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…