L-DS-4: Solutions — Variability and Spread

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

The numeric items in this section regenerate. Your lesson instance shows its own complete arithmetic after you commit an answer; the solutions below state the invariant method and the feedback targets for every current problem shape.

Problem 1 — Range and Outlier Sensitivity (C1)

For the generated dataset:

1. identify the minimum;
2. identify the maximum; and
3. subtract once.

The typed answer is the difference, not the maximum or minimum itself. The range uses only the two extremes, so one far-off observation can change it dramatically while the middle observations stay fixed.

Problem 2 — Sample Variance and Standard Deviation (C2 + C3)

The stepped generator uses this four-stage schedule:

1. compute ;
2. compute ;
3. divide by to obtain ; and
4. take the square root to obtain .

The mean still divides by . Bessel’s correction changes only the variance denominator. For independent observations drawn from the same population with finite variance, that correction makes unbiased for the population variance. Variance is in squared units; standard deviation returns to the data’s original units.

Problem 3 — Five-Number Summary and IQR Method (C4)

For each generated list:

1. sort the data;
2. find Q2;
3. split the list into lower and upper halves, excluding Q2 when is odd;
4. take the median of each half to obtain Q1 and Q3; and
5. subtract.

The five-number summary is , Q1, Q2, Q3, . The IQR is the width of the middle 50%, not the full range.

Problem 4 — Coefficient of Variation (C6)

The generated comparison asks for one CV as a typed percentage:

Divide the standard deviation by the matching mean, multiply by 100, and use the requested rounding. The larger CV identifies the variable with more spread relative to its own mean. Raw SD values with different units or scales are not directly comparable.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — Compute Sample Variance and Standard Deviation (C2 + C3)

The generated prompt asks for the sample standard deviation. Its full reveal shows the current data’s mean, deviation table, sum of squared deviations, sample variance, and final square root:

The two principal wrong-method values are the population SD obtained by dividing by and the variance obtained by stopping before the square root.

Problem 2 — Find the Five-Number Summary and IQR (C4)

Use the course’s exclusive median-of-halves convention on the generated sorted list: when is odd, omit the overall median before finding Q1 and Q3. The requested typed answer is

The reveal also reports the full five-number summary. A position-only quartile rule is a distractor because it is not the convention used in this course.

Problem 3 — Identify Outliers Using the Fence Method (C4 + C5)

Compute Q1, Q3, and IQR, then enter the fence pair:

Values outside the interval are potential outliers. They warrant investigation; the rule does not authorize automatic deletion.

Problem 4 — Synthesize Resistant and Non-Resistant Spread (C1 + C4 + C5)

For :

The median is . The lower half is , so ; the upper half is , so . Therefore:

The value 25 is above 11.25, so it is a potential outlier that warrants investigation. For the requested sensitivity check only, remove 25. The range becomes , while the IQR becomes . The range drops from 21 to 4; the IQR drops only from 2.5 to 2. This contrast shows why range is much more sensitive to an extreme value. It does not justify deleting 25 from a real analysis.

Problem 5 — Choose a Spread Measure from Context (C1 + C3 + C4 + C6)

1. Two symmetric mutual-fund return distributions in the same units: use standard deviation.
2. Strongly right-skewed household income: use IQR with the median.
3. Lizard length in cm versus mass in g: use CV.
4. Strongly right-skewed emergency waiting time: report median and IQR.
5. Same component made at typical lengths near 10 mm and 50 mm: when the stated target is variability relative to typical length, use CV. Absolute millimetre variability would use SD, while conformance would require engineering tolerances and a capability measure.

Problem 6 — Diagnose a Spread Calculation (C2 + C3)

Each regenerated worked solution contains exactly one named error from the current taxonomy:

• dividing by instead of for a sample variance;
• failing to square deviations before summing; or
• reporting as though it were .

Locate the first step where the work becomes invalid and repair that step. A changed numeric dataset does not change the diagnostic rule.

Mixed Review 1 — Central Tendency in a Skewed Distribution

Use the median for the typical home price. The $1.2M sale creates a long right tail and pulls the mean upward. Calling the mean “average” without showing the skew can be technically correct but practically misleading.

Mixed Review 2 — Histogram Shape and Measure Selection

The distribution is right-skewed because most observations occupy the lower-hour classes and a sparse tail extends to 41–45 hours. Report the median: the heavy-user values pull the mean to the right. A mean of 11.2 above a median of 8.5 is consistent with that shape.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Question 1 — Feynman Test (C2)

A sample uses its own mean as the centre, and that mean is fitted to the same observations. This makes the squared deviations a little too small, on average, for the population we want to describe. For independent observations drawn from the same population with finite variance, dividing by instead of makes the sample variance unbiased for the population variance. The correction matters most for small samples and becomes tiny for large ones.

Question 2 — Novel Application: Choose and Defend the Evidence

For Neighbourhood A, use the mean ($465) and standard deviation ($38): the distribution is symmetric with no outliers.

For Neighbourhood B, use the median ($438) and IQR ($74): luxury rentals pull the mean upward and inflate squared deviations.

Neighbourhood B has the wider typical middle spread because $74 exceeds $51. These summaries inform the decision but do not measure housing hardship or determine office placement by themselves.

Question 3 — Error Analysis: Variance vs. Standard Deviation

This question regenerates, so use the feedback attached to your lesson instance for its exact dataset. Every version contains one of these three first breaks:

1. Dividing by : the first invalid step divides the sum of squared deviations by . For a sample variance, divide by .
2. Using absolute deviations: the first invalid step makes deviations positive with absolute values instead of squaring them. Absolute deviation and squared deviation are different quantities.
3. Reporting variance as standard deviation: the variance step is correct, but the next step labels as . Take the square root of the exact variance before rounding, and report the result in the data’s original unit.

Downstream arithmetic can be internally consistent with an earlier mistake. Identify the first step where the reasoning changes to the wrong method, then repair that step without reusing a rounded intermediate.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

🔬 Path A — The Analyst

Task 1 — Comparable spread evidence

Engineering range:

Engineering IQR:

Engineering CV:

Marketing range:

Marketing IQR:

Marketing CV:

Marketing’s range, SD, and CV are much larger, but its IQR is slightly smaller. That pattern indicates that one extreme value is dominating the non-resistant measures.

Task 2 — Fence check

Engineering fences:

All Engineering values from 58 to 71 lie inside the fences.

Marketing fences:

The $95K observation exceeds 61.5 and is a potential outlier. Investigate it rather than deleting it automatically.

Task 3 — Limited HR conclusion

The data support a conclusion about observed salary dispersion: Engineering is more tightly clustered by SD and CV, while Marketing’s middle 50% is slightly tighter by IQR and one high value drives its non-resistant spread. Spread alone does not establish pay equity. A defensible equity analysis also needs like-for-like role, level, tenure, hours, performance criteria, and legally/privacy-reviewed demographic evidence.

🏗️ Path B — The Architect

Task 1 — Relative total-mass variability

Product S CV:

Product L CV:

Product L has the larger absolute SD but the smaller CV, so its total mass is slightly more consistent relative to its mean.

Task 2 — Fence rule

Product S IQR:

Product S fences:

Its observed 244.1–256.8 mg range is inside the fences.

Product L IQR:

Product L fences:

The 487.0 mg observation is below the lower fence and is flagged for record, equipment, and process investigation under the approved QC procedure.

Task 3 — Protocol boundary

Show mean and SD for routine total-mass consistency, CV for comparing the two scales, and median/IQR plus fence flags so an unusual tablet cannot hide inside an average. Document each investigation before recomputing summaries.

Low total-mass variability does not prove active-ingredient content uniformity. Total mass is a different measured variable; active-ingredient content requires its own validated assay and acceptance process.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Challenge 1 — Why the deviations sum to zero

Start from the definition of the sample mean:

Then

Because is constant within the sum,

Therefore,

Once deviations are known, the last is forced to make the sum zero. A sample needs at least for to be defined.

Challenge 2 — Comparing Spread Across Different Units with CV

For the generated variable requested in the prompt:

The inline reveal supplies the current instance’s arithmetic and identifies which variable is relatively more variable. A raw SD comparison is invalid when the units or meaningful scales differ.

Challenge 3 — The Unit-Invariance of CV

For centimetres:

For metres:

Changing units multiplies both the mean and SD by the same positive conversion factor, so their ratio and the CV do not change.