Lesson 3: Measures of Location

Cal

Reese

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Go To Vantage

ImportantVantage

Let’s go to Vantage and see if things are working.

Practical Exercise: Pulse Rate

TipData Collection

Let’s collect some real data!

  1. Find your pulse (wrist or neck)
  2. When instructed, count your heartbeats for 15 seconds
  3. Report your count to the instructor

We’ll use this data to practice everything we’ve learned in Lessons 1-3:

  • Lesson 1: What is our population? What is our sample? Is pulse rate categorical or quantitative?
  • Lesson 2: Create visual displays (stem-and-leaf, dotplot, histogram) and describe the distribution using S-C-S-O
  • Lesson 3: Calculate measures of center (mean, median, mode) and discuss which best represents a “typical” pulse rate

Lesson 2 Review

Pictorial and Tabular Methods

Last lesson we learned about visual displays for data:

Display Best For Preserves Exact Values?
Stem-and-leaf Small datasets (n < 50) Yes
Dotplot Small datasets with repeated values Yes
Histogram (Frequency) Any size, raw counts No
Histogram (Relative Freq) Comparing groups of different sizes No

Describing Distributions: S-C-S-O

  • Shape: Symmetric, skewed left, skewed right, unimodal, bimodal?
  • Center: Where is the “typical” value?
  • Spread: How much variability is there?
  • Outliers: Any unusual observations?

Today we formalize center with numerical measures.

Lesson 3 Content

Objectives

  • Calculate and interpret mean, median, and mode
  • Determine percentiles and quartiles
  • Compare measures of center for different distributions

Required Reading

Devore, Section 1.3: Measures of Location

Measures of Center

Why Measure Center?

Visual displays show us the shape of a distribution. But we often need a single number to summarize “where the data is.”

Three common measures:

  1. Mean (\(\bar{x}\)): The arithmetic average
  2. Median (\(\tilde{x}\)): The middle value
  3. Mode: The most frequent value

The Sample Mean

The sample mean is the arithmetic average of all observations:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \cdots + x_n}{n} \]

Properties:

  • Uses every data point
  • Sensitive to outliers (not resistant)
  • The “balance point” of the distribution

Example: Calculating the Mean

Suppose 7 cadets reported their study hours last week:

Study hours: 8, 10, 12, 9, 11, 8, 45 
Sum: 103 
n: 7 
Mean: 14.71 hours

Is 14.7 hours a good representation of a “typical” cadet’s study time?

WarningOutlier Effect

One cadet reported 45 hours. This outlier pulls the mean up significantly. Most cadets studied 8-12 hours, but the mean suggests nearly 15 hours.

The Sample Median

The sample median (\(\tilde{x}\)) is the middle value when data is ordered from smallest to largest.

Finding the median:

  1. Order the data from smallest to largest
  2. If \(n\) is odd: median = middle value (position \(\frac{n+1}{2}\))
  3. If \(n\) is even: median = average of two middle values

Example: Calculating the Median

Same study hours data:

Ordered data: 8, 8, 9, 10, 11, 12, 45 
n = 7 (odd)
Middle position: 4 
Median: 10 hours

The median (10 hours) is much more representative of typical study time than the mean (14.7 hours)!

The Mode

The mode is the value that occurs most frequently.

Study hours: 8, 10, 12, 9, 11, 8, 45 
Mode: 8 hours (appears twice)

Notes on the mode:

  • A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal, multimodal)
  • Less commonly used than mean or median
  • Most useful for categorical data or identifying clusters

Comparing Mean, Median, and Mode

Measure Formula/Method Resistant to Outliers? Uses All Data?
Mean \(\bar{x} = \frac{\sum x_i}{n}\) No Yes
Median Middle value Yes No
Mode Most frequent Yes No

Which Measure to Use?

Distribution Shape Relationship Best Measure
Symmetric Mean ≈ Median Mean (or either)
Skewed Right Mean > Median Median
Skewed Left Mean < Median Median
Outliers Present Mean pulled toward outliers Median
TipRule of Thumb

When in doubt, report both the mean and median. If they differ substantially, investigate why!

Percentiles and Quartiles

Percentiles

The \(p\)th percentile is a value such that approximately \(p\%\) of the data falls at or below it.

Common percentiles:

  • 25th percentile = First quartile (\(Q_1\))
  • 50th percentile = Median (\(Q_2\))
  • 75th percentile = Third quartile (\(Q_3\))

Quartiles

Quartiles divide ordered data into four equal parts:

  • \(Q_1\) (25th percentile): 25% of data below, 75% above
  • \(Q_2\) (50th percentile): The median
  • \(Q_3\) (75th percentile): 75% of data below, 25% above
Data: 61, 63, 65, 66, 66, 69, 69, 70, 70, 71, 71, 71, 71, 71, 72, 72, 75, 75, 75, 77 
Q1 (25th percentile): 68.25 
Q2 (Median): 71 
Q3 (75th percentile): 72

Computing Quartiles

Method (Devore’s approach):

  1. Order the data from smallest to largest
  2. Find the median (\(Q_2\)) - this divides data into lower and upper halves
  3. \(Q_1\) = median of the lower half
  4. \(Q_3\) = median of the upper half
NoteNote

Different software may use slightly different algorithms for quartiles. The differences are usually small for large datasets.

Example: Finding Quartiles

APFT push-up scores for 12 cadets:

Ordered data: 42, 45, 48, 49, 51, 52, 55, 58, 60, 63, 67, 72 
n = 12 
Lower half: 42, 45, 48, 49, 51, 52 
Upper half: 55, 58, 60, 63, 67, 72 
Q1 = median of lower half = 48.5 
Q2 = median of all data = 53.5 
Q3 = median of upper half = 61.5

The Five-Number Summary

The five-number summary provides a quick snapshot of a distribution:

  1. Minimum
  2. Q1 (25th percentile)
  3. Median (Q2, 50th percentile)
  4. Q3 (75th percentile)
  5. Maximum
Five-Number Summary:
Min: 42 
Q1: 48.75 
Median: 53.5 
Q3: 60.75 
Max: 72

Boxplots

A boxplot (box-and-whisker plot) visualizes the five-number summary:

Interpreting Boxplots

Feature What It Shows
Box width Interquartile range (IQR = Q3 - Q1) - middle 50% of data
Line in box Median
Whiskers Extend to min/max (or 1.5×IQR from box)
Points beyond whiskers Potential outliers

Practice Problems

Problem 1

The following data represents daily high temperatures (°F) for a week in January at West Point:

28, 32, 35, 31, 29, 33, 30

  1. Calculate the mean temperature
  2. Calculate the median temperature
  3. Which measure better represents a “typical” January day? Why?

  1. Mean = \(\frac{28 + 32 + 35 + 31 + 29 + 33 + 30}{7} = \frac{218}{7} = 31.14°F\)

  2. Ordered: 28, 29, 30, 31, 32, 33, 35 Median = 31°F

  3. Both are appropriate here! The data is roughly symmetric with no outliers, so mean ≈ median. Either measure works well.

Problem 2

Annual salaries (in thousands) for employees at a small company:

45, 48, 52, 55, 58, 62, 68, 72, 85, 250

  1. Calculate the mean salary
  2. Calculate the median salary
  3. Which measure would you report as the “typical” salary? Why?
  4. Find Q1 and Q3

  1. Mean = \(\frac{45 + 48 + ... + 250}{10} = \frac{795}{10} = 79.5\) thousand ($79,500)

  2. Ordered data (already sorted), n = 10 (even) Middle values are positions 5 and 6: 58 and 62 Median = \(\frac{58 + 62}{2} = 60\) thousand ($60,000)

  3. The median ($60,000) is more representative. The CEO’s salary of $250,000 is an outlier that pulls the mean up to $79,500, which is higher than 80% of the employees actually earn.

  4. Lower half: 45, 48, 52, 55, 58 → Q1 = 52 thousand Upper half: 62, 68, 72, 85, 250 → Q3 = 72 thousand

Problem 3

Two sections took the same quiz. Here are their scores:

Section A: 72, 75, 78, 80, 82, 85, 88 Section B: 60, 75, 78, 80, 82, 85, 100

  1. Calculate the mean and median for each section
  2. Which section performed better on average?
  3. Why do the means and medians tell different stories for Section B?
  1. Section A:
  • Mean = 560/7 = 80
  • Median = 80

Section B: - Mean = 560/7 = 80 - Median = 80

  1. Both sections have the same mean (80) and same median (80)!

  2. This is a trick question - the means and medians are the same for both sections. However, Section B has more variability (spread from 60 to 100) compared to Section A (spread from 72 to 88). This illustrates why we need measures of spread (next lesson!) in addition to measures of center.

Summary

Key Takeaways

  1. Mean (\(\bar{x}\)): Arithmetic average; uses all data but sensitive to outliers
  2. Median (\(\tilde{x}\)): Middle value; resistant to outliers
  3. Mode: Most frequent value; useful for categorical data
  4. Percentiles: Values that divide data into parts (e.g., 25th, 50th, 75th)
  5. Quartiles: Q1, Q2 (median), Q3 divide data into four parts
  6. Five-number summary: Min, Q1, Median, Q3, Max
ImportantRemember
  • For symmetric distributions: Mean ≈ Median
  • For skewed distributions: Median is usually preferred
  • Always consider outliers when choosing which measure to report

Before You Leave

Today

  • Mean, median, and mode as measures of center
  • When to use each measure
  • Percentiles, quartiles, and the five-number summary
  • Boxplots as visual summaries

Any questions?

Next Lesson

Lesson 4: Measures of Variability

  • Range and interquartile range (IQR)
  • Variance and standard deviation
  • Comparing spread across distributions

Upcoming Graded Events

  • WebAssign 1.4 - Due before Lesson 4
  • Exploratory Data Analysis - Due Lesson 9
  • WPR I - Lesson 16