Session 5 Notes, Section 3.3
Measures of Variation
This session will show some basic ways to summarize data with measures of variation or dispersion. These exploratory
techniques are used to examine data to see what they reveal. By combining
all of these techniques together, Chapter 3 shows the student how to collect,
organize, summarize and present data.
At the end of this session you will be able to:
If you have learned all of these objectives, then close this window to
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- Measures of average are also called measures of central tendency
and include the mean, median, mode, and midrange.
- Measures that determine the spread of data values are called measures
of variation or measures of dispersion and include the range,
variance, and standard deviation.
- A statistic is a characteristic or measure obtained by using the
data values from a sample.
- A parameter is a characteristic or measure obtained by using all
the data values for a specific population.
- When the data in a data set is ordered it is called a data array.
- In statistics the basic rounding rule is that when computations
are done in the calculation, rounding should not be done until the final
answer is calculated.
- Range
- The range is the highest value minus the lowest value in a data set.
- The symbol R is used for the range.
- Variance
- The variance is the average of the squares of the distance each value is from
the mean. The symbol for the population variance is
σ2.
- Standard Deviation
- The standard deviation is the square root of the variance. The symbol for the
population standard deviation is σ. Rounding
rule: The final answer should be rounded to one more decimal place than the
original data.
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- Coefficient of variation
- The coefficient of variation is the standard deviation divided by the mean. The
result is expressed as a percentage.
- The coefficient of variation is used to compare standard deviations when the
units are different for the two variables being compared.
- Spread of data
- Variances and standard deviations can be used to determine the spread of the
data. If the variance or standard deviation is large, the data are more
dispersed. The information is useful in comparing two or more data sets to
determine which is more variable.
- Consistency
- The measures of variance and standard deviation are used to determine the
consistency of a variable.
- The variance and standard deviation are used to determine the number of data
values that fall within a specified interval in a distribution.
- The variance and standard deviation are used quite often in inferential
statistics.
- Chebyshev’s Theorem
- The proportion of values from a data set that will fall within k standard
deviations of the mean will be at least 1 – 1/k2; where k is a number greater
than 1.
- This theorem applies to any distribution regardless of its shape.
- Empirical Rule for Normal Distributions
- The following apply to a bell-shaped distribution.
- Approximately 68% of the data values fall within one standard deviation of the
mean.
- Approximately 95% of the data values fall within two standard deviations of the
mean.
- Approximately 99.75% of the data values fall within three standard deviations of
the mean.
Return to objectives
- Some basic ways to summarize data include measures of central tendency,
measures of variation or dispersion, and measures of position.
- The three most commonly used measures of central tendency are the mean,
median, and mode. The midrange is also used to represent an average.
- The three most commonly used measurements of variation are the range,
variance, and standard deviation.
- Data values are distributed according to Chebyshev’s theorem and in special
cases, the empirical rule.
- The coefficient of variation is used to describe the standard deviation in
relationship to the mean.
- These methods are commonly called traditional statistics.
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