Module 9 Summary

Measures of center, the five-number summary, standard deviation, and the shape of a distribution on a single printable page.

MATH 114 Module 9: Describing Data

Reading: Chapter 6, Sections A and B  ·  Name: ______________________________

Due dates (11:59 p.m.): Homework 9, Sunday, Oct 25, 2026  ·  Quiz 9, Monday, Oct 26, 2026. Confirm all dates in Canvas.
Outcomes: compute and choose a measure of center; describe the shape of a distribution; find the range, five-number summary and standard deviation.

1. Measures of Center

  • Mean: the sum of the values divided by how many there are.
  • Median: the middle value once the data are sorted; with an even count, the mean of the two middle values.
  • Mode: the most common value (there may be none, or more than one).

Example (PR energy bar at eight stores): $1.09, 1.29, 1.29, 1.35, 1.39, 1.49, 1.59, 1.79 → mean ≈ $1.41, median (1.35 + 1.39)/2 = $1.37, mode $1.29.

Outliers pull the mean but not the median. Contracts $0, 0, 0, 0, $10,000,000: mean $2,000,000, median and mode $0.

2. Shape of a Distribution

  • Number of peaks: unimodal (one), bimodal (two), or more.
  • Symmetric: left and right halves are mirror images; mean = median.
  • Right-skewed: long tail to the right; the mean is pulled above the median.
  • Left-skewed: long tail to the left; the mean is pulled below the median.
  • Variation: how widely the data spread about the center.

3. Range and Quartiles

Range = highest value − lowest value. It can mislead: Quiz 1 (one score of 1, the rest 10) has range 9, larger than Quiz 2 (scores spread from 2 to 10, range 8), even though Quiz 1’s scores vary less.

Quartiles: the lower quartile Q1 is the median of the lower half; the upper quartile Q3 is the median of the upper half.

Five-number summary: low, Q1, median, Q3, high

Example (bank waits, minutes): Big Bank 4.1, 5.2, 5.6, 6.2, 6.7, 7.2, 7.7, 7.7, 8.5, 9.3, 11.0 → 4.1, 5.6, 7.2, 8.5, 11.0. Best Bank 6.6, 6.7, 6.7, 6.9, 7.1, 7.2, 7.3, 7.4, 7.7, 7.8, 7.8 → 6.6, 6.7, 7.2, 7.7, 7.8. Same mean and median (7.2), but Best Bank’s waits vary far less. A boxplot draws the five-number summary.

4. Standard Deviation

The standard deviation s measures how far values typically lie from the mean.

  1. Find the mean and each deviation (value − mean).
  2. Square each deviation and add the squares.
  3. Divide by (number of values − 1); take the square root.

Banks: Big Bank s ≈ 1.96 min; Best Bank s ≈ 0.44 min.

Range rule of thumb

s ≈ range ÷ 4

Low value ≈ mean − 2s; high value ≈ mean + 2s.

Resources

Lesson: andrewhvolk.github.io/courses/math114-module9.html  ·  Slides: andrewhvolk.github.io/pdfs/math114-module9-slides-2026-10-26.pdf  ·  Excel Online: =AVERAGE, =MEDIAN, =MODE.SNGL, =QUARTILE.INC, =STDEV.S

© Andrew H. Volk 2026