Module 6 — Dispersion

Data Description | Part II — Summarising one variable

Modified

October 8, 2026

The business question. “The Wandering Fork” has a mean daily revenue of €628.34 (Harbour, 30 days) — but two businesses can share the exact same mean and still be very different to run: one reliably steady, the other wildly erratic. Dispersion indicators measure exactly that variability, and they are what turns “the average day” into “the average day, ± how much”.

Learning objectives

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

Compute the range and the interquartile range.
Compute the sample variance and sample standard deviation, using both the definitional and the shortcut (computational) formula.
Compute and interpret the coefficient of variation.
Use the interactive calculator below to see how editing a sample changes its dispersion indicators live.

Time plan

Block	Minutes	What you do
Theory	75	Range, IQR, sample variance, sample standard deviation, coefficient of variation, shortcut formula
Excel lab	50	`VAR.S`, `STDEV.S`, `VAR.P`, `STDEV.P`, `SUMSQ`, CV formula
Exercises	55	Three by-hand and three Excel exercises
Total	180

Theory

Range and interquartile range

Definition — range and interquartile range

The range is \(\max - \min\): simple, but sensitive to a single extreme value. The interquartile range \(IQR = Q_3-Q_1\) (Module 5) measures the spread of only the central 50% of the data, making it more robust to extreme values.

In plain words: the range uses only the two most extreme values; the IQR uses the middle 50%, so a single outlier cannot distort it.

Sample variance and standard deviation

Formula — sample variance and standard deviation

For a sample of \(n\) values \(x_1,\dots,x_n\) with mean \(\bar{x}\), the sample variance is \[ s^2 = \frac{1}{n-1}\sum_{i=1}^{n} (x_i-\bar{x})^2, \] i.e. the average squared distance of each observation from the mean (with \(n-1\), not \(n\), correcting for the fact that \(\bar{x}\) is itself estimated from the same sample). The sample standard deviation \(s=\sqrt{s^2}\) returns to the original unit (€, not €²).

In plain words: measure how far each observation is from the mean, square those distances (so they are all positive), average them (with \(n-1\)), and take the square root to get back to the original unit.

Why \(n-1\) and not \(n\)?

The sample mean \(\bar{x}\) is computed from the same data, so the deviations \((x_i-\bar{x})\) are not fully independent — they sum to zero. Using \(n-1\) (Bessel’s correction) makes \(s^2\) an unbiased estimator of the true population variance. In this descriptive statistics course, we use the sample formulas (.S) by default because real business data are almost always samples. The population variants (.P) are shown for completeness.

Coefficient of variation

Formula — coefficient of variation

\[ CV = \frac{s}{\bar{x}} \times 100\%. \] Because it is unit-free, \(CV\) is the only dispersion indicator that lets you compare the relative variability of two variables measured on different scales or units (e.g., revenue in € vs. customer count).

In plain words: divide the standard deviation by the mean and express as a percentage. A \(CV\) of 20% means the typical deviation from the mean is about 20% of the mean itself.

Shortcut (computational) formula for variance

Formula — variance shortcut

\[ s^2 = \frac{1}{n-1}\left(\sum_{i=1}^{n} x_i^2 - n\bar{x}^2\right). \] This needs only \(\sum x_i^2\) and \(\bar{x}\) — one pass over the data instead of first computing every deviation \((x_i-\bar{x})\) and then squaring it. In Excel, use SUMSQ(range) for \(\sum x_i^2\).

Worked example

Using “The Wandering Fork”’s daily_revenue_eur (\(n=30\), \(\bar{x}=628.34\)), verified in data-model.md:

Statistic	Value
Range	478.97
Sample variance \(s^2\)	18,793.33
Sample std. dev. \(s\)	137.09
Coefficient of variation \(CV\)	21.82%

Step 1 — range. \(886.56 - 407.59 = 478.97\): over the 30-day period, the best day out-earned the worst by almost €480.

Step 2 — variance and standard deviation. Squaring each day’s deviation from \(\bar{x}=628.34\), summing, and dividing by \(n-1=29\) gives \(s^2 = 18{,}793.33\) €²; taking the square root gives \(s = 137.09\) €. On a typical day, revenue is about €137 away from the €628.34 average, in either direction.

Step 3 — coefficient of variation. \(CV = 137.09/628.34 \times 100\% = 21.82\%\) — using the guide below, a moderate level of relative variability: noticeably variable, but not wildly erratic day to day.

Step 4 — shortcut formula check. \(\sum x_i^2 = 12{,}389{,}403.98\), \(n\bar{x}^2 = 30 \times 628.3417^2 = 11{,}844{,}397.50\), difference \(= 545{,}006.48\), divided by \(29 = 18{,}793.33\) — identical to the definitional result. Keep the mean unrounded here: with \(\bar{x}=628.34\) the result drifts to about \(18{,}795\), because the shortcut subtracts two very large, nearly equal numbers.

Interpreting \(CV\)

\(CV\) range	Interpretation
\(CV < 15\%\)	Low relative variability — data is tightly clustered around the mean
\(15\% \le CV \le 35\%\)	Moderate relative variability — noticeable but not extreme
\(CV > 35\%\)	High relative variability — data is widely spread relative to the mean

“The Wandering Fork”’s \(CV = 21.82\%\) falls in the moderate band.

Using Excel

The complete list of function names, with their French equivalents, is on the Excel functions page.

Concept	Excel function / steps
Range	`=MAX(range)-MIN(range)`
Sample variance	`=VAR.S(range)`
Sample standard deviation	`=STDEV.S(range)`
Population variance	`=VAR.P(range)`
Population standard deviation	`=STDEV.P(range)`
Coefficient of variation	`=STDEV.S(range)/AVERAGE(range)`
Interquartile range	`=QUARTILE.EXC(range,3)-QUARTILE.EXC(range,1)`
Sum of squares (for shortcut)	`=SUMSQ(range)`

Sample vs population: This is a descriptive statistics course. We use the sample formulas (.S) by default because real business data are almost always samples. The population variants (.P) are shown for completeness.

For the full table (120 days per franchise) use the Excel Table tblFork and add the franchise as one more condition, for example =VAR.S(FILTER(tblFork[daily_revenue_eur], tblFork[franchise]="Harbour")).

Proof / derivation

Derivation — the variance shortcut (computational) formula (click to expand)

Starting from the definitional formula and expanding the square: \[ \sum_{i=1}^{n}(x_i-\bar{x})^2 = \sum_{i=1}^n x_i^2 - 2\bar{x}\sum_{i=1}^n x_i + n\bar{x}^2. \] Since \(\sum_{i=1}^n x_i = n\bar{x}\), the middle term becomes \(-2\bar{x}(n\bar{x}) = -2n\bar{x}^2\), so \[ \sum_{i=1}^{n}(x_i-\bar{x})^2 = \sum_{i=1}^n x_i^2 - 2n\bar{x}^2 + n\bar{x}^2 = \sum_{i=1}^n x_i^2 - n\bar{x}^2. \] Dividing by \(n-1\) gives the shortcut formula \[ s^2 = \frac{1}{n-1}\left(\sum_{i=1}^{n} x_i^2 - n\bar{x}^2\right), \] which needs only \(\sum x_i^2\) and \(\bar{x}\) — one pass over the data instead of first computing every deviation \((x_i-\bar{x})\) and then squaring it. Both formulas give the identical result, \(s^2=18{,}793.33\), for “The Wandering Fork”; the shortcut form is simply faster to compute by hand or with SUMPRODUCT/SUMSQ in Excel.

Visual intuition

Variance is literally the average squared distance from the mean: picture each of the 30 days as a point on a number line at its revenue value, and \(\bar{x}=628.34\) as a fixed pivot. Variance answers “on average, how far (squared) do the points sit from the pivot?” — a wide, flat spread of points gives a large variance; a tight cluster around the pivot gives a small one.

Interactive demo — central tendency & dispersion calculator

Edit the six values below and watch the mean, variance, standard deviation, and coefficient of variation update live. Starting values are a 6-day sample of “The Wandering Fork”’s daily_revenue_eur.

#| standalone: true
#| components: [viewer]
#| viewerHeight: 560

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

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

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 {{ 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("🧮 Central tendency & dispersion calculator", style=f"color:{FG}"),
    ui.p(
        "Move the sliders to change a 6-day revenue sample; the mean, "
        "variance, standard deviation, and CV recompute live.",
        style=f"color:{FG}; font-size:0.88em; margin-bottom:6px;",
    ),
    ui.input_slider("d1", "Day 1 (€):", min=300, max=950, value=460.48, step=1),
    ui.input_slider("d2", "Day 2 (€):", min=300, max=950, value=500.02, step=1),
    ui.input_slider("d3", "Day 3 (€):", min=300, max=950, value=418.65, step=1),
    ui.input_slider("d4", "Day 4 (€):", min=300, max=950, value=785.79, step=1),
    ui.input_slider("d5", "Day 5 (€):", min=300, max=950, value=719.66, step=1),
    ui.input_slider("d6", "Day 6 (€):", min=300, max=950, value=886.56, step=1),
    ui.output_ui("info"),
    ui.output_plot("plot", height="280px"),
)

def server(input, output, session):
    def sample():
        return np.array([input.d1(), input.d2(), input.d3(), input.d4(), input.d5(), input.d6()], dtype=float)

    @output
    @render.ui
    def info():
        x = sample()
        mean = x.mean()
        median = float(np.median(x))
        var = x.var(ddof=1)
        std = x.std(ddof=1)
        cv = std / mean * 100 if mean else 0
        return ui.HTML(f"""
            <div class='info-box'>
                Mean = <strong style='color:darkorange'>{mean:.2f}</strong> &nbsp;|&nbsp;
                Median = <strong style='color:darkorange'>{median:.2f}</strong><br>
                Variance (s²) = {var:.2f} &nbsp;|&nbsp;
                Std. dev. (s) = {std:.2f} &nbsp;|&nbsp;
                CV = {cv:.2f}%
            </div>
        """)

    @output
    @render.plot
    def plot():
        x = sample()
        mean = x.mean()

        fig, ax = plt.subplots(figsize=(7, 3.2))
        fig.patch.set_facecolor(BG)
        ax.set_facecolor(BG)
        for spine in ax.spines.values():
            spine.set_edgecolor(FG)
        ax.tick_params(colors=FG)

        days = np.arange(1, 7)
        ax.scatter(days, x, color='steelblue', zorder=5, s=70)
        ax.axhline(mean, color='darkorange', lw=1.8, linestyle='--', label=f'Mean = {mean:.2f}')
        for d, v in zip(days, x):
            ax.plot([d, d], [mean, v], color=FG, lw=1, alpha=0.5)
        ax.set_xticks(days)
        ax.set_xlabel("Day", color=FG)
        ax.set_ylabel("Revenue (€)", color=FG)
        ax.set_title("Deviations from the mean", color=FG)
        ax.legend(fontsize=8.5, facecolor=BG, edgecolor='none', labelcolor=FG)
        ax.grid(alpha=0.15, color=FG)
        plt.tight_layout()
        return fig

app = App(app_ui, server)

Exercises

Exercise 1 (by hand) — Shortcut formula (click for solution)

A sample of \(n=5\) values has \(\sum x_i = 100\) and \(\sum x_i^2 = 2140\). Find the sample variance.

Solution. \(\bar{x}=100/5=20\). Shortcut formula: \(s^2 = \frac{1}{n-1}\left(\sum x_i^2 - n\bar{x}^2\right) = \frac{1}{4}(2140 - 5\times400) = \frac{1}{4}(2140-2000) = 35\).

Exercise 2 (by hand) — Comparing variability across units (click for solution)

“The Wandering Fork”’s items_sold has mean \(74.40\) and sample standard deviation \(16.22\) (from data-model.md). Which variable — daily_revenue_eur or items_sold — has higher relative variability?

Solution. \(CV_{\text{revenue}} = 137.09/628.34 = 21.82\%\); \(CV_{\text{items}} = 16.22/74.40 = 21.80\%\). The two are almost identical (about 21.8%) even though the raw units (€ vs. items sold) are completely different — exactly the kind of comparison only \(CV\), not \(s\) alone, can make fairly.

Exercise 3 (by hand) — Effect of an outlier on the range (click for solution)

If one exceptionally bad day of \(\text{daily\_revenue\_eur}=150\) replaced the current minimum (407.59), how would the range change, and would the standard deviation be affected in the same direction?

Solution. New range \(= 886.56 - 150 = 736.56\) (up from 478.97) — the range, using only two values, reacts strongly to a single extreme point. The standard deviation would also increase (an extra-large deviation from the mean adds to \(\sum(x_i-\bar{x})^2\)), but by a much smaller relative amount, since it accounts for all 30 days rather than just the two most extreme.

Exercise 4 (Excel) — Dispersion for Campus revenue (click for solution)

In wandering-fork.xlsx, filter franchise to “Campus” and compute the range, sample variance, sample standard deviation, and \(CV\) for daily_revenue_eur (120 days). How do they compare to Harbour’s?

Solution. Campus 120-day: range = 1383.74, \(s^2 = 94{,}511.80\), \(s = 307.43\), \(CV = 67.56\%\). Over the same 120 days, Harbour has range = 998.79, \(s = 216.01\), \(CV = 52.24\%\). Campus has higher absolute and relative dispersion than Harbour (range 1384 vs 999, \(CV\) 67.6% vs 52.2%) — consistent with the right-skewed, event-driven revenue pattern.

Exercise 5 (Excel) — Shortcut formula with SUMSQ (click for solution)

Using the Harbour 30-day daily_revenue_eur data, verify the shortcut formula: compute =SUMSQ(range) and =AVERAGE(range), then apply the shortcut formula manually. Confirm it matches =VAR.S(range).

Solution. SUMSQ = 12,389,403.98, AVERAGE = 628.3417, \(n=30\). Shortcut: \((12{,}389{,}403.98 - 30 \times 628.3417^2) / 29 = 18{,}793.33\), matching VAR.S exactly (point to the AVERAGE cell rather than typing 628.34, or rounding shifts the result).

Exercise 6 (Excel) — Payroll skew: dispersion (click for solution)

Open payroll-skew.csv. Compute the range, sample standard deviation, and \(CV\) for the 11 salaries. Why is the \(CV\) so high?

Solution. Range = 154,834, \(s = 44{,}788\), \(CV = 94.8\%\). The \(CV\) is close to 100% because the standard deviation (driven by the CEO’s €180,000 salary) is almost as large as the mean (€47,231) — a hallmark of extreme right skew where the mean is pulled far above the bulk of the data.

Common mistakes

Watch out

Using VAR.P/STDEV.P when you need VAR.S/STDEV.S. The course standard is the sample formulas (denominator \(n-1\)). The population variants (denominator \(n\)) are for when you have the entire population — rare in business.
Forgetting the square root. Variance is in squared units (€²); standard deviation is in the original unit (€). Always report the one that matches your interpretation.
Computing \(CV\) with the median instead of the mean. \(CV = s/\bar{x}\), not \(s/Me\). The mean is the balance point that pairs with variance.
Using the range as your only dispersion measure. The range ignores everything between min and max; always report \(s\) or \(IQR\) alongside it.
Comparing standard deviations across different units. €137 vs 16 items — meaningless without \(CV\). Always use \(CV\) for cross-variable or cross-dataset comparisons.

Source	Where	What it adds	Time	Verified
OpenStax IBS 2e	§2.7 Measures of the Spread of the Data (book pages 82–90)	Sample variance, standard deviation, grouped formulas, Chebyshev, Empirical Rule, CV	~8 pages	checked against the PDF contents (T046)
Khan Academy	Unit Summarizing quantitative data in the Statistics and probability course; look for lessons on variance, standard deviation, and coefficient of variation	Short videos and practice on dispersion measures	~30–45 min (estimate)	link check pending (T048)