L-DS-3: Solutions — Central Tendency Measures

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 — Computation of the Arithmetic Mean (C1)

The generator supplies a fresh list of 5–7 values. For every version:

The first step checks the sum; the second checks division by the number of observations. Common errors are omitting one value, dividing by , reporting the median, or reporting the sum.

Problem 2 — Median Localization and Even/Odd Sample Sizes (C3)

Every generated version must be sorted before using its position:

The stepped version checks the position rule first and the resulting value second. Its solo mode asks for the complete result without those staged prompts.

Problem 3 — Influence of Skewness on Central Measures (C5)

• Household income, right-skewed: median.
• Symmetric petal lengths without outliers: use the mean as the primary summary; the median is also reasonable and should be similar.
• Preferred study location, nominal: mode.
• Marathon times with a low outlier: median.
• Symmetric absence counts without outliers: use the mean as the primary summary; the median is also reasonable and should be similar.

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 — Diagnosing Skewness via Central Measure Discrepancies (C2 + C5)

The mean sale price is greater than the median:

The histogram supplies the decisive shape evidence: its sparse tail extends toward multi-million-dollar homes, so the distribution is right-skewed. Those high values explain why the mean is above the median in this dataset. The inequality by itself would be consistent with right skew, but would not prove the shape or identify the mode.

Problem 2 — Defending Measure Choices for Skewed Environments (C2 + C4 + C5)

• Knee-surgery stays, right-skewed: median.
• Symmetric light-bulb lifetimes without outliers: use the mean as the primary summary; the median is also reasonable and should be similar.
• Streaming genre, nominal: mode.
• Mercury levels, right-skewed: median.
• Symmetric workout counts without outliers: use the mean as the primary summary; the median is also reasonable and should be similar.

Problem 3 — Algorithmic Computation of the Arithmetic Mean (C1)

For every generated list:

The answer is rounded to two decimal places. The feedback distinguishes dividing by , reporting the median, and reporting the midrange.

Problem 4 — Algorithmic Sorting and Location of the Median (C3)

Sort the generated list. Use the single middle value for odd , or average the two middle values for even . The generator deliberately presents the list unsorted.

Problem 5 — Algorithmic Identification of the Mode (C4)

Tally each generated value. Report the value with the greatest count. In the generator’s no-mode branch, every value occurs once, so at least two distinct values tie at the same frequency and none stands out. More generally, under this course’s convention, report no mode when at least two distinct values are present and every distinct value ties at the same frequency. A constant dataset is unimodal: its sole distinct value is the mode. Some formal definitions instead report every value as tied when multiple distinct values share the same frequency. The generator does not use the value’s frequency as the mode itself.

Problem 6 — Diagnosing a Flawed Central-Tendency Analysis (C2 + C3 + C5)

Each generated version shows eight sorted observations with one high outlier and exactly one flawed step. The error rotates among three authentic mistakes:

1. Shape error: calling the distribution symmetric after ignoring the separated high value. The outlier creates a long right tail, so the mean exceeds the median.
2. Median error: choosing only the lower of the two middle values. With even , average positions 4 and 5.
3. Selection error: choosing the mean merely because it uses every value. For the skewed dataset, the resistant median is the more representative centre.

The first faulty step changes across generated versions, so diagnose the displayed reasoning rather than memorizing a position.

Mixed Review 1 — Reading a Frequency Distribution (L-DS-2)

The generator rotates among three tables. Match the current table to the corresponding branch below.

Fitness-centre branch

The two sides of the central class are fairly balanced, so mean approximately equal to median is the defensible expectation. As a check on that shape judgment, class midpoints give an estimated mean of 9.70 visits and grouped interpolation gives an estimated median of about 9.64 visits.

Screen-time branch

The frequencies taper toward the higher classes, so the table is right-skewed and the mean is expected to be greater than the median.

Exam-score branch

The frequencies taper toward the lower classes, so the table is left-skewed and the mean is expected to be less than the median.

In every branch, grouped classes hide the exact observations inside each interval. The table supports a shape-based expectation, not exact values for either centre.

Mixed Review 2 — Variable Type and Appropriate Measure (L-DS-1)

The generator rotates among three contexts:

• School survey: GPA is quantitative, so a mean is meaningful; favourite academic subject is nominal, so its applicable centre is the mode. For the left-skewed GPA distribution, the median better represents a typical student.
• Commuter survey: One-way commute time is quantitative, so a mean is meaningful; usual transportation mode is nominal, so its applicable centre is the mode. For the right-skewed commute-time distribution, the median better represents a typical commute.
• Mobile-provider survey: Monthly data use is quantitative, so a mean is meaningful; phone brand is nominal, so its applicable centre is the mode. For the right-skewed data-use distribution, the median better represents a typical customer.

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

A strong explanation says that the mean shares the total equally, the median is the middle of a sorted list, and the mode identifies the value or category that stands out at the highest frequency. It also explains that outliers pull the mean, the median is resistant, and nominal categories support the mode but not arithmetic.

Question 2 — Novel Application: Delivery-Time Report

The delivery-time dataset regenerates. For the current instance:

1. Add all 12 delivery times.
2. Divide the sum by 12 to compute the mean.
3. Sort the values and average positions 6 and 7 to compute the median.
4. Lead with the median because the single unusually long delivery creates a right tail and pulls the mean upward.

Representative example: If the generated values are , then:

In this representative example, the 85-minute delivery creates a long right tail, so the advertisement should lead with the 24.5-minute median. The mean is a valid secondary summary, but it is not the best description of a typical delivery.

Question 3 — Error Analysis: The Wrong Centre

This question uses the same rotating generator as Independent Practice Problem 6. Every version contains eight sorted observations: seven consecutive clustered values and one separated high value. Exactly one of these errors appears:

1. Shape error: Calling the distribution symmetric ignores the separated high value. It creates a long right tail, so the mean exceeds the median.
2. Median error: With even , the median is not position 4 alone. Average positions 4 and 5.
3. Selection error: Using every magnitude does not automatically make the mean representative. For this right-skewed dataset, the resistant median is the better typical value; the mean can still be reported second.

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

The CAD 450K CEO salary creates a long right tail. Seven of ten employees earn below the CAD 128.3K mean, and six earn below CAD 100K. HR should lead with the CAD 86.5K median and explain the salary range and role differences.

Path B — The Architect

The 92-minute commute creates a right tail. The report should lead with the 26-minute median as the typical commute and include the 32.2-minute mean second to show the system-wide effect of long commutes.

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 — Trimmed Mean

The lesson uses the floor rule:

For and , . Remove 42 and 95.

As the trim approaches 50%, the trimmed mean approaches the median. For , a 10% trim removes observations from each end; a 25% trim removes two from each end. A judging system that removes its highest and lowest scores before averaging the rest uses a trimmed mean.

The regenerable rematch always provides 10 sorted scores. It removes exactly one score from each end, sums the remaining eight, and divides by 8.

Challenge 2 — Least-Squares Centre

For , .

The smaller value at illustrates the general derivative proof that the mean uniquely minimizes squared error.

Challenge 3 — Why Centre Is Not Enough

Both and have mean 50, but the second dataset is much more spread out. In both:

L-DS-5 introduces variance, standard deviation, and IQR so spread does not disappear through cancellation.