09/16 Lecture stat

Maple Center: Overview, Services, and Measures of Spread

  • Location and Identity

    • Maple Center stands for math, accounting, physics, and engineering. It also supports computer science and network IT.
    • Location: Student Affairs Building (SA), Room 202.
    • Not new, but a new location.

  • Contact and Hours

    • Hours:
    • Monday: 9:00–5:00
    • Tuesday–Thursday: 9:00–7:00
    • Friday: 9:00–4:00
    • Saturday: 10:00–3:00

  • Staffing

    • Besides the presenter, there are two other staff members, including a highly skilled statistician (referred to as the grand master of statistics).

  • What the Center Offers (for math, accounting, physics, engineering, CS/IT)

    • Free tutoring for all listed subjects.
    • Calculator lending for the semester (TI graphing calculators or scientific calculators). Free.
    • Textbooks available inside the center for use.
    • Computers available for student use, free of charge.
    • A printer in the center.
    • Printing from the student account: at the start of the semester, each student account gets $25. Printing cost:
    • Regular page:
      0.10=10 cents0.10 = 10\text{ cents}
    • Tutoring format:
    • Maple Center tutoring is walk-in; no appointment needed. You walk in, provide your MC (Maricopa College?) number, and study there during open hours.
    • Private study rooms in the Naples Center are first-come, first-served for quiet study (solo or group).
    • Virtual tutoring:
    • Starfish (formerly Academia).
    • Log in with MC credentials (same as RNC login).
    • Schedule a one-on-one tutoring appointment for up to an hour for virtual tutoring.
    • Virtual tutoring is by appointment; walk-in tutoring in the Maple Center is still available with no appointment.
    • Hiring at the Maple Center:
    • Students enrolled in Math 181 or other classes needing tutoring in higher-level math or CS can apply for paid tutoring positions.

  • Homework and Textbooks

    • If you have textbook issues, they are being handled.
    • You can turn in homework to the instructor; if not ready, you can submit later (e.g., Thursday).

  • Quick Q&A and Notices

    • Open floor for questions at the end of the session.

  • Preview of Today’s Topic: Measures of Spread

    • We are focusing on spread, a feature of distributions that helps identify data sources, data types, and characteristics of a dataset.
    • Previous focus (2.2) included recognizing symmetry of distributions.
    • Today’s focus includes:
    • Range (a rough measure of spread)
    • Quartiles (Q1 and Q3) and their role in understanding the middle 50% of data
    • The concept of the middle 50% captured by the interquartile range (IQR)
    • Five-number summary and box plots
    • Outliers and the 1.5 × IQR rule (to be covered in further depth on Thursday)

  • Definition: Range

    • The most basic measure of spread:
    • Range = max⁡(X)−min⁡(X)\max(X) - \min(X)
    • Note: It’s a rough measure; sensitive to outliers because it uses only the extremes.

  • Quartiles and Percentiles

    • Quartiles split data into four equal parts; Q1 is the first quartile (the 25th percentile), Q3 is the third quartile (the 75th percentile).
    • The median is sometimes called Q2.
    • Q1 and Q3 summarize the spread of the middle 50% of the data.
    • How to find Q1 and Q3 (using ordered data):
    • Arrange data in ascending order.
    • Q1 is the median of the lower half of the data; Q3 is the median of the upper half.
    • If the number of observations is odd, include the median appropriately when partitioning; if even, take the average of the two central values.

  • Example: Stem-and-Leaf Plot (New York commuter times)

    • Data: 20 observations representing commuter times in minutes (e.g., 10, 15, 85, etc.).
    • Task: Find the median (m), Q1, and Q3, and discuss the middle 50% of the data.
    • Process (as described in the session):
    • Order data in ascending order.
    • For an even number of observations (20), the median is the average of the 10th and 11th values:
      m=X<em>(10)+X</em>(11)2m = \frac{X<em>{(10)} + X</em>{(11)}}{2}
    • The instructor noted that this example yields a decimal median even though data are integers, illustrating robustness of the percentile concepts.
    • Notes from the shown example:
    • The speaker found the median to be a value around 22.5 minutes, illustrated by computing the average of the 10th and 11th values.
    • Q1 (first quartile) was found to be 15 minutes (the median of the lower half).
    • Q3 (third quartile) was discussed as a value in the 40s (e.g., 42 or 42.5 minutes depending on calculation).
    • The middle 50% of data lies roughly between Q1 and Q3 (e.g., between ~15 and ~42 minutes, depending on the exact calculation used in the example).
    • Observations:
    • The distribution appeared skewed to the right (long tail toward higher times).
    • The median was identified as a better measure of center than the mean in skewed data.

  • Five-Number Summary and Box Plots

    • Five-number summary (for the New York data example):
    • Min = 55
    • Q1 = 1515
    • Median (Q2) = 22.522.5
    • Q3 = 42.542.5
    • Max = 8585
    • This summary helps quickly describe center and spread and indicates potential skewness.
    • Box plot construction (described verbally):
    • Draw a box from Q1 to Q3.
    • Mark the median inside the box.
    • Draw whiskers from the box to the minimum and maximum, unless outliers are present.
    • When introducing multiple data groups, box plots can be used to compare distributions (e.g., commuter times by city).
    • Interpretation:
    • The gap between the whiskers and the quartiles indicates skewness. A larger gap between Q3 and the max compared to the gap between the min and Q1 suggests right-skewness.
    • Box plots provide a visual sense of symmetry vs. skewness and can highlight potential outliers.

  • Outliers and the Interquartile Range (IQR)

    • Interquartile Range (IQR):
    • IQR = Q<em>3−Q</em>1Q<em>3 - Q</em>1
    • It measures the spread of the middle 50% of the data and is resistant to outliers.
    • The 1.5 × IQR Rule for outliers: a data point x is considered an outlier if it falls outside the interval
    • Upper bound: Q3+1.5×IQRQ_3 + 1.5\times \mathrm{IQR}
    • Lower bound: Q1−1.5×IQRQ_1 - 1.5\times \mathrm{IQR}
    • Practical steps:
    • Compute IQR, then compute the upper and lower bounds using the formulas above.
    • Compare the data point(s) to these bounds to decide if they are outliers.
    • Worked example with the New York data (based on the provided numbers):
    • Given five-number summary: Min = $5$, Q1 = $15$, Median = $22.5$, Q3 = $42.5$, Max = $85$.
    • IQR = Q<em>3−Q</em>1=42.5−15=27.5Q<em>3 - Q</em>1 = 42.5 - 15 = 27.5
    • 1.5 × IQR = 1.5×27.5=41.251.5 \times 27.5 = 41.25
    • Upper outlier bound: Q3+1.5×IQR=42.5+41.25=83.75Q_3 + 1.5\times \mathrm{IQR} = 42.5 + 41.25 = 83.75
    • Lower outlier bound: Q1−1.5×IQR=15−41.25=−26.25Q_1 - 1.5\times \mathrm{IQR} = 15 - 41.25 = -26.25
    • Assessment:
    • Data point at 85 minutes exceeds the upper bound 83.75, so 85 is an outlier.
    • The minimum value 5 is above the lower bound (−26.25), so it is not considered an outlier under this rule.
    • Implications:
    • Outliers can significantly affect the mean but have less influence on the median; medians are more robust to outliers.
    • The presence of outliers and the degree of skewness can be visually assessed via box plots and the five-number summary.

  • Box Plots: Practical Use and Comparisons

    • Box plots facilitate quick comparisons across groups (e.g., commuter times in different cities).
    • Example interpretation provided in the session:
    • Raleigh, North Carolina box plot appeared more symmetric with a different range (e.g., min around 5, max around 60), suggesting a more balanced distribution than the New York data.
    • The comparison helps in assessing relative reliability of commute times when choosing job locations or evaluating data quality.

  • Summary of Key Takeaways for Today

    • Measures of spread give insight into distribution shape and robustness of statistics.
    • The range is simple but sensitive to outliers; quartiles and the IQR provide more robust summaries.
    • The five-number summary (min, Q1, median, Q3, max) is a compact descriptor that can indicate skew and guide outlier detection.
    • The 1.5 × IQR rule helps identify potential outliers and informs decisions about data cleaning and interpretation.
    • Box plots are a practical visualization that leverages the five-number summary to compare distributions and assess skewness.

  • Looking Ahead

    • Standard deviation will be introduced and discussed in depth on Thursday, along with further use of TI calculators to generate five-number summaries and box plots (referred to as creating a hotspot).
    • The instructor indicated that the topic is likely to appear on exams.

  • Final Note

    • If you have questions, bring them to class or ask during the Q&A period.

  • Quick Formulas (for quick reference)

    • Range: R=max⁡(X)−min⁡(X)R = \max(X) - \min(X)
    • IQR: IQR=Q<em>3−Q</em>1\mathrm{IQR} = Q<em>3 - Q</em>1
    • Outlier bounds: Q<em>3+1.5×IQR,Q</em>1−1.5×IQRQ<em>3 + 1.5\times \mathrm{IQR},\quad Q</em>1 - 1.5\times \mathrm{IQR}
    • Five-number summary example (New York data): Min = $5$, Q1 = $15$, Median = $22.5$, Q3 = $42.5$, Max = $85$.