Module 3: Measures of Variability – Ungrouped Data
The information in this module is contained in Chapter 3 in the text. The following is an outline of what we’ll cover in this module:
1.
The Range (Module 1)
2.
The Interquartile Range (Module 2)
3.
The Variance
a.
Population Variance (
b.
Sample Variance (s
2
)
4.
The Standard Deviation
a.
Population Standard Deviation (
b.
Sample Standard Deviation (s
5.
Coefficient of Variation (CV)
a.
Population Coefficient of Variation
b.
Sample Coefficient of Variation
6.
Interpreting the Standard Deviation
a.
The Empirical Rule
b.
Chebyshev’s Theorem
7.
The Z-Score

Module 3: Measures of Variability – Ungrouped Data
What’s the difference between a population and a sample?
Population
– The entire pool from which a sample is drawn.
Sample
-- A subset drawn from the population. In statistics, the sample should always be randomly drawn.
Example a: Assume we want to know the mean age of Kent State students.
We have two choices:
Choice 1. Take a census of all Kent State Students in the population(about 27,500). This would involve every
student enrolled
at Kent State and the mean age we calculate would be 100% accurate. However, taking a census
is expensive and the cost is
usually not justified.
Choice 2: Take a random sample. If done correctly, we can take a much smaller subset, a sample, of KSU
students. The
sample mean age will likely be slightly different than the actual mean age, but the difference is
unlikely to be statistically
significant.
Example b: Suppose we want to know the level of public approval for Obamacare.
Choice 1. Ask the entire US population (about 325 million).
Choice 2. Randomly sample 1,000 Americans.
Again, sample statistics by definition won’t be perfectly accurate like population parameters, but close enough to be useful and
much less costly to acquire.

Module 3: Measures of Variability – Ungrouped Data
Before we go on, let’s look at a few important symbols we’ll be using throughout the semester. Keep this page handy.
he Greek letter mu is the symbol used to denote the population mean.
-- X –bar is the symbol used to denote the sample mean
2
– The lower case Greek letter sigma (squared) is used to denote the population variance
–
The lower case Greek letter sigma is used to denote the population standard deviation
s
2
– This lower case letter “s” (squared) is used to denote the sample variance
s – This lower case letter “s” (squared) is used to denote the sample standard deviation
–
As in all mathematical applications, the upper case Greek letter sigma is used to indicate a
summation of summing of
the data
N – Upper case N denotes the size of the population
n – lower case n denotes the size of the sample

Module 3: Measures of Variability – Ungrouped Data
Measure
Population
Parameter
Sample
Statistic
Mean
Variance
Standard
Deviation
X
2
S
S
2

Module 3: Measures of Variability – Ungrouped Data
3.
The Variance -- A common measurement of the variability between numbers in a data set.

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- Spring '14
- DebraACasto
- Standard Deviation, Ungrouped Data