Frequency Distributions: Concepts, Formulas and Examples

A frequency distribution is a table that shows how often each value, or each group of values, occurs in a set of data. Instead of listing hundreds of raw numbers one by one, you group them and count how many fall into each group, so the pattern in the data becomes visible at a glance.

It matters because raw data is almost impossible to interpret — a column of 200 test scores tells you nothing until you organise it. Frequency distributions are the first step in statistics; every chart, average and standard deviation starts from this organisation.

Key Concepts

Raw data is data as collected, unorganised. Class is a group of values, e.g. marks from 40 to 49. Frequency (f) is the number of observations in a class. Class limits are the stated boundaries of a class (40 and 49); class boundaries are the true continuous boundaries (39.5 and 49.5). Class width = upper boundary − lower boundary. Class mark (midpoint) = (lower limit + upper limit)/2, used as the representative value of the class. Relative frequency = f/Σf, the fraction of the total in each class.

Formulas

Number of classes (Sturges' rule): k ≈ 1 + 3.322·log₁₀(n), where n is the total number of observations. Class width: w ≈ (maximum − minimum)/k, rounded up to a convenient value. Relative frequency: r = f/n, which always satisfies Σr = 1. Percentage frequency: p = (f/n) × 100. These use no units beyond the units of the measured variable, since frequencies are pure counts.

Worked Example

Twenty students scored: 42, 55, 61, 48, 73, 58, 66, 51, 77, 69, 44, 62, 59, 71, 53, 64, 57, 68, 75, 50. Build a frequency distribution.

Step 1: n = 20. Sturges: k ≈ 1 + 3.322·log₁₀(20) ≈ 1 + 3.322 × 1.301 ≈ 5.32, so use 5 classes. Step 2: Range = 77 − 42 = 35. Width ≈ 35/5 = 7, round up to 10 for convenience. Step 3: Start at 40. Classes: 40–49, 50–59, 60–69, 70–79. Step 4: Tally: 40–49 contains 42, 48, 44 → f = 3; 50–59 contains 55, 58, 51, 59, 53, 57, 50 → f = 7; 60–69 contains 61, 66, 69, 62, 64, 68 → f = 6; 70–79 contains 73, 77, 71, 75 → f = 4. Step 5: Relative frequencies: 3/20 = 0.15, 7/20 = 0.35, 6/20 = 0.30, 4/20 = 0.20 (sum = 1.00). The data clearly centres around the 50s and 60s.

Common Misconceptions

1. "Class limits and class boundaries are the same." Wrong — for the class 40–49 the limits are 40 and 49, but the boundaries are 39.5 and 49.5, which matter when drawing histograms. 2. "More classes are always better." Too many classes make each one nearly empty and hide the pattern; too few merge everything together. Use the Sturges guideline. 3. "The class mark is an observed value." It is only a representative midpoint used in grouped calculations, not an actual data point.

Real-World Applications

Frequency distributions are used to summarise exam scores into grade bands, group customers by age for marketing, report income brackets in census data, monitor the distribution of product weights in quality control, and prepare data before computing the mean and standard deviation of grouped data.

A frequency distribution turns a pile of raw numbers into an organised table of classes, counts and relative frequencies. It reveals where data clusters, guides every statistical calculation that follows, and is the foundation on which histograms, averages and dispersion measures are built.