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
