What Is a Frequency Distribution?
A frequency distribution organizes data to show how often each value (or range of values) occurs. It can be presented as a table, a chart (histogram, bar chart, ogive), or specialized trader charts (point-and-figure). Frequency distributions make raw data easier to interpret, reveal shape and central tendency of a dataset, and support probability or risk assessment.
Key takeaways
– A frequency distribution counts observations by value or interval and displays the pattern of those counts.
– Formats include ungrouped (individual values), grouped (intervals/classes), relative (proportions), cumulative, and graphical forms (histogram, ogive, frequency polygon, point-and-figure).
– Proper class selection (nonoverlapping, exhaustive) is essential; rules such as Sturges’ or Freedman–Diaconis can guide bin choice.
– Frequency distributions are widely used in statistics and are applied in business, demographics, and some trading techniques (e.g., point-and-figure charts).
Understanding frequency distributions
– Frequency: how many times a value or values in an interval occur.
– Distribution: the overall pattern of those frequencies across values or ranges.
– Ungrouped vs. grouped:
– Ungrouped frequency distribution lists distinct values and their counts (best for small discrete datasets).
– Grouped frequency distribution groups values into intervals/classes (best for large or continuous datasets).
Types of frequency distribution
– Ungrouped frequency distribution: each distinct data value listed with its frequency.
– Grouped frequency distribution: data grouped into fixed intervals (classes) with a frequency for each class.
– Relative frequency distribution: frequency divided by total sample size; shows proportions.
– Cumulative frequency distribution: running total of frequencies up to and including each class.
– Relative cumulative frequency distribution: cumulative frequency divided by total sample size (useful for percentiles and quantiles).
Visual representations
– Histogram: contiguous bars whose heights equal class frequencies (for continuous data and grouped distributions). Useful for seeing shape (normal, skewed, multimodal).
– Bar chart: separate bars for discrete categories (bars are separated because categories are distinct).
– Frequency polygon: line connecting midpoints of histogram bars — helps compare distributions.
– Ogive (cumulative frequency graph): plots cumulative frequency at class boundaries — useful for percentiles.
– Point-and-figure chart: specialist chart used by some traders (Wyckoff followers) using Xs and Os to show price direction without fixed time scale.
Why frequency distributions matter
– Summarize large data sets into accessible form.
– Reveal central tendency, spread, skewness, multimodality, and outliers.
– Support probability estimates (e.g., percent of observations within a range).
– Enable comparisons across groups, time periods, or assets.
– In business: sales bands, customer age ranges, defect counts; in finance: return distributions, risk buckets; in research: test scores, demographic breakdowns.
How to construct a frequency distribution — practical, step‑by‑step
A. For a small discrete dataset (ungrouped)
1. List each distinct value.
2. Count how many times each value occurs.
3. Optionally compute relative frequency = frequency / total observations and cumulative frequency.
Example (ungrouped)
Data: [2, 3, 3, 4, 2, 5, 3]
Value | Frequency | Relative freq.
2 | 2 | 2/7 ≈ 0.286
3 | 3 | 3/7 ≈ 0.429
4 | 1 | 1/7 ≈ 0.143
5 | 1 | 1/7 ≈ 0.143
B. For continuous or large datasets (grouped)
1. Decide number of classes (k). Common rules:
– Sturges’ rule: k ≈ 1 + 3.322 log10(n)
– Square-root rule: k ≈ √n
– Freedman–Diaconis for bin width: width = 2 * IQR / n^(1/3)
Choose a rule based on sample size and data shape.
2. Compute class width ≈ (max − min) / k and round to a convenient number.
3. Define class boundaries so they are mutually exclusive and exhaustive (no overlaps, cover all observations).
4. Tally data into classes and get frequencies.
5. Compute relative frequency = frequency / n and cumulative frequency by summing frequencies up to each class.
6. Create a table and one or more visualizations (histogram for grouped frequencies, ogive for cumulative).
Worked example — heights of 50 children (illustrative)
– Suppose min = 110 cm, max = 150 cm, n = 50.
– Use k ≈ √50 ≈ 7 classes.
– Class width ≈ (150 − 110) / 7 ≈ 5.7 → choose width = 6 cm.
– Define classes: 110–115.9, 116–121.9, 122–127.9, 128–133.9, 134–139.9, 140–145.9, 146–151.9
– Tally counts, then compute relative and cumulative frequencies.
– Draw a histogram: x-axis = height class, y-axis = frequency (or relative frequency).
Practical steps in software
– Excel:
– Use FREQUENCY(array, bins) to compute counts for specified bins.
– Or use PivotTable: group numeric values by intervals.
– Create histograms via the Analysis ToolPak or Insert > Chart > Histogram.
– Python (pandas / numpy / matplotlib):
– numpy.histogram(data, bins=…) returns counts and bin edges.
– pandas.cut(data, bins=…) then value_counts(sort=False) to get grouped frequencies.
– matplotlib.pyplot.hist(data, bins=…) plots histogram directly.
Interpreting frequency distributions
– Shape: symmetric (normal-like), skewed (left/right), or multimodal.
– Spread: width of the distribution indicates variability.
– Center: modal class (highest frequency) and median (via cumulative distribution).
– Outliers: classes with very low counts far from bulk of data.
– For probabilities: relative frequencies estimate probabilities for future draws (empirical probabilities).
Common pitfalls and best practices
– Do not let intervals overlap; ensure classes are exhaustive.
– Be aware that different bin choices can change perceived shape; test alternatives.
– For small samples, prefer ungrouped displays or fewer bins.
– Label axes clearly and state whether vertical axis is frequency count or relative frequency.
– When comparing distributions, use equal-width bins (or normalized relative frequencies) to avoid misleading comparisons.
Frequency distribution in trading
– Not commonly used as a formal statistical tool across all trading, but certain techniques use frequency-like methods.
– Point-and-figure charts (Wyckoff followers) are a form of frequency chart: they record price movements with Xs and Os and identify trends by runs of three Xs (uptrend) or three Os (downtrend). These charts strip out time to focus on price action and supply/demand balance.
– Traders may also examine return frequency distributions or histograms of price returns to assess volatility, skew, and tail risk.
The bottom line
Frequency distributions are foundational tools for summarizing and visualizing how data are distributed. Whether you need to summarize exam scores, analyze customer age ranges, evaluate return distributions, or apply specialized trading charts, constructing a correct frequency distribution (appropriate classes, nonoverlapping intervals, proper visualizations) is crucial to accurate interpretation.
References
– Investopedia — “Frequency Distribution” (Investopedia). https://www.investopedia.com/terms/f/frequencydistribution.asp
– Toppr — “Frequency Distribution and Data: Types, Tables, and Graphs.” https://www.toppr.com/guides/maths/statistics/frequency-distribution/
– Nasdaq — “Point & Figure Basics.” https://www.nasdaq.com/articles/point-and-figure-basics
If you’d like, I can:
– Build a frequency table and histogram for your dataset (paste data or upload CSV).
– Show Excel formulas or a ready-to-run Python snippet to generate and plot distributions.