Organize raw data into frequency distribution tables with classes and cumulative frequencies.
| Class Interval | Frequency (f) | Relative f (%) | Cumulative f | Midpoint |
|---|---|---|---|---|
| 12.0 – 20.0 | 4 | 22.2% | 4 | 16.0 |
| 20.0 – 28.0 | 5 | 27.8% | 9 | 24.0 |
| 28.0 – 36.0 | 6 | 33.3% | 15 | 32.0 |
| 36.0 – 44.0 | 1 | 5.6% | 16 | 40.0 |
| 44.0 – 52.0 | 2 | 11.1% | 18 | 48.0 |
A frequency distribution is a systematic organization of raw data into a table showing how often each value or range of values (class) appears in a dataset. Rather than working with individual data points, frequency distributions summarize data into meaningful groups, making patterns visible and calculations tractable. This is especially valuable for large datasets where listing every value would be impractical and obscure important trends.
The table typically includes class intervals (ranges), absolute frequency (count in each interval), relative frequency (proportion or percentage), and cumulative frequency (running total). Class width is determined by dividing the range by the number of desired classes, balancing between too few classes (losing detail) and too many (fragmentation). Frequency distributions form the foundation for histograms, which visualize the distribution graphically.
This technique is used across statistics, business analytics, quality control, and scientific research to transform raw measurements into actionable insights quickly.
Organize exam scores: 45, 48, 52, 55, 58, 62, 65, 68, 72, 75, 78, 82, 85, 88, 92 (n=15) into 3 classes.
Step 1: Find Range
Range = 92 − 45 = 47
Step 2: Calculate Class Width
Width = ⌈47 / 3⌉ = 16 (rounded up)
Step 3: Create Classes
Class 1: 45–61 (includes 45, 48, 52, 55, 58) = 5 observations
Class 2: 61–77 (includes 62, 65, 68, 72, 75) = 5 observations
Class 3: 77–93 (includes 78, 82, 85, 88, 92) = 5 observations
Step 4: Build Table
Class 1: f=5, Rel f=33.3%, Cum f=5
Class 2: f=5, Rel f=33.3%, Cum f=10
Class 3: f=5, Rel f=33.3%, Cum f=15
Result: The frequency distribution shows perfectly equal distribution across classes—each contains 33.3% of data, suggesting uniform spacing of test scores.
How do I choose the right number of classes?
Use the rule: k ≈ √n or Sturges' rule: k ≈ 1 + 3.3 log(n). For n=100, this gives 5–8 classes. Aim for about 5% data per class on average.
What's the difference between frequency and relative frequency?
Frequency (f) is the count of observations in a class. Relative frequency is f/n, often expressed as percentage. Relative frequency enables comparison across datasets of different sizes.
How should I handle class boundaries to avoid overlap?
Use intervals like [lower, upper) so each value falls in exactly one class. The last class should be [lower, upper] to include the maximum value.
What does cumulative frequency tell me?
Cumulative frequency at class i shows how many observations are less than the upper boundary of class i. It helps find medians, quartiles, and percentiles quickly.
Can I have unequal class widths?
Yes, but it complicates interpretation. Use equal widths by default unless data has natural groupings (like age ranges 0–18, 18–65, 65+).
What if I have decimal data?
Works fine! The calculator handles decimals in all calculations. Class boundaries will also be decimal. Sort data first to avoid boundary issues.
How do I visualize frequency distribution?
Create a histogram using the class intervals on x-axis and frequencies (or relative frequencies) on y-axis. Each bar's height represents frequency for that class.
What if one class has zero frequency?
That's okay—it indicates no data fell in that range. Some classes may be empty, especially if class width is large or distribution is sparse.
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