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The Frequency Distribution Calculator computes the parameters needed to construct a frequency distribution table from your raw data: the minimum, maximum, range, class width, and the recommended number of classes using Sturges' formula. A frequency distribution organizes raw data into classes (intervals) and counts the number of observations in each class.
Frequency distributions are foundational in statistics. They transform an unorganized list of numbers into a structured summary that reveals the data's pattern, shape, center, and spread. Every histogram, frequency polygon, cumulative frequency curve (ogive), and relative frequency table is built from a frequency distribution.
Constructing a frequency distribution involves several decisions: how many classes to use, how wide each class should be, and where the class boundaries should fall. The number of classes ($$k$$) is typically chosen between 5 and 20, depending on the dataset size. This calculator applies Sturges' formula as a guideline: $$k = \lceil 1 + 3.322 \log_{10}(n) \rceil$$
The class width is then computed as: $$w = \left\lceil \frac{\text{Range}}{k} \right\rceil = \left\lceil \frac{\max - \min}{k} \right\rceil$$ The ceiling function ensures that all classes are wide enough to cover the entire data range. In practice, the class width is often rounded to a convenient number for readability.
Once you have the class width and starting point (usually the minimum value or a round number just below it), you can construct the complete frequency distribution table: list each class interval, tally the observations in each, compute frequencies, relative frequencies (frequency/total), and cumulative frequencies. This table forms the backbone of statistical summarization.
Frequency distributions are used in virtually every data-driven field: quality control (monitoring manufacturing tolerances), demographics (age group distributions), finance (return frequency analysis), meteorology (temperature and rainfall patterns), medicine (dosage response distributions), and education (grade distributions). Understanding how to create and interpret them is an essential statistical skill.
A frequency distribution table is constructed through these steps:
The resulting table compactly summarizes the entire dataset's distribution and serves as the basis for graphical displays.
Use the computed values to build your frequency distribution table: start the first class at the minimum, make each class exactly class width units wide, and create the specified number of classes. The Sturges' recommended classes value is a starting point; adjust as needed for clarity.
Once the table is built, look for: (1) Which class has the highest frequency — this is the modal class. (2) How frequencies change across classes — symmetric distributions peak in the middle, skewed ones peak toward one end. (3) Classes with zero frequency — gaps in the distribution. (4) Cumulative frequencies — useful for finding percentiles and the median class.
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Range = 55-22 = 33 years. With 5 classes, width = ceil(33/5) = 7 years. Classes: [22-29], [29-36], [36-43], [43-50], [50-57]. Tally data into these intervals to complete the frequency table.
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Results
Range = 108-98 = 10g. With 4 classes, width = ceil(10/4) = 3g. Classes: [98-101], [101-104], [104-107], [107-110]. Sturges' rule also suggests 4 classes for 8 observations.
A frequency distribution is a table that organizes raw data into classes (intervals) and shows the number of observations (frequency) in each class. It provides a structured summary of the data's distribution pattern, making it easy to identify the most common values, the spread, and the overall shape. It is the foundation for histograms and other frequency-based visualizations.
Use a guideline such as Sturges' formula ($$k = 1 + 3.322 \log_{10}(n)$$), the Square Root rule ($$k = \sqrt{n}$$), or Rice's rule ($$k = 2n^{1/3}$$) as a starting point. Then adjust: aim for 5-20 classes, with each class containing at least a few observations. The goal is a table that reveals the distribution's shape without being too coarse (too few classes) or too granular (too many).
Class width is the size of each interval in the frequency distribution: $$w = \frac{\text{Range}}{k}$$ where Range = max - min and $$k$$ is the number of classes. The result is typically rounded up to a convenient number. All classes should have equal width for a standard frequency distribution. The class width determines how finely the data range is divided.
Frequency is the raw count of observations in each class. Relative frequency is the proportion: $$\text{Relative frequency} = \frac{\text{frequency}}{\text{total count}}$$. Relative frequencies sum to 1 (or 100%). They allow meaningful comparison between datasets of different sizes, since a frequency of 15 means different things in a sample of 50 vs. a sample of 1000.
Cumulative frequency is the running total of frequencies up to and including each class. The first class's cumulative frequency equals its frequency; each subsequent class adds its frequency to the previous cumulative total. The final cumulative frequency always equals the total number of observations. Cumulative frequency is plotted in an ogive (cumulative frequency curve) and is used to find percentiles and the median class.
A histogram is the graphical representation of a frequency distribution. Each class interval becomes a bar whose height equals the class frequency (or relative frequency or density). The horizontal axis shows the class boundaries, and bars are adjacent with no gaps. Every histogram is built from an underlying frequency distribution table.
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