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Parameters and Statistics
A statistic is a descriptive measure
computed from a sample of data.
A parameter is a descriptive measure
computed from an entire population of data.
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Measures of Central Tendency
- Arithmetic Mean -
The arithmetic mean of a set of data is
the sum of the data values divided by
the number of observations.
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Sample Mean
If the data set is from a sample, then the
sample mean, , is:
X
n
X
x
i 1
n
i
x1  x2    xn

n
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Population Mean
If the data set is from a population, then
the population mean,  , is:
N
x
x1  x2    xn


N
N
i 1
i
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Measures of Central Tendency
- Median -
An ordered array is an arrangement of data
in either ascending or descending order.
Once the data are arranged in ascending
order, the median is the value such that 50%
of the observations are smaller and 50% of
the observations are larger.
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Measures of Central Tendency
- Median -
If the sample size n is an odd number, the
median, Xm, is the middle observation. If the
sample size n is an even number, the
median, Xm, is the average of the two middle
observations. The median will be located in
the 0.50(n+1)th ordered position.
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Measures of Central Tendency
- Mode -
The mode, if one exists, is the most
frequently occurring observation in
the sample or population.
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Shape of the Distribution
The shape of the distribution is said
to be symmetric if the observations
are balanced, or evenly distributed,
about the mean. In a symmetric
distribution the mean and median are
equal.
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Shape of the Distribution
A distribution is skewed if the observations are
not symmetrically distributed above and below
the mean. A positively skewed (or skewed to the
right) distribution has a tail that extends to the
right in the direction of positive values. A
negatively skewed (or skewed to the left)
distribution has a tail that extends to the left in
the direction of negative values.
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Shapes of the Distribution
Frequency
Symmetric Distribution
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9
8
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4
3
2
1
0
1
2
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8
Negatively Skewed Distribution
Positively Skewed Distribution
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12
10
10
8
8
Frequency
Frequency
9
6
4
6
4
2
2
0
0
1
2
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9
1
2
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Measures of Variability
- The Range -
The range is in a set of data is
the difference between the
largest and smallest observations
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Measures of Variability
- Sample Variance -
The sample variance, s2, is the sum of the squared
differences between each observation and the
sample mean divided by the sample size minus 1.
n
s 
2
 (x  X )
i 1
2
i
n 1
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Measures of Variability
- Short-cut Formulas for s2
Short-cut formulas for the sample variance, s2, are:
( xi ) 2
xi 

n
2
i 1
s 
n 1
n
or
s2 
2
2
x

n
X
i
n 1
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Measures of Variability
- Population Variance The population variance, 2, is the sum of the
squared differences between each observation and
the population mean divided by the population size,
N.
N
 
2
 (x  )
i 1
2
i
N
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Measures of Variability
- Sample Standard Deviation -
The sample standard deviation, s, is the positive
square root of the variance, and is defined as:
n
s s 
2
 (x  X )
i 1
2
i
n 1
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Measures of Variability
- Population Standard Deviation-
The population standard deviation, , is
N
  
2
 (x  )
i 1
2
i
N
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The Empirical Rule
(the 68%, 95%, or almost all rule)
•
•
•
For a set of data with a mound-shaped histogram, the
Empirical Rule is:
approximately 68% of the observations are contained with a
distance of one standard deviation around the mean;  1
approximately 95% of the observations are contained with a
distance of 2 standard deviations around the mean;  2
almost all of the observations are contained with a distance
of three standard deviation around the mean;  3
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Coefficient of Variation
The Coefficient of Variation, CV, is a measure
of relative dispersion that expresses the
standard deviation as a percentage of the
mean (provided the mean is positive).
The sample coefficient of variation is
s
CV  100
X
if X  0
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Coefficient of Variation
The population coefficient of variation is

CV  100

if   0
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Percentiles and Quartiles
Data must first be in ascending order.
Percentiles separate large ordered data sets
into 100ths. The Pth percentile is a number
such that P percent of all the observations are
at or below that number.
Quartiles are descriptive measures that
separate large ordered data sets into four
quarters.
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Percentiles and Quartiles
The first quartile, Q1, is another name for the
25th percentile. The first quartile divides the
ordered data such that 25% of the observations
are at or below this value. Q1 is located in the
.25(n+1)st position when the data is in ascending
order. That is,
(n  1)
Q1 
ordered position
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Percentiles and Quartiles
The third quartile, Q3, is another name for the
75th percentile. The first quartile divides the
ordered data such that 75% of the observations
are at or below this value. Q3 is located in the
.75(n+1)st position when the data is in
ascending order. That is,
3(n  1)
Q3 
ordered position
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Interquartile Range
The Interquartile Range (IQR) measures
the spread in the middle 50% of the data;
that is the difference between the
observations at the 25th and the 75th
percentiles:
IQR  Q3  Q1
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