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TABLE 20: Formulas for Descriptive Statistics
The formulas above are for ungrouped data. For grouped data, see below.
Basic Moments
Population Data
Sample Data
Mean
x
x

x
n
N
2
Variance
 x  x 2
x   
2
2
Definitional
s 
 
n 1
N
Formula




Variance
Computational
Formula
Coefficient of
Variation
Skewness
Definitional
Formula
Skewness
Computational
Formula
Relative Skewness
Kurtosis
Definitional
Formula
Coefficient of
Excess
2 
C
x
2
N
 2
s2 


 x   

3
3
1 
1
N
 x
3
 3
x
2
 2N  3

3

3
 x   

4
4
n 1
k 3
n
(n  1)( n  2)
 x  x 
k 3
n
(n  1)( n  2)
 x
g1 
k3
k4 
Moments or Fractiles
Moments
g2 
Ungrouped Data
As above
3
3
 x  x
4
n


3n  13 s 4 

n2

k4
s4


Location of Fractile
position  pn  1  a.b
position  pn  1
Value of Fractile
x1 p  xa  .bxa 1  xa 
 pN  F 
x1 p  L p  
w
 f p 
Other Basic Formulas
Px    k 
Standard Deviation
variance
Pearson's Measures of Skewness SK1 
1
k2
mean  mode ,
std .deviation
Pg. 52
SK 2 
x
n2

n  1n  2n  3
Grouped Data
In above formulas
f for
substitute
Tchebyschev Relation
 3x
s3

n  1


4
3
4
 nx 2
s
x
N
2 
2
C
N
3 
x
3mean  median
std .deviation
.
2
 2nx 3

Other Means
Geometric Mean
x g  x1  x 2  x 3  x n  n  n
Harmonic Mean
1
1

xh n
Root Mean Square
x rms 
1
x
or
1
n
x
 
ln x g 
1
n
 ln( x)
x
1
1
n
x
2
or
x rms 2 
2
Measure of relation between two variables
Sample Covariance (For later)
s xy 
 x  x  y  y    xy  nx y
n 1
n 1
Notes and Warnings
The mode is the data point that occurs most or the midpoint of the largest class.
For the median, use the formulas for fractiles with p  .5
If you have a sample, you must compute sample statistics. If you have a population made up of
data points and (perhaps) observed frequencies. If you have a population given in the form of points and
probabilities, you cannot use the formulas above and must go to materials on random variables.
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