SECTION 3.1 - MEASURES OF CENTRAL TENDENCY
Definitions
The arithmetic mean of a variable is computed by determining the sum of all the
values of the variable in data set and dividing by the number of observations. The
population arithmetic mean, ., is computed using all the individuals in a population. If
B" ß B# ß Þ Þ Þ ß BR are the R observations of a variable from the population, then the
population mean, ., is
B3
B"  B#  â  BR
.œ
œ
R
R
The population mean is a parameter.
The sample arithmetic mean, B, is computed using sample data. If B" ß B# ß Þ Þ Þ ß
B8 are the 8 observations of a variable from a sample, then the sample mean, B, is
B3
B"  B#  â  B8
Bœ
œ
8
8
The sample mean is a statistic.
Example 1
Use the divdend data from Example 1 in Section
dividend.
1.7
0
1.15 0.62 1.06 2.45
2.83 2.16 1.05 1.22 1.68 0.89
2.59 0
1.7
0.64 0.67 2.07
2.04 0
0
1.35 0
0
NOTE:
2.2 to calculate the mean
2.38
0
0.94
0.41
Round the value of the mean to one more decimal place than that of
the raw data.
Definition
The median of a variable is the value that lies in the middle of the data when
arranged in ascending order. The book will use Q to represent the median
Example 2
Use the divdend data from Example 1
dividend.
1.7
0
1.15 0.62
2.83 2.16 1.05 1.22
2.59 0
1.7
0.64
2.04 0
0
1.35
in Section 2.2 to calculate the median
1.06
1.68
0.67
0
2.45
0.89
2.07
0
2.38
0
0.94
0.41
Definitions
The mode of a variable is the most frequent observation of the variable that
occurs in the data set. If no observation occurs more than once, the data set has no mode.
If the data set has two modes, the data is said to be bimodal, and if the data set has three
or more modes, it is said to be multimodal.
Example 3
Use the divdend data from Example 1 in Section 2.2 to calculate the mode.
1.7
0
1.15 0.62 1.06 2.45 2.38
2.83 2.16 1.05 1.22 1.68 0.89 0
2.59 0
1.7
0.64 0.67 2.07 0.94
2.04 0
0
1.35 0
0
0.41
Definition
A numerical summary of data is said to be resistant if extreme values (very large
or small) relative to the data do not affect its value substantially.
Relation Between the Mean, Median, and Distribution Shape
Distribution Shape
Mean versus Median
Skewed left
Mean substantially smaller than median
Symmetric
Mean roughly equal to median
Skewed right
Mean substantially larger than median