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Measures of Variation
Basic Concepts
Other Concepts
What is variation?
How data is spread or dispersed
Range of a data set...
Difference between
the maximum data
value and the
minimum data value
RANGE  (Maximum Data Value) - (Minimum Data Value)
What does Range tell us about
variation
Uses only max and min values therefore...
Use the same round off rule as MOC
STANDARD DEVIATION (s)
MOV most commonly used in stats
Measures variation of values about the
mean
Deviation of sample values away from the
mean
Finding Standard Deviation of a
Data Set
Sample Standard
Deviation:
( x  x )
s
n 1
2
Shortcut Formula:
n( x )  (x)
s
n(n  1)
2
2
Characteristics of
Standard Deviation?
s is usually positive and NEVER negative
s is 0 only when all data values are the
same number
the larger value for (s) the greater amount
the data varies
s can increase dramatically with the
inclusion of outliers
the units (minutes, feet, etc...) are the
same as the units of original values
Procedure for Calculating (s)
Find the mean
Subtract the mean from each ind. Value
Square each result
Add all the squares
Divide the sum by (n-1)
Find the square root
Example
From Excel STATS SD
Calculate the standard deviation of 1, 3
and 14
Using Calculators to Find Standard
Deviation
Input your set of data into a List by using
STAT and EDIT
Hit STAT and right arrow to CALC
Choose option 1-VarStats and enter L1 (or
whatever list you put the data in) and
ENTER
Find the Standard Deviation
Given the sample below, determine the standard
deviation:
Nitrate deposits as a result of acid rain for
Massachusetts from July to Sept.:
6.40
5.53
5.41
5.21
8.23
4.66
5.24
6.96
6.80 5.78
6.00
Finding the Standard Deviation of a
Frequency Table
In L1 input all the class marks. InL2
input the frequencies .
Use STAT, CALC, and 1-VarStats.
On your screen you will see
1-VarStats
Input L1 , L2 and hit ENTER.
Variation and Standard Deviation
Understanding Variation
Range Rule of Thumb
Empirical (68-95-99) Rule for Data
Chebyshev’s Theorem
Range Rule of Thumb
For typical data sets, the range of a set of
data is approximately 4 standard deviations
wide.
To approximate the standard deviation,
range
standard deviation 
4
Range Rule of Thumb
If we know, or approximately know, the
value of the standard deviation, we can find
estimates of the minimum and maximum
scores.
minimum  (mean)  2  (standard deviation)
maximum  (mean) + 2  (standard deviation)
Empirical Rule for Data
This rule applies to a data set that is
approximately bell-shaped.
68-95-99 Rule:
About 68% of all scores fall within 1 standard deviation of
the mean.
About 95% of all scores fall within 2 standard deviations
of the mean.
About 99.7% of all scores fall within 3 standard deviations
of the mean.
Examples
Use the range rule of thumb to estimate
the standard deviation of 100 credit rating
scores. The minimum is 444 while the
max is 850.
Empirical Rule for Data
Chebyshev’s Theorem
A proportion (or fraction) of any data set
lying within K standard deviations of the
mean is always at least
1
1
K
2
where K is any number greater than 1.