Part II
Sigma Freud & Descriptive
Statistics
Chapter 3    
Viva La Difference:
Understanding Variability
What you will learn in Chapter 3
• Variability is valuable as a descriptive tool
• Difference between variance & standard
•
deviation
How to compute:
• Range
• Inter-quartile Range
• Standard Deviation
• Variance
Why Variability is Important

Variability

• Spread
• Dispersion
What is the “score” of interest here?

• how different scores are from one particular score
• Ah ha!!
It’s the MEAN!!
So…variability is really a measure of how each
score in a group of scores differs from the
mean of that set of scores.

Measures of Variability
Four types of variability that examine the
amount of spread or dispersion in a group of
scores…
• Range
• Inter-quartile Range
• Standard Deviation
• Variance

Typically report the average and the
variability together to describe a distribution.
Computing the Range


Range is the most “general” estimate of
variability…
Two types…
• Exclusive Range
•R=h-l
• Inclusive Range
•R=h–l+1
(Note: R is the range, h is the highest score, l is the lowest score)
Measures of variation
Range

Range
• The difference between the highest and
lowest numbers in a set of numbers.
2, 35, 77, 93, 120, 540
540 – 2 = 538
Chapter 3
6
Measures of variation
Range

What is the range of:
2, 3, 3, 3, 4, 5, 6, 6, 7, 9, 11, 13, 15, 15, 15, 16
24, 57, 81, 96, 107, 152, 179, 211
1001, 1467, 1479, 1680, 1134
Chapter 3
7
Interquartile range


Difference between upper (third) and
lower (first) quartiles
Quartiles divide data into four equal
groups
• Lower (first) quartile is 25th percentile
• Middle (second) quartile is 50th percentile and
•
is the median
Upper (third) quartile is 75th percentile
Calculating the interquartile
range for high temperatures
Date
7-Jan
8-Jan
6-Jan
10-Jan
5-Jan
4-Jan
9-Jan
11-Jan
2-Jan
3-Jan
High
Temperature
32
32
35
41
42
43
46
52
59
60
<===Bottom Half Middle Value = First Quartile = 35
<===Middle Value
<===Middle Value
Median = Second Quartile = 42.5
<===Top Half Middle Value = Third Quartile = 52
interquartile range = 52 – 35 = 17
Stem and Leaf 0730 Q1 Fall 2010
(N=22)







2|349
3|03344555666677779
4|01
Q1= .25 (22)=5.5 data point round up to 6th data
point=value of 33
Q2= n+1/2=23/2=11.5 = avg of 11th and 12th data
pt = 35.5
Q3= .75(22)=16.5 =round up to17th data point=
Value of 37
Chapter 3
10
Interquartile range and outliers


Value can be considered to be an outlier if it falls
more than 1.5 times the interquartile range
above the upper quartile or more than 1.5 times
the range below the lower quartile
Example for high temperatures
• Interquartile range is 17
• 1.5 times interquartile range is 25.5
• Outliers would be values
• Above 52 + 25.5 = 77.5 (none)
• Below 35 – 25.5 = 9.5 (none)
Review: Steps to Quartiles,
Interquartile Range,and Checking for
Outliers






1) Put values in ascending OR descending
order
2) Multiply .25 (n) for Q1
3) Multiply .75 (n) for Q3
4) Q3 - Q1 = IQR
5) Q1 – 1.5 (IQR)= value below smallest value
in data set;
6) Q3 + 1.5 (IQR)= value above largest value
in data set;
Let’s practice Finding Outliers



What is the median, Q1, Q3, range, and IQR
for the following? Then check for outliers.
10, 25, 35, 65, 100, 255, 350, 395 (n=8)
10, 65, 75, 99, 299 (n=5)
5, 39, 45, 59, 64, 74 (n=6)
Chapter 3
13
Computing Standard Deviation



Standard Deviation (SD) is the most
frequently reported measure of variability
SD = average amount of variability in a
set of scores
What do these symbols represent?
Why n – 1?

The standard deviation is intended to be
an estimate of the POPULATION
standard deviation…
• We want it to be an “unbiased estimate”
• Subtracting 1 from n artificially inflates the
SD…making it larger

In other words…we want to be
“conservative” in our estimate of the
population
Things to Remember…




Standard deviation is computed as the
average distance from the mean
The larger the standard deviation the
greater the variability
Like the mean…standard deviation is
sensitive to extreme scores
If s = 0, then there is no variability among
scores…they must all be the same value.
Computing Variance

Variance = standard deviation squared

So…what do these symbols represent?
Does the formula look familiar?
Standard Deviation or Variance

While the formulas are quite similar…the
two are also quite different.
• Standard deviation is stated in original units
• Variance is stated in units that are squared
• Which do you think is easier to interpret???
Same mean, different standard
deviation; Sample variance and Sample
standard deviation: {20,31,50,69,80}
Each number x1
Mean
Distance from Mean
20
50
-30
31
50
-19
50
50
0
69
50
19
80
50
30
Chapter 3
19
Then square each distance from
mean and add together…






(-30)2 + (-19)2 + (0)2+ (19)2 + (30)2
900+ 361+ 0+ 361 +900=
2522
Divide by N-1 (N=5)
2522/4=630.5= Sample Variance
To find sample standard deviation, take
square root of variance= 25.11
Chapter 3
20
Same mean, different standard
deviation: {39,44,50,56,61}
Each number x1
Mean
Distance from Mean
39
50
-11
44
50
-6
50
50
0
56
50
6
61
50
11
Chapter 3
21
Which data set has more
variability?






(-11)2 + (-6)2 + (0)2 + (11)2 + (6)2
121+ 36+ 0+ 121+ 36=
314
Divide by N-1 gives us sample variance
314/4=78.5
Square root of 78.5 gives us sample
standard deviation=8.86
Chapter 3
22
Measures of variation
Standard deviation

How about a more user-friendly equation?

x

x  N
2
2
S
Chapter 3
N 1
23
Using Excel’s VAR Function
Using the Computer to Compute
Measures of Variability
Glossary Terms to Know

Variability
• Range
• Standard deviation
• Mean deviation
• Unbiased estimate
• Variance