Seminar onStandard Deviation

Presented by:

Jiban Ku. SinghM. Sc Part-I (2015-16)

P. G. DEPARTMENT OF BOTANYBERHAMPUR UNIVERSITY

BHANJA BIHAR, BERHAMPUR- 760007GANJAM, ODISHA, INDIA

E-mail- jibansingh9@gmail.com

Summary Measures

Central Tendency

Mean

MedianMode

Summary Measures

Variation

Quartile deviation

Standard Deviation

Mean DeviationRange

Standard Deviation

• While looking at the earlier measures of dispersion all of them suffer from one or the other demerit i.e.• Range –it suffer from a serious drawback considers only 2 values and

neglects all the other values of the series.• Quartile deviation considers only 50% of the item and ignores the

other 50% of items in the series.• Mean deviation no doubt an improved measure but ignores negative

signs without any basis.

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Standard Deviation• The concept of standard deviation was first introduced by Karl Pearson in

1893.• Karl Pearson after observing all these things has given us a more scientific

formula for calculating or measuring dispersion. While calculating SD we take deviations of individual observations from their AM and then each squares. The sum of the squares is divided by the Total number of observations. The square root of this sum is knows as standard deviation.• The standard deviation is the most useful and the most popular measure of

dispersion.• It is always calculated from the arithmetic mean, median and mode is not

considered.

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Definition:

• Standard Deviation is the positive square root of the average of squared

deviation taken from arithmetic mean.

• The standard deviation is represented by the Greek letter (sigma).

• Formula.

• Standard deviation ==

Formula

• Standard deviation = =

• Alternatively =

¿√ ∑ (𝒙𝟐−𝟐 𝒙 𝒙+𝒙𝟐 )𝒏

¿√ ∑𝒙𝟐

𝒏 −𝟐 𝒙∑ 𝒙𝒏 +∑𝒙

𝟐

𝒏¿√ ∑𝒙𝟐

𝒏 −𝟐 𝒙 𝒙+𝐧 𝒙𝟐

𝒏¿√ ∑𝒙𝟐

𝒏 −𝟐 𝒙𝟐+𝒙𝟐

¿√ ∑𝒙𝟐

𝒏 − 𝒙𝟐

CALCULATION OF STANDARD DEVIATION-INDIVIDUAL OBSERVATION

Two Methods:-

By taking deviation of the items from the actual mean.

By taking deviation of the items from an assumed mean.

CASE-I. When the deviation are taken from the actual mean.

DIRECT METHOD

Standard deviation ==or

=value of the variable of observation,= arithmetic mean = total number of observations.

Example : Find the mean respiration rate per minute and its standard deviation when in 4 cases the rate was found to be : 16, 13, 17 and 22.

• Solution:Here Mean =

16131722

Standard deviation == ==

-1-405

1160

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Short-Cut Method Standard deviation ==

CASE-II. When the deviation are taken from the Assumed mean.

= = =

= = 16.398

Example: Blood serum cholesterol levels of 10 persons are as under: 240, 260, 290, 245, 255, 288, 272, 263, 277, 251.

calculation standard deviation with the help of assumed mean.

ValueA=264

240260290245255288272263277251

57616

67636181

576641

169169

-24-426-19-9248-11313

Here, Mean= = = 9

= 263.9 is a fraction.

CALCULATION OF STANDARD DEVIATION- DISCERETE SERIES OR GROUPED DATA

Three Methods

a) Actual Mean Method or Direct Method

b)Assumed Mean Method or Short-cut Method

c) Step Deviation Method

a) Actual Mean Method or Direct Method

• The S.D. for the discrete series is given by the formula.=

Where is the arithmetic mean, is the size of items, is the corresponding frequency and

b) Assumed Mean Method or Short-cut Method

Standard deviation= =

Where is the assumed mean,

is the corresponding frequency and

Example:

Periods: 10 11 12 13 14 15 16

No. of patients: 2 7 11 15 10 4 1

Solution:Period

s:(x)No. of

patients()

10111213141516

27

11151041

Total N==50

-3-2-10123

-6-14-110

1083

=-10

9410149

1828110

10169

=92

Mean= =

= 13

= 12.8 is a fraction.

===== 1.342

c) Step Deviation Method

• We divide the deviation by a common class interval and use the following formula

Standard deviation= =×

Where common class interval,

is assumed mean f is the respective frequency.

=× = × = × = × = 1.235× =4.94 mm Hg.

Example:

B.P.(mmHg): 102 106 110 114 118 122 126

No. of days: 3 9 25 35 17 10 1Solution:B.P.(mmHg) No. of days ()

102106110114118122126

39

253517101

Total N=100

-3-2-10123

-9-18-250

17203

=-12

2736250

17409

=154

A. M = ×

mm Hg

CALCULATION OF STANDARD DEVIATION- CONTINUES SERIES

S.D. of Continues Series can be calculated by any one of the methods discussed

for discrete frequency distribution But Step Deviation Method is mostly used.

Standard deviation= =×

Where common class interval,

is assumed mean f is the respective frequency.

Example: I.Q. 10-20 20-30 30-40 40-50 50-60 60-70 70-80No. of students: 5 12 15 20 10 4 2Solution:

I.Q. No. of students:()

Mid-value

10-2020-3030-4040-5050-6060-7070-80

51215201042

Total =N=68

-3-2-10123

-15-24-150

1086

=-30

4548150

101618

=152

Standard deviation= = ==

15253545556575

CALCULATION OF COMBINED STANDARD DEVIATION

It is possible to compute combined mean of two or more than two groups.

Combined Standard Deviation is denoted by

=

Wherecombined standard deviation ,

a) Combined S.D. =

combined Mean = =

= = 55

The following are some of the particulars of the distribution of weight of boys and girls in a class:a) Find the standard deviation of the combined datab) which of the two distributions is more variable

Boys GirlsNumbers 100

50Mean weight 60 kg45 kgVariance() 9

4==

b)

C.V (Boys)=

C.V (Girls)=

MERITS OF STANDARD DEVIATION

Very popular scientific measure of dispersion

From SD we can calculate Skewness, Correlation etc

It considers all the items of the series

The squaring of deviations make them positive and the

difficulty about algebraic signs which was expressed in case

of mean deviation is not found here. 22

DEMERITS OF STANDARD DEVIATION

• Calculation is difficult not as easier as Range and QD• It always depends on AM• Extreme items gain great importance

The formula of SD is =Problem: Calculate Standard Deviation of the following series X – 40, 44, 54, 60, 62, 64, 70, 80, 90, 96

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USES OF STANDARD DEVIATION

It is widely used in biological studies .

It is used in fitting a normal curve to a frequency distribution.

It is most widely used measure of dispersion.

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THANK YOU