📊 Session 6.3 - Histogram – Compressive Strength

What is a histogram? A histogram is a bar graph that shows how many data values fall in each interval (bin). Good for seeing the shape (symmetric / skewed), the center, and the spread.

Use it when: you have many data points (20+) for one quantitative variable.

🔢 Raw Data (80 Values)

105, 221, 183, 186, 121, 181, 180, 143, 97, 154, 153, 174, 120, 168, 167, 141, 245, 228, 174, 199, 181, 158, 176, 110, 163, 131, 154, 115, 160, 208, 158, 133, 207, 180, 190, 193, 194, 133, 156, 123, 134, 178, 76, 167, 184, 135, 229, 146, 218, 157, 101, 171, 165, 172, 158, 169, 199, 151, 142, 163, 145, 171, 148, 158, 160, 175, 149, 87, 160, 237, 150, 135, 196, 201, 200, 176, 150, 170, 118, 149

📊 Histogram with Wide Bins (9 Bins)

📊 Histogram with Narrow Bins (17 Bins)

🧮 Practice Question

Using the wide-bin histogram, estimate what percent of specimens have compressive strength between 130 and 190 psi?

🧱 Constructing a Histogram (Equal Bin Widths)

Label bin boundaries along the horizontal axis (e.g., 70–90, 90–110, …).
Label the vertical axis with frequency (counts) or relative frequency (percent).
Draw a rectangle for each bin whose height equals the frequency (or relative frequency) in that bin.

Equal bin width is important so height comparisons are fair.

🧭 How to Read a Histogram

Center: where the bulk of the bars sit (median/mean are around here).
Spread: how wide the bars extend; wider = more variability.
Skewness: long tail to the right → right‑skewed (mean > median > mode); long tail to the left → left‑skewed (mode > median > mean); balanced tails → symmetric.
Modes: one hump (unimodal), two humps (bimodal).
Outliers: isolated bars far from the rest.

🖼️ Shape Gallery (toy examples)

Left‑skewed (tail on the left)

Symmetric

Right‑skewed (tail on the right)

🟦 Excel (no formulas): Insert → Histogram

Select your numbers only (e.g., A2:A81).
Go to Insert → Statistic Chart → Histogram.
Right‑click the horizontal axis → Format Axis → set Bin width (e.g., 20) or Number of bins (e.g., 9–12 for n≈80), and optional Overflow/Underflow bins.
To show relative frequency: compute percents in a helper column and chart that column; or change the axis to %.
Make it look like a histogram: Format Data Series → Gap Width = 0%.

📏 Picking a Bin Size (Width) — quick rules

Keep 6–15 bins for most classes. For n = 80, try 9–12 bins.
Square‑root rule: bins ≈ √n → √80 ≈ 9.
Sturges: bins ≈ 1 + log₂(n) → ≈ 1 + 6.32 ≈ 7–9.
Freedman–Diaconis (nice for outliers): width h = 2·IQR·n^(−1/3).

Too wide → hides shape; too narrow → looks noisy. Adjust until the pattern is clear.

🙋 Student Q&A (super short answers)

Q: How many bins should I use?
A: Not too big, not too small — aim for 6–15; try √n and adjust.

Q: Why are very wide bins bad?
A: They hide shape (modes/outliers disappear).

Q: Why are tiny bins bad?
A: They look noisy; random spikes hide the pattern.

Q: Histogram vs. stem‑and‑leaf?
A: Histogram is faster for large data; stem‑and‑leaf keeps exact values (better for small sets).

Q: Which center to report if skewed?
A: Use the median (mean gets pulled by the tail).

Q: Frequency vs. relative frequency histogram?
A: Heights are counts vs. percents (sum to 100%). Shape is the same.

📘 Session 6.3 – Histogram & Frequency Distribution