Measures of central tendency & measures of dispersion

Prepared by:Dr. Namir Al-Tawil

There are four basic properties to describe any frequency

distribution:

Central Tendency. Dispersion. Skewness. Kurtosis.

Measures of Central Tendency

1. Arithmetic mean __ ∑X

X = nAdvantage -Simple to compute. -All values are included. - Amenable for tests of statistical significanceDisadvantage - Presence of extreme values (very high or very low values).

Measures of Central Tendency cont.

2. Median (50th percentile)Position of the median -For odd number of observations ( n+1/2 ) -For even number of observations ( n/2) & ( n/2 +1)Advantage of computing the median: -It is unaffected by extreme values.Disadvantage: -Provides no information about all values (observations). -Less amenable than the mean to tests of statistical

significance.

Measures of Central Tendency cont.

3. Mode It is the value that is observed most frequently in

a given data set.Advantage -Sometimes gives a clue about the

aetiology of the disease.Disadvantage -With small number of observations,

there may be no mode. -Less amenable to tests of

statistical significance.

Choice of measures of Central Tendency

For continuous variables with unimodal ( single peaked ) & symmetrical distribution; the mean, median & mode will be identical.

For skewed distribution, the median may be more informative descriptive measure.

For tests of statistical significance; the mean is used.

Measures of Dispersion

1. The Range Calculated by subtraction the lowest

observed value from the highest.

2. The Variance & the Standard Deviation

The variance: the sum of the squareddeviation of the values from the meandivided by sample size minus one. (∑x) 2

∑(x-x)2 ∑x2 - n

V= V=

n – 1 n - 1

The Standard Deviation (s.d.) = √v

Note :- The term ( n–1 ) rather than ( n ) is used in thedenominator to adjust for the fact that we are workingwith sample parameters rather than populationparameters, n–1 is called the number of

Degrees of freedom (d.f.) of the variance. The number is n-1 rather than n since only n-1 of thedeviations (x-x) are independent from each other. TheLast one can always be calculated from the othersbecause all n of them must add up to zero.

3. Coefficient of Variation

s.d.CV = X 100 X

Advantage: When two distributions have means of different magnitude, a comparison of the C.V. is therefore much more meaningful than a comparison of their respective s.d.

4. Standard Error of the Sample mean ( S.E. )

The sample mean is unlikely to beexactly equal to the population mean. The standard error measures the variabilityof the mean of the sample as an estimate ofthe true value of the mean for the populationfrom which the sample was drown. s.d. S.E. = √n

So

Standard Error is the standard deviation of the sample means.

Or SD of M1, M2, M3, M4 etc…