study

1. MEASURES OF
DISPERSION, SKEWNESS
AND KURTOSIS
 STANDARD DEVIATION (𝝈) AND ROOT MEAN SQUARE DEVIATION (S)
 RELATION BETWEEN ☌ AND S
Chapter 3
Reporters: JAZER B. LEUTERIO
RINA CRIS DE MESA
JULIAH ANGELICA B. CRISTOBAL
MPA STUDENTS

2. STANDARD DEVIATION (𝝈) AND ROOT MEAN SQUARE
DEVIATION (S)
Standard deviation, usually denoted by the
Greek letter small sigma (𝜎), is the positive
square root of the arithmetic mean. For the
frequency distribution
xi /𝑓𝑖, 𝑖 = 1, 2, … , 𝑛,
𝜎 =
1
𝑁
𝛴∫ 𝑖(𝑥 − 𝑥 )
2
where 𝑥 is the arithmetic mean of the distribution and 𝑡 𝑓𝑖 = 𝑁

3. STANDARD DEVIATION (𝝈) AND ROOT MEAN SQUARE
DEVIATION (S)
The step of squaring the deviations (xi -
x) overcomes the drawback of ignoring the
signs in mean deviation. Standard deviation
is also suitable for further mathematical
treatment (∮ 3.7.3). Moreover of all the
measures, standard deviation is affected least
by fluctuations of sampling.

4. STANDARD DEVIATION (𝝈) AND ROOT MEAN SQUARE
DEVIATION (S)
Thus we see that standard deviation
satisfies almost all properties laid down for
an ideal measure of dispersion except for the
general nature of extracting the square root
which is not readily comprehensible for non-
mathematical person.

5. STANDARD DEVIATION (𝝈) AND ROOT MEAN SQUARE
DEVIATION (S)
It may also be pointed out that standard
deviation gives greater weight to extreme
values and such has not found favor with
economists or businessmen who are more
interested in the results of the modal class.

6. STANDARD DEVIATION (𝝈) AND ROOT MEAN SQUARE
DEVIATION (S)
Taking into consideration the pros and cons
and also the wide applications of standard
deviation in statistical theory, we may regard
standard deviation as the best and the most
powerful measure of dispersion.

7. STANDARD DEVIATION (𝝈) AND ROOT MEAN SQUARE
DEVIATION (S)
The square of standard deviation is called the
variance and is given by
𝜎2
=
1
𝑁 𝑖 𝑓𝑖(𝑥𝑖 - 𝐴)2

8. STANDARD DEVIATION (𝝈) AND ROOT MEAN SQUARE
DEVIATION (S)
Root mean square deviation, denoted by ‘s’ is
given by
𝑠=
1
𝑁 𝑖 𝑓𝑖(𝑥𝑖- 𝐴)2
where A is any arbitrary number. 𝑠2
is called
mean square deviation.

9. RELATION BETWEEN 𝝈 AND 𝒮
By definition we have
𝑠=
1
𝑁 𝑖 𝑓𝑖(𝑥𝑖- 𝐴)2
=
1
𝑁 𝑖 𝑓𝑖 (𝑥𝑖 - 𝑥 + 𝑥 - 𝐴)2
=
1
𝑁 𝑖 𝑓𝑖 [ (𝑥𝑖 -𝑥 )2 + (𝑥 - 𝐴)2
+ 2(𝑥 - A) (𝑥𝑖 - 𝑥 ) ]
=
1
𝑁 𝑖 𝑓𝑖 [ (𝑥𝑖 -𝑥 )2 + (𝑥 - 𝐴)2 1
𝑁 𝑖 𝑓𝑖 + 2 (𝑥 - A) 𝑖 𝑓𝑖 [ (𝑥𝑖 -𝑥 )

10. RELATION BETWEEN 𝝈 AND 𝒮
(𝑥 - A), being constant is taken outside, the
summation sign. But 𝑖 𝑓𝑖 [ (𝑥𝑖 -𝑥 )= 0, being
the algebraic sum of the deviations of the
given values from their mean.
Thus……….
𝑠2 = 𝜎2 + ( 𝑥 - 𝐴)2= 𝜎2 + 𝑑2, where d= 𝑥 - A

11. RELATION BETWEEN 𝝈 AND 𝒮
Obviously s2 will be least when d=0, i.e., 𝑥 = A.
Hence mean square deviation and consequently
root mean square deviation is least when the
deviations are taken from A= 𝑥, i.e., standard
deviation is the least value of root mean square
deviation.

12. RELATION BETWEEN 𝝈 AND 𝒮
The same result could be obtained alternatively
as follows:
Mean square deviation is given by
𝑠2
=
1
𝑁 𝑖 𝑓𝑖( 𝑥𝑖 - 𝐴)2
It has been shown in ∮ 2.5.1property 2 that
𝑖 𝑓𝑖( 𝑥𝑖 - 𝐴)2
is minimum when

13. RELATION BETWEEN 𝝈 AND 𝒮
A= 𝑥. Thus mean square deviation is minimum
when A= 𝑥 and its minimum value is
(𝑠2
) min =
1
𝑁 𝑖 𝑓𝑖 [ (𝑥𝑖 -𝑥 )2 = 𝜎2
Hence variance is the minimum value of mean square deviation or
standard deviation is the minimum value of root mean square
deviation.

14. Thank you 😊