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# Quantitative Methods - Chapter 4: Measures of variability

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## Jean-Michel Sotiron

on 6 September 2016

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#### Transcript of Quantitative Methods - Chapter 4: Measures of variability

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

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.

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
?!
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
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?
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