Module 3 — Organising quantitative data: histograms and cumulative curves

Data Description | Part I — Organising data

Modified

October 7, 2026

The business question. The owner of the Harbour truck is planning how much food to prepare. Each day brings a different revenue, to the cent, so a list of 30 or 120 numbers tells her nothing at a glance. How is daily revenue distributed? Are most days clustered around €600, or spread all over the place? And on how many days do we earn less than €500? This module turns a long column of continuous numbers into a short table and three pictures that answer these questions.

Learning objectives

By the end of this module, you will be able to:

Explain why a continuous variable must be grouped into classes before it can be summarised in a frequency table.
Choose a sensible number of classes and a class width, and write the classes with the half-open convention $[L_i, L_i + h[$.
Build a class table with counts, relative frequencies and cumulative frequencies, by hand and in Excel.
Draw and read a histogram, a frequency polygon and a cumulative frequency curve, including a histogram with unequal class widths.
Read a percentile or a share of observations off the cumulative curve.

Time plan

Block	Minutes	What you do
Theory	70	Classes, frequencies, histogram, polygon, cumulative curve
Excel lab	60	`COUNTIFS` class table, histogram chart, PivotTable grouping
Exercises	50	Three by-hand and two Excel exercises
Total	180

Theory

Definition — class (grouped) frequency table

When almost every value of a continuous variable is different, a plain frequency table lists each value once and says nothing useful. The fix is to cut the range of values into $k$ consecutive classes (intervals) of width $h$ and count how many observations fall into each one. Classes are written half-open, $[L_i, L_i + h[$: the lower bound belongs to the class, the upper bound belongs to the next class. So every value lands in exactly one class.

In plain words: a class table trades precision for readability. You lose the exact values, but you can finally see where the data sit.

How many classes, how wide?

Rule of thumb — Sturges’ rule

For $n$ observations, a good number of classes is about \[ k \approx 1 + 3.3 \log_{10}(n), \qquad h \approx \frac{\text{max} - \text{min}}{k}, \] then round $h$ to a convenient number (10, 50, 100 …) and start the first class at a round number at or below the minimum.

In plain words: more data can support more classes, but slowly. Thirty days give about 6 classes; 120 days give about 8. Too few classes hide the shape; too many make a ragged picture where each bar holds only one or two days.

For the first 30 Harbour days, revenue runs from €407.59 to €886.56, a range of €478.97. Sturges gives $k \approx 5.9$ and $h \approx 82$, which we round to $h = 100$ with the first class starting at 400. That gives five classes.

Frequencies in a class table

Formulas — counts, shares and running totals

For class $i$, with $n_i$ the number of observations in it and $n$ the total number of observations: \[ f_i = \frac{n_i}{n}, \qquad N_i^{+} = \sum_{j=1}^{i} n_j, \qquad F_i^{+} = \frac{N_i^{+}}{n}. \]

In plain words: $n_i$ is a count; $f_i$ is its share of the total (often written as a percentage); $N_i^{+}$ and $F_i^{+}$ are the running totals of counts and shares up to and including class $i$. The last one answers “how many days earned less than the upper bound of this class?”.

The histogram

Definition — histogram

A histogram draws one rectangle per class, with the rectangles touching each other (the variable is continuous, so there are no gaps). The area of a rectangle is proportional to the number of observations in the class. When all classes have the same width $h$, area is proportional to height, so the height is simply $n_i$. When the widths differ, the height must be the density \[ \text{height}_i = \frac{n_i}{h_i}. \]

In plain words: your eye reads area, not height. A wide class holds more days than a narrow one of the same height, so for unequal widths we divide the count by the width to keep the picture honest.

Frequency polygon and cumulative curve

The frequency polygon joins the points (class midpoint, $n_i$) with straight lines. It carries the same information as the histogram but is easier to superimpose for two distributions.
The cumulative frequency curve (also called the ogive) joins the points (class upper bound, $F_i^{+}$) and starts at $0\%$ on the lower bound of the first class. It never goes down and ends at $100\%$.

Formula — reading a percentile on the cumulative curve

If the share $p$ you are looking for lies between $F_{i-1}^{+}$ and $F_i^{+}$ in class $[L_i, L_i + h[$, linear interpolation gives \[ x_p \approx L_i + h \times \frac{p - F_{i-1}^{+}}{F_i^{+} - F_{i-1}^{+}}. \]

In plain words: inside a class we assume the observations are spread evenly, so the curve is a straight line there. Find the class where the curve crosses $p$, then move along that straight segment.

Worked example

We use the first 30 trading days of Harbour (sheet harbour_30d). The class table for daily_revenue_eur with $h = 100$ and $n = 30$ is:

Class (€)	$n_i$	$N_i^{+}$	$f_i$ (%)	$F_i^{+}$ (%)
[400, 500[	6	6	20.0	20.0
[500, 600[	6	12	20.0	40.0
[600, 700[	8	20	26.7	66.7
[700, 800[	7	27	23.3	90.0
[800, 900[	3	30	10.0	100.0

Step 1 — the histogram. All five classes have width 100, so the bar heights are the counts 6, 6, 8, 7, 3. The tallest bar is [600, 700[: that is where the largest group of trading days sits.

Step 2 — the frequency polygon. The class midpoints are 450, 550, 650, 750 and 850. Plot (450, 6), (550, 6), (650, 8), (750, 7), (850, 3) and join them with straight lines.

Step 3 — the cumulative curve. Plot the upper bounds against $F_i^{+}$: (500, 20%), (600, 40%), (700, 66.7%), (800, 90%), (900, 100%), starting from (400, 0%). Reading at 600 tells us that 40% of the days earned less than €600.

Step 4 — reading the curve. Where does the curve cross 50%? Between 600 (40%) and 700 (66.7%), so \[ x_{50} \approx 600 + 100 \times \frac{50 - 40}{66.7 - 40} \approx 637.5. \] The exact median of the 30 values is €643.63, so the curve reading is within €6. For the third quartile, $p = 75\%$ lies between 700 (66.7%) and 800 (90%): $x_{75} \approx 700 + 100 \times \frac{75 - 66.7}{90 - 66.7} \approx 735.7$ (the exact value, computed in Module 5, is €729.99).

Exam tip

A cumulative curve is read at upper bounds. A polygon is read at midpoints. Mixing them up is the most common error on this topic.

Using Excel

The sheet harbour_30d of wandering-fork.xlsx has the same 16 columns as the main table; daily_revenue_eur is column N, rows 2 to 31. The complete list of function names, with their French equivalents, is on the Excel functions page.

Concept	Excel function or steps
Class bounds	Type the lower bounds 400, 500, … in column R and the upper bounds in column S (each upper bound is the next lower bound)
Count $n_i$ (column T)	`=COUNTIFS($N$2:$N$31,">="&R2,$N$2:$N$31,"<"&S2)`, then copy down
Relative frequency $f_i$ (column U)	`=T2/COUNT($N$2:$N$31)`, formatted as a percentage
Cumulative count $N_i^{+}$ (column V)	In the first row `=T2`; below, `=V2+T3` (running sum), then copy down
Cumulative share $F_i^{+}$ (column W)	`=V2/COUNT($N$2:$N$31)`, formatted as a percentage
Same count in one step	`=COUNTIFS($N$2:$N$31,"<"&S2)` gives $N_i^{+}$ directly
Histogram	Select the revenue column → Insert → Chart → Histogram, then Format Axis → Bin width = 100
PivotTable grouping	Insert a PivotTable, put `daily_revenue_eur` in Rows and `day` in Values (Count), then right-click a row → Group with Starting at 400, Ending at 900, By 100
Frequency polygon	Insert → Chart → Line (or Scatter with straight lines) using the midpoints and counts
Cumulative curve	Insert → Chart → Scatter with Straight Lines using the upper bounds and $F_i^{+}$

For the full table (120 days per franchise) use the Excel Table tblFork and add the franchise as one more condition, for example:

=COUNTIFS(tblFork[franchise],"Harbour",
          tblFork[daily_revenue_eur],">="&R2,
          tblFork[daily_revenue_eur],"<"&S2)

(Type it on one line in Excel; it is split here only for reading.)

Excel’s Histogram chart draws its bins the other way round

In many Excel versions the built-in Histogram chart labels its bins as $(400, 500]$: the upper bound belongs to the class. Our convention is $[400, 500[$. With revenue to the cent a value of exactly 500.00 is rare, but a number such as 500 or 600 can occur in other data. When in doubt, trust your COUNTIFS table, not the chart labels. The PivotTable grouping follows our convention (a value of exactly 500 goes into the class starting at 500).

Proof / derivation

Derivation — the cumulative curve must reach 100% (click to expand)

By definition $N_k^{+} = \sum_{j=1}^{k} n_j$, summed over all $k$ classes. The classes are consecutive and half-open, so every observation falls in exactly one class and the class counts add up to the total: \[ N_k^{+} = \sum_{j=1}^{k} n_j = n \quad\Longrightarrow\quad F_k^{+} = \frac{N_k^{+}}{n} = \frac{n}{n} = 100\%. \] For the first 30 Harbour days, $N_5^{+} = 6+6+8+7+3 = 30 = n$, so $F_5^{+} = 100\%$. Whatever class bounds you choose, the cumulative curve always finishes in the top-right corner. If it does not, a class is missing or an observation fell outside your classes — a quick way to check your own table.

Visual intuition

Think of the histogram as a pile of sand poured on the revenue axis. The area of each bar is the amount of sand in that class, and the total area is the whole sample (the 30 days). If every class has the same width, a taller bar means more sand. If a class is twice as wide and you keep the same height, you have silently doubled its sand: the picture lies. Dividing the count by the width ($n_i/h_i$) keeps the amount of sand honest.

The cumulative curve is the same pile seen from another angle: at any revenue $x$, its height is the share of sand lying to the left of $x$. Where the histogram has a tall bar, the curve climbs steeply; where the histogram is nearly empty, the curve is nearly flat.

Interactive demo — build your own histogram

Drag the bin-width slider and watch the same days regroup into wider or narrower classes. Switch between the 30-day Harbour subset, all 120 Harbour days, and all 120 Campus days to see how the shape changes. Notice that Campus has a long tail to the right: a few very high-revenue days stretch the picture.

#| standalone: true
#| components: [viewer]
#| viewerHeight: 600

from math import floor, log10

from shiny import App, render, ui
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import numpy as np

BG = '#1C2E22'
FG = '#D2CCC0'

HARBOUR_120 = [
    460.48, 500.02, 418.65, 508.80, 646.19, 785.79, 719.66, 478.04,
    669.00, 666.42, 614.83, 520.99, 725.66, 886.56, 691.50, 744.92,
    614.75, 700.41, 692.60, 764.59, 856.52, 641.07, 407.59, 437.31,
    434.58, 587.06, 742.99, 841.83, 559.53, 531.91, 387.93, 328.04,
    68.26, 302.75, 324.42, 365.85, 417.09, 446.87, 435.07, 569.91,
    416.06, 246.84, 630.10, 374.12, 1000.00, 504.27, 206.82, 180.95,
    70.19, 406.16, 273.57, 384.50, 51.21, 314.63, 328.44, 517.13,
    147.56, 275.44, 469.52, 333.34, 343.64, 185.77, 382.81, 84.07,
    440.31, 261.95, 61.17, 441.58, 664.67, 644.39, 75.42, 453.04,
    623.77, 261.15, 104.94, 401.36, 389.79, 270.44, 321.17, 1050.00,
    311.41, 393.14, 66.17, 413.06, 484.42, 80.44, 204.27, 227.49,
    291.01, 382.51, 119.97, 549.51, 401.30, 137.45, 271.02, 267.04,
    329.19, 634.78, 297.20, 166.48, 164.03, 363.36, 247.18, 462.55,
    657.88, 67.08, 474.32, 607.24, 178.05, 346.90, 317.60, 208.79,
    331.38, 315.26, 335.38, 273.53, 227.29, 187.58, 620.20, 116.27,
]

CAMPUS_120 = [
    1089.40, 1335.48, 540.03, 598.76, 377.74, 170.84, 23.85, 368.83,
    599.66, 432.71, 478.40, 572.40, 299.09, 186.26, 556.11, 356.11,
    658.89, 697.39, 361.09, 277.62, 172.24, 870.03, 1407.59, 448.33,
    715.64, 267.68, 114.21, 323.14, 396.69, 742.01, 1114.89, 741.91,
    275.82, 245.21, 55.06, 346.24, 598.12, 989.43, 295.85, 138.44,
    267.00, 72.34, 898.08, 384.09, 291.95, 472.05, 667.05, 79.47,
    26.57, 1135.89, 951.18, 403.89, 427.84, 156.30, 181.88, 30.58,
    241.99, 380.60, 318.68, 446.42, 413.63, 159.43, 121.60, 577.01,
    493.92, 436.74, 529.14, 524.11, 35.80, 234.09, 359.34, 269.31,
    246.10, 270.87, 747.84, 436.33, 180.67, 402.41, 348.79, 717.81,
    399.38, 864.08, 171.61, 213.40, 333.03, 566.94, 519.92, 439.65,
    568.41, 251.73, 49.02, 950.12, 345.89, 569.65, 483.54, 251.10,
    227.23, 198.44, 849.86, 1000.56, 621.45, 441.41, 574.90, 93.29,
    204.87, 521.04, 268.06, 859.15, 949.10, 175.90, 179.84, 208.10,
    1062.98, 761.06, 773.08, 278.86, 245.06, 127.26, 63.49, 1362.93,
]

DATASETS = {
    "harbour30": ("Harbour, first 30 days", HARBOUR_120[:30]),
    "harbour120": ("Harbour, all 120 days", HARBOUR_120),
    "campus120": ("Campus, all 120 days", CAMPUS_120),
}


def make_bins(values, width):
    lo = int(floor(min(values) / width) * width)
    hi = int((floor(max(values) / width) + 1) * width)
    return np.arange(lo, hi + width, width)


def draw(values, width, title):
    bins = make_bins(values, width)
    fig, ax = plt.subplots(figsize=(7, 4))
    fig.patch.set_facecolor(BG)
    ax.set_facecolor(BG)
    for spine in ax.spines.values():
        spine.set_edgecolor(FG)
    ax.tick_params(colors=FG)
    ax.hist(values, bins=bins, color='steelblue', edgecolor=BG, alpha=0.9)
    ax.set_xlabel("Daily revenue (EUR)", color=FG)
    ax.set_ylabel("Number of days", color=FG)
    ax.set_title(title, color=FG)
    ax.grid(alpha=0.15, color=FG, axis='y')
    fig.tight_layout()
    return fig


app_ui = ui.page_fluid(
    ui.tags.style(f"""
        body {{ background-color: {BG}; color: {FG}; padding: 12px;
               font-family: sans-serif; margin: 0; }}
        .form-label, label {{ color: {FG} !important; }}
        .form-range {{ accent-color: steelblue; width: 100%; }}
        .info-box {{ background: #152119; border-radius: 6px;
                    padding: 10px 14px; margin: 8px 0;
                    border-left: 3px solid steelblue;
                    font-family: monospace; font-size: 0.9em; }}
    """),
    ui.h5("Histogram builder: The Wandering Fork's daily revenue",
          style=f"color:{FG}"),
    ui.input_radio_buttons(
        "data", "Data:",
        {key: label for key, (label, _) in DATASETS.items()},
        selected="harbour30", inline=True,
    ),
    ui.input_slider("bin_width", "Bin width (EUR):",
                    min=25, max=400, value=100, step=25),
    ui.output_ui("info"),
    ui.output_plot("plot", height="340px"),
)


def server(input, output, session):
    @output
    @render.ui
    def info():
        label, values = DATASETS[input.data()]
        w = input.bin_width()
        n = len(values)
        k = len(make_bins(values, w)) - 1
        sturges = 1 + 3.3 * log10(n)
        suggested = (max(values) - min(values)) / sturges
        return ui.HTML(f"""
            <div class='info-box'>
                n = <strong style='color:darkorange'>{n}</strong> days |
                bin width = <strong style='color:darkorange'>{w}</strong> EUR
                &rarr; <strong style='color:darkorange'>{k}</strong> classes<br>
                Sturges suggests about {sturges:.1f} classes, i.e. a width
                near {suggested:.0f} EUR.
            </div>
        """)

    @output
    @render.plot
    def plot():
        label, values = DATASETS[input.data()]
        return draw(values, input.bin_width(), label)


app = App(app_ui, server)

Exercises

Exercise 1 (by hand) — Reading the cumulative curve (click for solution)

Use the worked-example table (Harbour, first 30 days). What percentage of trading days earned at least €700?

Solution. The curve gives $F^{+}(700) = 66.7\%$ of days below €700, so at least €700 means $100\% - 66.7\% = 33.3\%$. Check with the counts directly: $(7 + 3)/30 = 33.3\%$.

Exercise 2 (by hand) — Unequal class widths (click for solution)

Suppose the last two classes are merged into one class [700, 900[ of width 200, which holds $7 + 3 = 10$ days. The first class [400, 500[ has width 100 and holds 6 days. What heights should the two bars have so that their areas are comparable? Which bar looks taller?

Solution. Use the density $n_i/h_i$.

[400, 500[: $6/100 = 0.06$ days per euro.
[700, 900[: $10/200 = 0.05$ days per euro.

The merged bar is drawn at height $0.05$ — about 83% of the height of the first bar, even though it holds more days (10 versus 6), because it is twice as wide. Had we plotted the raw count 10, the merged class would look almost twice as tall as it should.

Exercise 3 (by hand) — Polygon points (click for solution)

What is the midpoint of the class [600, 700[, and which point does the frequency polygon plot for it? What is the point for the cumulative curve at the end of that class?

Solution. The midpoint is $(600 + 700)/2 = 650$ and $n_{[600,700[} = 8$, so the polygon plots (650, 8). The cumulative curve plots the upper bound against the running share: (700, 66.7%).

Exercise 4 (Excel) — A skewed distribution (click for solution)

In wandering-fork.xlsx, build a class table for the Campus daily_revenue_eur over all 120 days with classes of width 200 starting at 0. Use COUNTIFS on tblFork. Add relative and cumulative frequencies. What share of days earned less than €600, and how does the shape differ from Harbour’s?

Solution. The formula for the class in row 2 is:

=COUNTIFS(tblFork[franchise],"Campus",
          tblFork[daily_revenue_eur],">="&R2,
          tblFork[daily_revenue_eur],"<"&S2)

You should obtain:

Class (€)	$n_i$	$N_i^{+}$	$f_i$ (%)	$F_i^{+}$ (%)
[0, 200[	25	25	20.8	20.8
[200, 400[	37	62	30.8	51.7
[400, 600[	30	92	25.0	76.7
[600, 800[	11	103	9.2	85.8
[800, 1000[	9	112	7.5	93.3
[1000, 1200[	5	117	4.2	97.5
[1200, 1400[	2	119	1.7	99.2
[1400, 1600[	1	120	0.8	100.0

76.7% of Campus days earned less than €600. The tallest class is [200, 400[ (37 days), and more than three days in four sit below €600, but eight days (6.7%) earned €1,000 or more, up to €1,407.59. The histogram has a tall peak on the left and a long tail to the right. Harbour’s revenue is more compact, with only a mild tail. We will give this shape a name in Module 7.

Exercise 5 (Excel) — Group with a PivotTable (click for solution)

Using all 120 Harbour days, build the class table with a PivotTable that groups daily_revenue_eur from 0 to 1,100 by steps of 100. What share of days earned between €500 and €700?

Solution. Filter franchise = Harbour, put daily_revenue_eur in Rows and day in Values (set to Count), then right-click → Group (Starting at 0, Ending at 1100, By 100). The counts are:

Class (€)	0–99	100–199	200–299	300–399	400–499	500–599
Days	9	11	17	24	21	10

Class (€)	600–699	700–799	800–899	900–999	1000–1099
Days	16	7	3	0	2

They add up to 120 (the PivotTable does not show the empty 900–999 group). Days between €500 and €700 are the classes 500–599 and 600–699: $(10 + 16)/120 = 21.7\%$.

Common mistakes

Watch out

Plotting the raw count for unequal widths. Use the density $n_i/h_i$, otherwise wide classes look too large.
Mixing up the plotting positions. Polygon points use the class midpoint; the cumulative curve uses the class upper bound.
Overlapping or gappy classes. Write them half-open, $[L, L+h[$, so each value belongs to exactly one class.
Too few or too many classes. Two classes hide the shape; 40 classes on 30 observations draw noise. Start from Sturges’ rule and adjust.
Leaving gaps between histogram bars. The variable is continuous: bars touch. Gaps belong to bar charts of categories (Module 2).

Source	Where	What it adds	Time	Verified
OpenStax IBS 2e	§2.1 Display Data, the parts on histograms, frequency polygons and time series graphs (book pages 46–65)	A second explanation of class tables and histograms with more examples	about 20 pages	checked against the PDF contents (T046)
Khan Academy	Unit Displaying and comparing quantitative data in the Statistics and probability course; look for the lessons on histograms and cumulative graphs	Short videos and practice questions on reading and building histograms	about 30–45 min (estimate)	link check pending (T048)