This document discusses measures used to describe the central tendency and dispersion of a frequency distribution. It outlines four key properties: central tendency, dispersion, skewness, and kurtosis. For measures of central tendency, it describes the arithmetic mean, median, and mode. For measures of dispersion, it discusses range, variance, standard deviation, coefficient of variation, and standard error of the sample mean.

1. Measures of central tendency &
measures of dispersion
Prepared by:
Dr. Namir Al-Tawil

2. There are four basic properties
to describe any frequency
distribution:
 Central Tendency.
 Dispersion.
 Skewness.
 Kurtosis.

3. 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).

4. 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.

5. 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.

6. 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.

7. Measures of Dispersion
 1. The Range
Calculated by subtraction the lowest
observed value from the highest.

8. 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

9. 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.

10. 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.

11. 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

12. So
 Standard Error is the standard deviation of
the sample means.
 Or SD of M1, M2, M3, M4 etc…