**Quantitative Methods**

**Measures of variability**

**Roll call**

**Today's Plan**

Last week: Measures of Central Tendency.

**Measures of Variability**

Who's been to Vancouver?

**Example:**

Vancouver & Montreal

Vancouver & Montreal

**Measures of Variability**

**Variance**

**Standard Deviation**

What were the three measures

we saw last week?

Mode

Median

Mean

What does each mean?

About finding a number that is typical of a distribution

Problem?

A little too much of a good thing. Incomplete.

What province is it in?

What's the climate like?

Let's take a look at the blackboard

Another Example:

Salaries

You're looking for a job and hey, you just found two! Both at a grocery store.

John's Grocery: 13$

Abbott's Market: 13$

Mean hourly wage

John's Grocery: Everyone makes 13$/h

Abbott's Market: 10.15-15$/h

Where do you want to work?

Central tendency is limited.

What else do we need?

An index of how data is scattered, a measure of variability (aka spread, width or dispersion)

Looking back at our examples:

Montreal greater weather variability than Vancouver

Abbott Market greater salary variability than John's Grocery

Measures of Variability

Standard deviation

Interquartile range or Interdecile range

Range

Mean difference

Median absolute deviation

Average absolute deviation

Distance standard deviation

Coefficient of variation

Quartile coefficient of dispersion

Relative mean difference

Variance

Variance-to-mean ratio

**Range**

Measures of Variability

Help us understand:

-A more complete picture of our data

-How our data is scattered around

our "averages."

**Tells us how far apart scores are from one another**

**To find the range:**

**In a distribution, subtract**

the lowest score from the highest score.

the lowest score from the highest score.

**R = H - L**

**range**

**highest score**

**lowest score**

R = H - L

Ex: 14, 7, 28, 18, 5, 36, 21

What's the highest score?

What's the lowest score?

What do you do?

R = 36 - 5 = 31

You are now range masters

Another Use for Ranges

Frequency charts

Frequency charts with categories?

Knowing the range helps you decide how to group your intervals

Practice:

1. Range

2. Classes

The Range

Gives us a simple estimate of how ALL scores are different from one another

...but not how INDIVIDUAL scores differ

Can be greatly influenced by outliers

Ex: Boban MARJANOVIC in classroom

What do we need?

A "standard value" to compare scores against...

Deviation from the Mean

A more useful measure of variability is how far scores are from the mean

Why not just do this?

1. Find the mean

What's the mean here?

3

2. Subtract each from the mean

3. Add the scores together

?!

**Let's stay positive!**

Let's square the differences.

Then we can divide our total squared deviation by the total number of scores = VARIANCE

10/5 =2

VARIANCE = 2

Steps to Variance

1. Calculate mean

Practice:

Variance

POSITIVE: We have found a way to get rid of the ZERO SUM of deviations from the mean

NEGATIVE: Variance gives us our average deviation from the mean in units squared which is less practical.

What should we do?

Get back to our

original unit of measurement

Standard Deviation

Standard deviation is simply taking the square root of the variance

This will bring us back to our original unit of measurement

Let's take a closer look

Standard Deviation

1. Calculate mean

Mean = 12

2. Calculate deviations from mean

3. Square

deviations

4. Sum the squared values

5. Divide the sum by the number of scores/values

Variance = 106/5 = 21.2

6. Take the square root of the variance

Square root of 21.2 = 4.6

Formulas

Variance

Standard Deviation

Why N-1?

We use N for a population and n-1 for a sample.

We usually work with samples...

Using N-1 increases the Standard Deviation to better reflect the population

The bigger the N, the less of a difference. Bigger sample = better

Standard Deviation

What is SD really telling us?

Each unit on the graph below indicates one standard deviation from the mean value

So, SD is the average size of a deviation from the mean

Standard Deviation

Practical example:

**3/37**

29/37

34/37

Standard Deviation: Raw Score

**Weekly Schedule**

**Talk about next week's test**

**Measures of Variability**

**In-class work**

4

1

0

1

4

10

2. Calculate deviations from mean

3. Square deviations

4. Sum the squares

5. Divide the sum by the number of scores/values

2.22m or 7'3 1/2"

How old are you? (In months)

Group 01

Group 02

228

216

203

215

229

222

220

204

218

227

228

201

276

212

207

223

216

215

208

211

Find the range, variance and standard deviation and draw a mock graph of both groups.

Use the raw score formula

What's the problem with this method?