L-DS-4 · GUIDED INQUIRY Introduction to Probability and Statistics

L-DS-4 · Guided Inquiry

Same average. Different story.

Two players can share the same mean and still feel completely different to a coach.

What might an average hide?

Anonymous check

Which claim is most defensible?

Player A: 16, 17, 18, 19, 22. Player B: 8, 12, 18, 24, 30. Both players average 18.4 points.

  1. They are equally consistent because the means match

  2. The first player is more consistent despite the equal means

  3. The second player must have the larger mean

  4. Consistency cannot be described numerically

The inquiry map

What must an honest spread report answer?

Six ideas, each solving a limitation exposed by the one before it.

C1 Range

How wide is the full span?

C2 → C3 Variance → SD

How far are values from the mean, in usable units?

C4 CV

How large is spread relative to the mean?

C5 → C6 IQR → fences

What survives extremes, and what deserves investigation?

The final decision depends on purpose, shape, extremes, and scale.

C1 · Predict

What can one extreme value do to range?

Original data: \(10,\,11,\,12,\,13,\,14\), so the range is \(14-10=4\).

Discuss: Without changing the other four values, how much larger will the range become? Which observations control the answer?

Four values stay \(10,\,11,\,12,\,13\)

Most of the data do not move.

One value changes \(14\rightarrow40\)

Only the upper endpoint changes.

Revised range: \(40-10=?\)

C1 · Reveal and critique

One extreme makes the range jump from 4 to 30.

Four of the five observations are unchanged.

  1. 1 · Original
    \(10,\,11,\,12,\,13,\,14\)

    \(14-10=4\)

  2. 2 · Revised
    \(10,\,11,\,12,\,13,\,40\)

    \(40-10=30\)

  3. 3 · Effect
    \(4\text{ of }5\text{ values unchanged}\)

    Yet the range becomes 7.5 times as large.

Range is not resistant: one extreme value can dominate the entire measure.

Anonymous check

What is the average signed deviation from the mean?

A signed deviation is \(x-\bar{x}\), and its average is \(\frac{\sum(x-\bar{x})}{n}\). For 3.8, 5.3, 6.8, 8.3, 9.8 hours, \(\bar{x}=6.8\text{ hours}\).

  1. Exactly 0

  2. 1.5

  3. 1.8

  4. 3.0

C2 · Variance

Why square deviations?

Compare a 1-hour deviation with a 3-hour deviation.

Discuss: A 3-hour miss becomes 9 while a 1-hour miss becomes 1. What does squaring gain—and what does it cost?

  1. Why square?
    \(1^2=1,\qquad 3^2=9\)

    A three-hour miss counts nine times as much after squaring.

  2. Population
    \(\sigma^2=\frac{\sum(x-\mu)^2}{N}\)

    Use when the data are the entire population.

  3. Sample
    \(s^2=\frac{\sum(x-\bar{x})^2}{n-1}\)

    Use when the data are a sample.

Squaring prevents cancellation and emphasizes large departures; the tradeoff is squared units.

C2 · Derive

Build one sample variance together.

Data: \(3.8,\,5.3,\,6.8,\,8.3,\,9.8\text{ hours}\) with \(\bar{x}=6.8\text{ hours}\).

  1. 1 · Deviations
    \(-3.0\)\(-1.5\)\(0\)\(1.5\)\(3.0\)

    Distances balance around the mean.

  2. 2 · Squares
    \(9\)\(2.25\)\(0\)\(2.25\)\(9\)

    Each square sits directly below its deviation.

  3. 3 · Variance
    \(s^2=\frac{22.5}{5-1}=5.625\text{ h}^2\)

    Divide by \(n-1\) for a sample.

One computation per line makes the reasoning inspectable.

C3 · Standard deviation

Take the root to return home.

Standard deviation is the square root of variance.

Formula \(s=\sqrt{s^2}\)

The square root restores the original units.

Our data \(s=\sqrt{5.625}\approx2.37\text{ hours}\)

Scores typically sit about 2.37 hours from the mean.

SD uses every value and is sensitive to extremes.

C4 · Create the need

Did variability change—or only the unit?

The same bottle-fill data are reported first in grams, then in kilograms.

Discuss: Did the filling process become more consistent when we changed grams to kilograms? What should a unit-free comparison preserve?

In grams \(\bar{x}=100\text{ g},\ s=10\text{ g}\)

The SD is 10 grams.

In kilograms \(\bar{x}=0.100\text{ kg},\ s=0.010\text{ kg}\)

The SD is 0.010 kilograms.

The physical spread is identical even though the numerical SD changes with the unit.

C4 · Coefficient of variation

CV preserves the comparison when units change.

\(\mathrm{CV}=\frac{s}{\bar{x}}\times100\%\) is unit-free.

In grams \(\frac{10\text{ g}}{100\text{ g}}\times100\%=10\%\)

The units cancel.

In kilograms \(\frac{0.010\text{ kg}}{0.100\text{ kg}}\times100\%=10\%\)

The same physical spread gives the same CV.

A unit conversion changes the mean and SD together, so CV stays the same.

C4 · Apply the ratio

Now compare consistency relative to scale.

Two filling processes use the same unit but have different target amounts.

Discuss: Which filling process is less consistent relative to its own mean? Why is comparing 4 g with 10 g alone not enough?

Small fill \(\bar{x}=20\text{ g},\ s=4\text{ g}\)

\(\mathrm{CV}=\frac{4}{20}\times100\%=20\%\)

Large fill \(\bar{x}=100\text{ g},\ s=10\text{ g}\)

\(\mathrm{CV}=\frac{10}{100}\times100\%=10\%\)

The large fill has the larger SD, but the small fill has the larger variability relative to its mean.

C4 · Boundary

CV needs a meaningful zero.

\(\mathrm{CV}=\frac{s}{\bar{x}}\times100\%\) compares spread relative to the mean.

Use it Ratio scale

A true zero means absence, and the mean is safely away from zero.

Do not use it Arbitrary zero or mean near 0

Temperature in °C is the classic warning.

A percentage is not automatically meaningful; the scale must justify it.

C5 · Quartiles

Sort before splitting the data.

Course convention: median of the lower and upper halves.

  1. Sorted scores
    \(9,\,12,\,14,\,16,\,18,\,19,\,21,\,24,\,35\)

    \(Q_2=18\)

  2. Lower half
    \(9,\,12,\,14,\,16\)

    \(Q_1=\frac{12+14}{2}=13\)

  3. Upper half
    \(19,\,21,\,24,\,35\)

    \(Q_3=\frac{21+24}{2}=22.5\)

For odd \(n\), exclude the overall median before finding \(Q_1\) and \(Q_3\).

C5 · Interquartile range

IQR measures the resistant middle half.

Five-number summary: \((9,\,13,\,18,\,22.5,\,35)\).

Box-and-whisker plot from the five-number summary min = 9 Q₁ = 13 median = 18 Q₃ = 22.5 max = 35
Formula \(\mathrm{IQR}=Q_3-Q_1\)

A resistant measure of spread.

Our data \(22.5-13=9.5\text{ points}\)

The middle half spans 9.5 points.

Median pairs naturally with IQR; mean pairs naturally with SD.

C6 · Potential outliers

Fences flag values for investigation.

The whiskers stop at observed unflagged values; fences are decision boundaries.

Box-and-whisker plot with fences and a flagged value at 40 lower fence -1.25 upper fence 36.75 40 flagged
Lower fence \(13-1.5(9.5)=-1.25\)

Values below −1.25 are flagged.

Upper fence \(22.5+1.5(9.5)=36.75\)

Values above 36.75 are flagged.

A fence is a screening convention—not a deletion command.

Anonymous check

What should the analyst do first?

A recorded score lies above the calculated upper fence.

  1. Delete it automatically

  2. Replace it with the mean

  3. Investigate its accuracy and context

  4. Ignore every other observation

Decision tool

Match the measure to the evidence.

Purpose → shape and extremes → scale → measure → qualified claim.

Roughly symmetric; no influential extremes Mean + SD

Uses all values and preserves original units.

Skewed or affected by outliers Median + IQR

Resists extreme observations.

Compare valid ratio scales CV

Compares spread relative to a meaningful mean.

Quick full-span description Range

Readable, but fragile.

No spread measure is “best” without a purpose and a distribution.

Anonymous check

Which centre-and-spread pair best describes a typical listing?

Apartment rents are strongly right-skewed because a few luxury units are extremely expensive.

Then explain: Write 1–2 sentences that name the feature of the rent distribution supporting your choice and explain why the main alternative is less trustworthy.

  1. Mean and range

  2. Mean and SD

  3. Median and IQR

  4. Median and CV

Transfer · Solo, then pair

Which team was more consistent in these games?

Use the summaries below to write one evidence-based sentence.

Write: Identify the more consistent team, cite the same spread measure for both teams, and limit your conclusion to the observed games.

  1. Team North
    \(\bar{x}=18.4\text{ points},\ s=2.2\text{ points},\ \mathrm{IQR}=3\text{ points}\)

  2. Team South
    \(\bar{x}=18.4\text{ points},\ s=8.1\text{ points},\ \mathrm{IQR}=12\text{ points}\)

Commit alone, compare reasoning with a partner, then revise one sentence.

Transfer · Model

A complete comparison is short and bounded.

Both teams average 18.4 points, so centre does not separate them.

  1. Evidence
    2.2 points vs 8.1 points

    Team North has the smaller observed SD.

  2. Claim
    North was more consistent

    Its scores typically stayed closer to the common mean.

  3. Boundary
    Observed games only

    The summaries do not prove future performance.

Evidence, interpretation, and a boundary make an honest statistical statement.

Anonymous check

Which report is statistically defensible?

For a sample of delivery times, \(s^2=4.2\text{ min}^2\) and \(s\approx2.05\text{ min}\). One value above the upper fence is verified as a real weather delay.

  1. Typical distance is 4.2 min; delete the weather delay

  2. Typical distance is 2.05 min; ignore the delay because it is real

  3. Report \(s\approx2.05\text{ min}\); retain the verified delay and add median + IQR to show spread resistantly

  4. Replace the delay with the upper fence and report only the range

Optional enrichment · optional

Why does a sample lose one degree of freedom?

Once \(n-1\) deviations are known, the final deviation is forced.

Constraint \(\sum(x-\bar{x})=0\)

Deviations around the sample mean must balance.

Consequence \(\text{Only }n-1\text{ deviations vary freely}\)

Bessel’s correction offsets systematic underestimation.

Required learning: use \(n-1\) correctly. Enrichment: explain why the correction is needed.

Synthesis

Centre is only half the story.

A trustworthy spread report makes five deliberate choices.

1 Purpose

What decision must the summary support?

2 Shape and extremes

Do we need sensitivity or resistance?

3 Scale

Are original units or relative spread meaningful?

4–5 Measure and claim

Calculate correctly; interpret without overclaiming.

Range · variance · SD · CV · IQR · fences—six tools, one honest decision.

Anonymous exit reflection · confidence, not mastery

How confident are you with each idea?

First use the exit question as evidence. Then rate whether you could explain each idea—not merely recognize it.

C1

I can compute and critique a range.

C2

I can choose the population or sample variance formula.

C3

I can compute and interpret standard deviation in original units.

C4

I can use coefficient of variation only when its ratio is meaningful.

C5

I can find quartiles and interpret the interquartile range.

C6

I can use fences to investigate—not automatically delete—potential outliers.

0 Not yet 1 Very unsure 2 Beginning 3 Mostly understand 4 Confident 5 I could explain it