4|CENTRAL TENDENCY

Overview

Mode
Median
Mean
Distributions
Learning checks

Central tendency

A number to describe the center of a distribution
Purpose:
- To organize & summarize scores
- Find the single score that is most typical or best represents the entire group
- Allow comparisons

Finding the center

Where is the center of the distribution?

Finding the center

Where is the center of the distribution?

Finding the center

Where is the center of the distribution?

Measures of central tendency

Imagine you get the following grades:
- 90, 0, 80, 85, 90
How do you fairly describe all these scores with a single number?
Three ways:
- Mode: grade you get most often
- Median = grade that divides lowest 50% of scores from highest 50% of scores
- Mean = sum of grades / # of grades = \(\dfrac{\Sigma X} N\)

Measures of central tendency

Imagine you get the following grades:
- 90, 0, 80, 85, 90
Three ways:
- Mode = 90, 0, 80, 85, 90
- Median = 0, 80, 85, 90, 90
- Mean = (90 + 0 + 80 + 85 + 90) / 5 = 69

Mode

The score/category with the greatest frequency
- What occurs most often?

\(X\)	\(f\)
5	1
4	2
3	4
2	5
1	3

\(X\)	\(f\)
90-99	7
80-89	4
70-79	5
60-60	3
50-59	0
40-49	1

Mode: generic examples

Mode: realistic example

Median

Definition:
- The midpoint of the scores in a distribution when they are listed in order from smallest to largest
- Divides the scores into two groups of equal size
- Equal number of data points either side of the median
- 50% below, 50% above

Locating the median

Put scores in order
Find the number that gives and equal number of scores on either side
Odd number of scores
- Median is the center score

1 2 3 4 5

Locating the median

Put scores in order
Find the number that gives and equal number of scores on either side
Even number of scores:
- Average the 2 numbers either side of center

1 2 3 | 4 5 6

(3 + 4) / 2 = 3.5

65 70 70 80 90 90

65 70 80 80 80 90 92 95

Mean

What is the “average”?
- Take a set of scores
- Add them up
- Divide by how many there are
Developed in the 16th century
- Mainly used by astronomers
Adolphe Quetelet (1796-1874)
- Applied the concept to people
- Size measurements (BMI), divorce, crime, suicide
- See The Atlantic: How the Idea of a ‘Normal’ Person Got Invented

History

American Civil War
- Mass production of uniforms
- Small, Medium, Large
- Also food rations, weapon design, etc
1926: Plane cockpits
- Based on average measurements
- By WW2 worked terribly
- Didn’t fit most pilots
- Nobody is average on all dimensions

Calculating the mean

Sum of scores divided by number of scores
Represented by a symbol (unlike mode & median)

Sample: \(M = \dfrac{\Sigma X} n\)

(sometimes \(\overline{X}\))

Population: \(\mu = \dfrac{\Sigma X} N\)

Visualizing the mean

Another way of thinking about the mean
- The balance point for the distribution

Distributions

mean, median, mode

Distributions: income

Sensitivity to outliers
- Extreme values; observations far from the center
- Mean is more influenced by outliers than median

Distributions: income

Summary

Mode
- Most useful for nominal/ordinal data
- Reflects the single most common score
- Or more complicated (bimodal) distributions
Median
- Most useful for interval/ratio data
- Represents the midpoint in terms of number of scores
- Less influenced by outliers; helpful for skewed data
Mean
- Most useful for interval/ratio data
- Represents the midpoint in terms of distances of scores
- More influenced by outliers; best for normal data

Learning checks

True or False: It is possible for more than 50% of the scores in a distribution to have values above the:
1. mode
2. mean
3. median

What shape is this distribution?
- What can you predict about its mode, mean, and median?

people_illustration = {

  const w = 650
  const h = 400
  const personWidth = 80
  
  const data = [
    {id: 0, income: "$33,000"},
    {id: 1, income: "$34,000"},
    {id: 2, income: "$35,000"},
    {id: 3, income: "$35,000"},
    {id: 4, income: "$35,000"},
    {id: 5, income: "$36,000"},
    {id: 6, income: "$37,000"},
  ]
      
  const svg = d3.select("#distributions-people-container")
    .append("svg").attr("width", w).attr("height", h).attr("class", "invertable")
  
  const imgs = svg.selectAll("image")
    .data(data)
    .enter()
      .append("svg:image")
      .attr('x', d => 50 + d.id * personWidth)
      .attr('y', 50)
      .attr('width', personWidth)
      .attr("xlink:href", "media/person.svg")
      
  const incomes = svg.selectAll("text")
    .data(data)
    .enter()
      .append("text")
      .attr('text-anchor', 'middle')
      .attr('x', d => 50 + personWidth/2 + d.id * personWidth)
      .attr('y', 30)
      .style('font-size', '0.5em')
      .text(d => d.income)

  const mean1 = svg.append("g").selectAll("text")
    .data(['median: $35,000', '<tspan style="font-family: Times; font-style: italic">M</tspan> = $35,000'])
    .enter()
      .append("text")
      .attr('text-anchor', 'middle')
      .attr('x', personWidth * 7/2)
      .attr('y', function(d,i){return 250 + i*50})
      .style('font-size', '0.8em')
      .html(d => d)
}

monopoly_man = {
  const w = 400
  const h = 400
  const personWidth = 80
  
  
  const svg = d3.select("#distributions-monopoly-man-container")
    .append("svg").attr("width", w).attr("height", h).attr("class", "invertable")
    
  svg.append("svg:image")
      .attr('x', 100)
      .attr('y', 30)
      .attr('width', 200)
      <!-- .attr('height', 210) -->
      .attr("xlink:href", "media/monopoly-man.svg")
      
  svg.append("text")
      .attr('text-anchor', 'middle')
      .style('font-size', '0.5em')
      .attr('x', w/2)
      .attr('y', 30)
      .text("$1 billion")
      
  const new_mean = svg.append("g").selectAll("text")
    .data(['new median: $35,000', 'new <tspan style="font-family: Times; font-style: italic">M</tspan> = $125 million'])
    .enter()
      .append("text")
      .attr('text-anchor', 'middle')
      .attr('x', w/2)
      .attr('y', function(d,i){return 250 + i*50})
      .style('font-size', '0.8em')
      .html(d => d)
}