3.1
Measures
of Central
Tendency
PARAMETER VS. STATISTIC
●Analyzing populations versus analyzing samples
●For populations
 We know all of the data
 Descriptive measures of populations are called
parameters
 Parameters are often written using Greek letters ( μ )
●For samples
 We know only part of the entire data
 Descriptive measures of samples are called statistics
x
 Statistics are often written using Roman letters ( )
ARITHMETIC MEAN
●The arithmetic mean of a variable is often
what people mean by the “average” … add up
all the values and divide by how many there
are
●Compute the arithmetic mean of
6, 1, 5
●Add up the three numbers and divide by 3
(6 + 1 + 5) / 3 = 4.0
●The arithmetic mean is 4.0
MEAN
●The arithmetic mean is usually called the
mean
●For a population … the population mean
 Is computed using all the observations in a
population
 Is denoted μ
 Is a parameter
●For a sample … the sample mean
 Is computed using only the observations in a sample
 Is denoted x
 Is a statistic
LET’S TRY IT
Whole Class (Population)
 Height in Inches
 Find the arithmetic mean
5 Students (Sample…statistic)
 Height in Inches
 Find the arithmetic mean
MEDIAN
●The median of a variable is the “center”
●When the data is sorted in order, the median
is the middle value
●The calculation of the median of a variable is
slightly different depending on
 If there are an odd number of points, or
 The middle number
 If there are an even number of points
 Take the 2 middle numbers and find the mean
MEDIAN
●An example with an odd number of
observations (5 observations)
●Compute the median of
6, 1, 11, 2, 11
●Sort them in order
1, 2, 6, 11, 11
●The middle number is 6, so the median is 6
MEDIAN
● An example with an even number of observations (4
observations)
● Compute the median of
6, 1, 11, 2
● Sort them in order
1, 2, 6, 11
● Take the mean of the two middle values
(2 + 6) / 2 = 4
● The median is 4
MODE
● The mode of a variable is the most frequently
occurring value
● Find the mode of
6, 1, 2, 6, 11, 7, 3
● The values are
1, 2, 3, 6, 6, 7, 11
● The value 6 occurs twice, all the other values occur
only once
● The mode is 6
WEIRD MODE
●Qualitative data
 Values are one of a set of categories
 Cannot add or order them … the mean and median do
not exist
 The mode is the only one of these three
measurements that exists
●Find the mode of
blue, blue, blue, red, green
●The mode is “blue” because it is the value that
occurs the most often
MODE (NO REPEATS)
●Quantitative data
 The mode can be computed but sometimes it is not
meaningful
 Sometimes each value will only occur once (which
can often happen with precise measurements)
●Find the mode of
5.1, 6.6, 6.8, 9.3, 1.9
●Each value occurs only once
●The mode is not a meaningful measurement
●Mode is what is used in elections!
SHAPE
The mean and the median are often different
This difference gives us clues about the shape
of the distribution
 Is it symmetric?
 Is it skewed left?
 Is it skewed right?
 Are there any extreme values?
SHAPE
 Symmetric – the mean will usually be close to the
median
 Skewed left – the mean will usually be smaller than
the median
 Skewed right – the mean will usually be larger than
the median
SYMMETRIC
●If a distribution is symmetric, the data values
above and below the mean will balance
 The mean will be in the “middle”
 The median will be in the “middle”
●Thus the mean will be close to the median, in
general, for a distribution that is symmetric
SKEWED LEFT
● If a distribution is skewed left, there will be some
data values that are larger than the others
 The mean will decrease
 The median will not decrease as much
● Thus the mean will be smaller than the median, in
general, for a distribution that is skewed left
SKEWED RIGHT
● If a distribution is skewed right, there will be some
data values that are larger than the others
 The mean will increase
 The median will not increase as much
● Thus the mean will be larger than the median, in
general, for a distribution that is skewed right
FINDING MEAN AND MEDIAN ON
CALCULATOR
Birth Weights
5.8
7.4
9.2
7.0
8.5
7.6
7.9
7.8
7.9
7.7
9.0
7.1
8.7
7.2
6.1
7.2
7.1
7.2
7.9
5.9
7.0
7.8
7.2
7.5
7.3
6.4
7.4
8.2
9.1
7.3
•Find the mean and median
•Make a Histogram to discuss the shape of the
data
•How to sort data in lists
SUMMARY (IMPORTANT STUFF!)
 Mean
 The center of gravity
 Useful for roughly symmetric quantitative data
 Median
 Splits the data into halves
 Useful for highly skewed quantitative data
 Mode
 The most frequent value
 Useful for qualitative data