x

Sample mean

x

Population mean

x



Weighted mean

x

i

n i N

w x
w
i

i

i

Geometric mean

x g  (x1 )(x 2 ) (x n )   (x1 )(x 2 ) (x n ) n Calculation of the pth percentile

1 n  p  i 
n
100



If i is not an integer, round up. The next integer greater than i denotes the position of the pth percentile. If i is an integer, the pth percentile is the average of the values in positions i and i + 1.

1

Quartiles
Q1 = 1st quartile, or 25th percentile
Q2 = 2nd quartile, or 50th percentile, or median
Q3 = 3rd quartile, or 75th percentile
Range = largest value – smallest value
Interquartile range

Population variance

IQR = Q3 – Q1

2 

  xi   
N



Population standard deviation

Sample variance

Sample standard deviation

2

s 
2

s

 x

i  

2

N

  xi  x 

2

n 1

 x

i

 x

n 1

x


2 i  nx 2

n 1

2



x

2 i  nx 2

n 1
2

Coefficient of variation

Skewness

z-score

 standard deviation 

100% mean 


 xi  x 
 n  1 n  2   s  n zi 

xi  x s Chebyshev’s Theorem: At least 1 

3

1 of the data values will be within z standard deviations of the z2 mean, where z is any value grater than 1.
Empirical Rule:

For data having a bell-shaped distribution:
•

Approximately 68% of the data values will be within one standard deviation of the mean

•

Approximately 95% of the data values will be within two standard deviations of the mean

•

Approximately 99.7% of the data values will be within three standard deviation of the mean

3

Detecting outliers:

z-score approach: z  3
Q1 ,Q3 and IQR approach :

Lower limit: Q1  1.5 IQR 
Upper limit: Q3  1.5 IQR 

Sample covariance s xy 

Sample correlation

 x

rxy 

i

s xy sxsy  x  yi  y  n 1

where s x 

  xi  x  n 1

2

and s y 

  yi  y 

2

n 1

4

Counting rule for