7-3 Statistics: Mean, Median, and Mode

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8-2 Statistics: Mean, Median, and Mode
Example:
We want to estimate the mean (average) beak-lengths of all
yellow-cheeked warblers in Texas)
Discussion:
 if we could capture and measure every individual
 we could obtain the population mean  (“mu”)
 instead, we will select a sample from the population
 and compute the sample mean x (“x-bar”)
Suppose we capture a sample of 10 and get measurements:
2, 3, 3, 4, 5, 5, 5, 6, 6, 7 cm.
Then x = (the average of these numbers) = 4.6 cm.
 if we make n measurements: x1, x2, x3, … , xn
 the sample mean is given by the formula:
x = (x1+ x2+ x3+ … + xn)/n
Summation ( = “sigma”) notation:
 write x1+ x2 + x3 + … + xn as
n
 xi
i 1
 read “the sum for i equals 1, 2, … up to n of the xi’s”
 using this, the formula for the sample mean is
n
 xi
x =
8-2
i 1
n
p. 1
Some more terminology:
 we will frequently use the term score as the generic word
for any kind of measurement or count
 things like sample mean that are computed based on a
sample, are called statistics
 the mean is a measure of central tendency
 a single number that summarizes …
 some characteristic of a population
 there are other measures of central tendency …
MEDIAN
 the median is the middle score of a set of scores …
 that have been placed in order
 if there is no middle score (i.e. there are an even number
of scores)
 take the average of the two middle scores
Example: 1, 3, 5, 6, 7, 8 has median 5.5
MODE
 the mode is the most frequently occurring score
 there may be more than one mode
Example:
1, 3, 3, 3, 5, 6, 6
1, 3, 3, 3, 5, 6, 6, 6
8-2
has mode 3
has modes 3 and 6
p. 2
ANOTHER VIEW OF DISTRIBUTIONS
For measurements (as opposed to counts) the graphical
representation of the distribution of a r.v. X can look like:
2.5
3.0
x
 this kind of distribution is called a density function
 areas under the curve represent relative frequency of
occurrence of ranges of values
 if the shaded area is 25% of the total area under the curve
 then 25% of the scores lie between 2.5 and 3.0
25%
2.5
8-2
3.0
x
p. 3