(n) $$
Where $n$ is the total number of data points. We always round $k$ to the nearest whole number.
3. Calculating Class Width (The Round-Up Rule)
Once you know the number of classes, you need a Class Width Calculator. This determines the size of each bin.
Class Width Formula:
$$ \text{Width} \approx \frac{\text{Max} – \text{Min}}{k} $$
⚠️ PROFESSOR’S WARNING: Always Round UP to the next whole number (even if it’s 4.1, round to 5). If you round down, your last data point won’t fit in the Frequency Table!
⚠️ PROFESSOR’S WARNING: Always Round UP to the next whole number (even if it’s 4.1, round to 5). If you round down, your last data point won’t fit in the Frequency Table!
4. Class Limits vs. Class Boundaries
This is where most students lose points on homework. You must know the difference between the “written limits” and the “mathematical boundaries” in a Frequency Distribution.
🛑 Class Limits
10 – 19
20 – 29
20 – 29
- Concept: The numbers you write in the Frequency Table.
- Gaps: Yes! There is a gap between 19 and 20.
- Use Case: Human reading.
🟢 Class Boundaries
9.5 – 19.5
19.5 – 29.5
19.5 – 29.5
- Concept: The true mathematical edges.
- Gaps: No! 19.5 connects both classes.
- Use Case: Drawing Histograms (Bars must touch).
5. Step-by-Step Construction Guide
Let’s build a table for a coffee shop’s wait times (in minutes) using our Frequency Distribution Calculator logic:
Data: $\{2, 5, 18, 12, 14, 7, 9, 22, 5, 8\}$ ($n=10$)
Step 1 Find Range
Min = 2, Max = 22.
Range = $22 – 2 = 20$.
Range = $22 – 2 = 20$.
Step 2 Determine Width
Assume we want 5 classes ($k=5$).
Width = $20 / 5 = 4$.
Tip: Add 1 to ensure coverage, so Width = 5.
Width = $20 / 5 = 4$.
Tip: Add 1 to ensure coverage, so Width = 5.
Step 3 Create Intervals
Start at Min (2). Add width (5).
• Class 1: 2 – 6
• Class 2: 7 – 11
• Class 3: 12 – 16…
• Class 1: 2 – 6
• Class 2: 7 – 11
• Class 3: 12 – 16…
6. Histogram Maker: From Table to Chart
Once the Frequency Table is complete, creating a Histogram is straightforward.
- X-Axis: Use the Class Boundaries (e.g., 1.5, 6.5, 11.5). This ensures histogram bars touch each other.
- Y-Axis: Use the Frequency ($f$). The height of the bar represents the count.
- Shape: The histogram reveals if the data is Symmetric (Bell Curve), Skewed Left, or Skewed Right.
7. Grouped vs. Ungrouped Frequency Distribution
Should you use a Grouped Frequency Distribution or an Ungrouped one?
- Ungrouped: Use when data range is small (e.g., number of pets: 0, 1, 2, 3). No intervals needed.
- Grouped: Use when data range is large (e.g., salaries: $30k – $150k). You must use intervals (bins).
8. Professor’s FAQ Corner
Why do we use Midpoints ($x$)?
Midpoints represent the “average” value of a class in a Frequency Distribution Table. We need them to calculate the Mean of Grouped Data later on, because we’ve lost the specific raw values during grouping.
What is Relative Frequency useful for?
Relative Frequency converts counts into percentages. This allows you to compare two datasets of different sizes (e.g., comparing test scores from a class of 30 vs a class of 300).
Can classes have different widths?
In a standard Frequency Distribution, No. All classes must be equal width to make the Histogram accurate. If widths vary, you must use “Frequency Density” (advanced statistics).
References
- Sturges, H. A. (1926). “The choice of a class interval”. Journal of the American Statistical Association.
- Triola, M. F. (2018). Elementary Statistics. Pearson. (Chapter on Frequency Distributions).
- Rice, J. A. (2006). Mathematical Statistics and Data Analysis. (Alternative rules for binning).
Organize Your Data
Paste your raw data below to generate a Frequency Table & Histogram Data instantly.
Create Frequency Table