Measures of Variation1.)Range ( R ) - the difference in value

between the highest(maximum) and the lowest(minimum) observation.

• R = Highest – Lowest

2.)Mean Deviation – the sum of all the absolute values of the deviations from the mean divided by the number of observations.

MAD = X – X

n

3.) Sample Variance ( S2 ) – the sum of the square of each deviation from the mean divided by the number of observations minus 1.

S2 = X – X )2

n – 1

Standard Deviation ( S ) – the square root of variance.

S = ( X- X )2

n – 1

Equations for Population and Sample Means and Standard Deviations

Quantity Sample Population

Mean X = X = X

n N

Variance S2 = X – X )2 2 = ( X – X )2

n - 1 N

Standard S = ( X – X )2 ( X – X )2

Deviation n – 1 N

Sample Standard Deviation FormulaUngrouped data

S = X2 X 2

n n

Grouped data

S = nfX2 - (fX)2 LONG METHOD

n(n-1)

S = c nfd2 - (fd)2 CODED METHOD

n(n-1)

Averaging Standard Deviation

2 2 2 2 2

ST = nA(XA+SA ) + nB(XB+SB) XT

nA+nB

where nA,nB – sample sizes

XA,XB – mean of the groups

XT - mean of the two groups combined

SA,S B - standard deviation of the two groups being combined

When there are more than two groups, the above formula may be modified by placing additional terms in the numerator and denominator.

Coefficient of Variation– Measure of relative dispersion or percentages– Relates the standard deviation to its mean and convert it

to percentage.

C.V. = s ( 100 ) X

Skewness

- is the degree of asymmetry or departure from symmetry exhibited by a frequency distribution.

Sk = Mean – Mode = 3 ( Mean – Median)

Standard Deviation Standard Deviation

• Kurtosis– the degree of peakedness or humpedness’

• Leptokurtic- slender pointed curve.

• Platykurtic- flat or broad curve.

• Mesokurtic- not too pointed nor too flat.

For a mesokurtic distribution, the value of Kurtosis is 3. For a leptokurtic distribution, the value is greater than 3.

For a platykurtic distribution, the value is less than 3.

Probability