MEDIAN

MEAN

Mode

Median

Mean

Exercise

**Measures of Central Tendency**

**Quantitative Methods Lecture IV-Chapter 3**

What does this have to do with QM?

A lot of it is about wealth and income inequality

Thinking about averages. Asking questions.

What is the average income?

Who is above it?

Who is below it?

What are measures of central tendencies?

Measures of Central Tendency

What are they?

What is the average income earned by graduates of high school vs. university?

How many cigarettes are smoked by the average teenage smoker?

How do female and male grade averages compare at cegep?

Measures of Central Tendency

The value that best represents a data set

A single element that is typical

Generally located at the centre of a distribution

Center of data distribution

Values greater than central tendency

Values lower than central tendency

Measures of Central Tendency

The average person uses them in an average way

We are not average!

Precise

Numerical

Many types:

Arithmetic mean

Median

Mode

Geometric mean

Harmonic mean

Weighted mean

Truncated mean

Midrange

Midhinge

Trimean

Winsorized mean

Are we average?

AKA

Mo

What is the mode?

It is the value that appears the Mo-st

Ex: Religion in Canada

MODE

How do you find the mode?

Inspection, not calculation

1. Put numbers in order of smallest to largest

2. Look for numbers that appear most often

Example:

My set of numbers

7, 9, 11, 6, 13, 6, 6, 3, 11

Put them in order

3, 6, 6, 6, 7, 9, 11, 11, 13

Find the most common element

MODE

Why use it?

Let's check out its characteristics

-Least precise measurement of central tendency

-Important in understanding sets of scores with fewer values that occur quite often

What level of measurement?

-Most often used with nominal data (only one)

-No formula

-You can have more than one mode. Ex: Bimodal

5, 8, 1, 78, 4, 4, 36, 17, 4, 36, 25, 36

1, 4, 4, 4, 5, 8, 17, 25, 36, 36, 36, 78

Set of numbers:

What do you do now?

MEDIAN

What is the median?

It is the middle value

What is a median in "real" life?

How do you find it?

1. Order the numbers from smallest to largest

2. Find the number in the middle of the list

My set of numbers: 7, 9, 11, 6, 13, 6, 6, 3, 11

Put them in order: 3, 6, 6, 6, 7, 9, 11, 11, 13

What's the number in the middle?

MEDIAN

Used less often than the mean

When is it useful?

Cases with extreme values

Ex: 13, 9, 27, 22, 17, 18, 20, 907

What about if there's an even number of values?

Take the average of the two middle values

Ex: 17, 8, 21, 25, 13, 15, 32, 45

What do you do?

8, 13, 15, 17, 21, 25, 32, 45

Now?

17 + 21 = 38

38/2 = 19 = MEDIAN

AKA

Mdn

MEAN

What is the mean?

It's the average. If every element were the same, what would it be?

How do you calculate it?

1. Add up all the numbers

2. Divide the sum by the number of elements

7, 9, 11, 6, 13, 6, 6, 3, 11

7 + 9 + 11 + 6 + 13 + 6 + 6 + 3 + 11

= 72

Then what?

72 / 9 = 8 = MEAN

MEAN

It is the most straight forward and widely used central tendency measure

X = ∑Xi / N

_

Remember: Formulas are like recipes

X = Mean

∑ = Sum (Sigma)

Xi = list of values

N = Number of cases (Population)

Ex: 4, 6, 5, 7

4 + 6 + 5 + 7

5.5

4, 6, 5, 7

4

= 22

22/ 4

Deviation from Mean

What does deviation "mean"?

Distance and direction from the mean

aka "How far is this number from the mean"

The greater the deviation, the greater the distance

How do you find it?

Score - mean = deviation

Penultimate set: 7, 9, 11, 6, 13, 6, 6, 3, 11

What was the mean?

8

Score - mean = deviation

What is the deviation of 3 from the mean?

(AKA How far is 3 from the mean?)

3 - 8 = -5

3 is -5 away from (and below) the mean

**Comparing Mode, Median and Mean**

**Comparing Mode, Median and Mean**

**Mode**

Median

Mean

Median

Mean

**Comparing Mode, Median and Mean**

**Comparing Mode, Median and Mean**

**Sometimes MODE is most useful:**

**With nominal data**

**Ex: Ethnicity**

**Sometimes MEDIAN is most useful:**

**When wanting to disregard outliers**

**Example: 1, 2, 3, 5, 8, 9, 10, 12, 98**

**Sometimes MEAN is most useful:**

**When we want our “average” to reflect all values in our data set**

**Ex: 35, 60, 65, 67, 71, 73, 74, 75, 77, 79, 80, 81, 84, 97**

**Mean: 72.7**

**Practice**

**Determine the mean, median, and mode for the following set of numbers:**

**3, 7, 5, 4, 5, 6, 7, 8, 6, 5**

**Mean: 5.6**

**Mode: 5**

**Median: 5.5**

**Video Time**

**Do your exercises!**

**...and enjoy your day**

What is the most common wake-up time for young adults?

Troy 22

Virginie 80

Jean-George 55

Ali 33

Christopher 31

Elizabeth 130

Toni 41

Miley 29

Nicole 22

Joshua 39

Jerome 58

Ben 80

Paul 29

Carole 33

Julia 33

(Salary in '000s)

What level of measurement is this?

If you were accountant for the union, which measure of central tendency would you use to represent salaries? Why? What would be your result?

If you were accountant for the owners, which measure of central tendency would you use to represent salaries? Why? What would be your result?

Salary negotiation time!

Median: 19

Mean: 129.125

**What's this?**

**Let's look at a race.**

Nominal

Ordinal

Scale

SCALE

Countries represented

Order

Time