Question 1

How many bins should a histogram have?

Accepted Answer

Common rules for choosing bin count: (1) Sturges' rule: k = ⌈1 + log₂(n)⌉ — simple, works well for n < 200. (2) Square root rule: k = ⌈√n⌉ — tends to give more bins. (3) Scott's rule: h = 3.49σ/n^(1/3) — based on data spread, better for normal-like data. (4) Freedman-Diaconis: h = 2×IQR/n^(1/3) — robust to outliers, best for non-normal data. This calculator uses Sturges' rule by default, with a manual override slider.

Question 2

What does histogram shape tell you?

Accepted Answer

Bell-shaped (symmetric): data is normally distributed, mean ≈ median ≈ mode. Right-skewed (positive skew): long right tail — mean > median, common in income, prices, wait times. Left-skewed (negative skew): long left tail — mean < median. Bimodal: two peaks — suggests two distinct subgroups in the data. Uniform: all bars similar height — data spread equally. Spiky or irregular: small sample or data with natural clusters. The histogram shape directly informs which statistical tests are appropriate.

Question 3

What is relative frequency vs cumulative frequency?

Accepted Answer

Relative frequency = count / total = proportion of data in that bin (sums to 1.0). Cumulative frequency = sum of all relative frequencies up to and including that bin (reaches 1.0 at the last bin). Use relative frequency to compare datasets of different sizes. Use cumulative frequency (ogive) to answer "what percentage of values are below x?" — equivalent to the empirical CDF.

Question 4

How do I read a frequency distribution table?

Accepted Answer

Each row represents one bin (class interval). Columns: lower/upper boundary define the class (typically left-closed: [lower, upper)). Count = number of data points in that class. Relative freq = count / n = probability of falling in that class. Cumulative freq = percentage of data at or below the upper boundary. The midpoint of each class = (lower + upper) / 2, used to represent the class in calculations.

Question 5

Why does the last bin include the maximum value?

Accepted Answer

Bins are typically left-closed intervals: [lower, upper). This means the right endpoint is excluded. However, the last bin is closed on both sides: [lower, max] so that the maximum value is included. Without this adjustment, the maximum value would fall outside all bins.

Question 6

When should I use a histogram vs a box plot?

Accepted Answer

Use a histogram when you want to see the full shape of the distribution — symmetry, modality, skew, and gaps. Use a box plot when comparing multiple datasets side by side, detecting outliers, or showing the five-number summary concisely. Histograms require more data (n ≥ 20 ideally); box plots work for small samples. For a single dataset, use both: the histogram reveals shape, the box plot reveals outliers.

Rule	Formula	Best for
Sturges (default)	k = ⌈1 + log₂(n)⌉	Normal-like data, n < 200
Square root	k = ⌈√n⌉	Simple, general purpose
Scott's rule	h = 3.49σ/n^(1/3)	Normal data, minimises MSE
Freedman-Diaconis	h = 2×IQR/n^(1/3)	Skewed data, robust to outliers
Rice rule	k = ⌈2n^(1/3)⌉	Moderate sample sizes

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