STA 220 Section 3-1 1 .pdf - STA 220 Guided Notes for...

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STA 220
Guided Notes for Section 3-1
Measures of Center
A measure of center is a value at the center or middle of a data set.
• Mean (or Arithmetic Mean)
– The mean (or arithmetic mean ) of a set of data is the measure of center found
by adding all of the data values and dividing the total by the number of data
values.
Important Properties of the Mean
• Sample means drawn from the same population tend to vary less than other
measures of center.
• The mean of a data set uses every data value.
• A disadvantage of the mean is that just one extreme value (outlier) can change the
value of the mean substantially. (Using the following definition, we say that the
mean is not resistant. )
– A statistic is resistant if the presence of extreme values (outliers) does not cause
it to change very much.
Notation
∑ (Capital Greek letter sigma) denotes the addition of a set of values
x is the variable usually used to represent the individual data values
n represents the number of values in a sample (i.e. n is the sample size )
N represents the number of values in a population . (i.e. N is the population size)
x ( x-bar ) is the mean of a set of sample values. Sample Mean =

x
x =
n
μ ( mu ) is the mean of all values in a population. Population Mean =

x
μ =
N
Example: Mean
Data Set 32 “Airport Data Speeds” in Appendix B includes measures of data speeds of
smartphones from four different carriers. Find the mean of the first five data speeds for
Verizon: 38.5, 55.6, 22.4, 14.1, and 23.1 (all in megabits per second, or Mbps).

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• Caution
• Never use the term average when referring to a measure of center. The word
average is often used for the mean, but it is sometimes used for other measures of
center.
• The term average is not used by statisticians.
• The term average is not used by the statistics community or professional journals.
• Median
– The median of a data set is the measure of center that is the middle value when
the original data values are arranged in order of increasing (or decreasing) order.
Important Properties of the Median
• The median does not change by large amounts when we include just a few extreme
values, so the median is a resistant measure of center.