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 =

n

Advantage -Simple to compute.

-All values are included.

- Amenable for tests of

statistical significance

Disadvantage - 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 squared

deviation of the values from the mean

divided 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 the

denominator to adjust for the fact that we are working

with sample parameters rather than population

parameters, 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 the

deviations (x-x) are independent from each other. The

Last one can always be calculated from the others

because 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 be

exactly equal to the population mean.

The standard error measures the variability

of the mean of the sample as an estimate of

the true value of the mean for the population

from 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…