Chapter 3 – Descriptive Statistics: Numerical Measures

Measures that are computed for data from a sample are
sample statistics
, if they are
computed for data from a population, they are
population parameters
. In statistical
inference, a sample statistic is referred to as the
point estimator
or the corresponding
population parameter
Measures of Location
Mean

The
mean
, or average value, is a measure of location for a variable

The mean provides a measure of central location for the data

The mean is denoted by x bar for a sample, and by the Greek letter μ for a population

The value of variables is denoted by a subscript; the value of the variable x for the first
observation is denoted by x
1
, and the second by x
2
, etc.

The formula for the sample mean is:
Xbar = Σx
1
N

The formula for computing the mean of a population remains the same, but we use
different notation to indicate that we are working with the entire population (Equation
3.2)
Median

The
median
is another measure of central location for a variable; the median is the
largest value in the middle when the data are arranged in ascending order (smallest value
to largest value)

An even number of observations has no single middle value (the median is the average of
the values for the middle two observations)

The median is sometimes preferred over the mean because the mean is easily influenced
by extremely small and large data values (in general, if a data set contains extreme
values, the median is often the preferred measure of central location
Mode

The
mode
is the value that occurs with greatest frequency

IF data contains exactly two modes, the data is
bimodal
; if it contains more than two
modes, the data is
multimodal
Percentiles

A
percentile
provides information about how data is spread over the interval from the
smallest value to the largest value

The pth percentile divides data into two parts:
1.
Approximately
p
percent of the observations have values less than the pth percentile
2.
Approximately (100p) percent of the observations have values greater than the pth
percentile
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Calculating the pth percentile:
Step 1 – Arrange the data in ascending order (smallest value to largest value)
Step 2 – Compute an index i
I = (p/100)n
Where p is the percentile of interest and n is the number of observations
Step 3 – A. if I is not an integer, round up. The next integer greater than I denotes the
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 Fall '07
 Thornton
 Standard Deviation, Mean

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