# Range, SD and Variance

```S519: Evaluation of
Information Systems
Social Statistics
Ch3: Difference
This week




Range
Standard deviation
Variance
Using Excel to calculate them
The whole story

Descriptive statistics



Centrality tendency (average)
Measurement of variability (variability)
Average+Variability = describe the
characteristics of a set of data
Measures of variability

Variability


Three sets of data



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How scores differ from one another
7, 6, 3, 3, 1
3, 4, 4, 5, 4
4, 4, 4, 4, 4
Variability = the difference from the mean
Measures of variability
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Three ways
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

Range
Standard deviation
Variance
Range


The most general measure of variability
How far apart scores are from one another
Range = highest score – lowest score
What is the range for
98, 86, 77, 56, 48
Standard deviation

Standard deviation (SD)


Average deviation from the mean (average
distance from the mean)
Represents the average amount of variability
s
 ( x  x)
n 1
2
Lab
Exercise

Calculate standard deviation

5, 8, 5, 4, 6, 7, 8, 8, 3, 6

By hand
Using excel (STDEV())
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STDEV and STDEVP
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STDEV is standard deviation for sample (biased
SD)
STDEVP is standard deviation for population
(unbiased SD)
If your dataset is the whole population, use
STDEVP to calculate standard deviation
If you dataset is the sample of something, use
STDEV to calculate standard deviation
STDEV and STDEVP
s
s
2
(
x

x
)

n 1
2
(
x

x
)

n
STDEV
STDEVP
Why n or n-1?


To be conservative
STDEV



This is the standard deviation for sample
Take n-1 in order to make STDEV a bit larger
than it would be.
If we have err, we compensate by overestimating
the STDEV
Why n or n-1?
Sample size
Numerator in
standard
deviation
formula
Denominator
Population
standard
deviation
STDEVP
(dividing by
n)
Denominator
Sample
standard
deviation
STDEV
(dividing by
n-1)
Difference
between
STDEVP and
STDEV
10
500
7.07
7.45
0.38
100
500
2.24
2.25
0.01
1000
500
0.7071
0.7075
0.0004
What to remember
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Standard Deviation (SD) = the average
distance from the mean
The larger SD, the more different data are
from one another
Since mean is sensitive to extreme scores,
so do SD
If SD=0, this means that there is no variability
in the set of scores (they are all identical in
value) – this happens very rarely.
Variance

Variance = (Standard Deviation)^2
s2 
2
(
x

x
)

n 1
Lab
Exercise

Calculate variance in Excel



8, 8, 8, 7, 6, 6, 5, 5, 4, 3
Var()  STDEV
Varp()  STDEVP
SD vs. variance

Often appears in the “Results” sections of
journals

They are quite different

Variance is squared SD
SD vs. variance
9
8
7
mean
6
5
4
3
2
1
0
1
Average distance to
mean=(2+2+2+1+1
+1+2+3)/10=1.4
2
3
4
5
6
7
SD = 1.76
Variance = 3.1
8
9
10
Lab
Exercise 1 (S-p78-problem2)

Calculate range, STDEV and STDEVP and
variance by hand or calculator

31, 42, 35, 55, 54, 34, 25, 44, 35
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Use Excel to do that.
Lab
Exercise 2 (S-p79-problem4)
Height


Problem 4 in S-p79
Calculate the
variation measures
for height and weight
Weight
53
156
46
131
54
123
44
142
56
156
76
171
87
143
65
135
45
138
44
114
57
154
68
166
65
153
66
140
54
143
66
156
51
173
58
143
49
161
48
131
Lab
Exercise 3 (S-p79-problem5)
Western Airlines Flight
Report
Morning Flights
Number of passengers
Evening Flights
Number of passengers
Thursday
To Kansas
Friday
To Kansas
258
Thursday
To Kansas
Friday
Thursday
Friday
To Philadelphia To Providence To Providence
303
312
166
176
Thursday
Friday
Thursday
Friday
To Philadelphia To Providence To Providence
321
331
210
274
251
Friday
To Kansas
312
Thursday