“STANDARD DEVIATION”

Standard Deviation:

• Std deviation is the best and scientific method of dispersion.

• It is widely used method used in statistical analysis.

• Std deviation is that method of dispersion where deviations are taken from

mean and while taking deviations algebraic signs are kept in mind.

• Also known as root mean square deviation.

• The greater the standard deviation, the greater will be the magnitude of the values from their mean.

• A small standard deviation means a high degree of uniformity of the observations as well as homogeneity of the series.

• The standard deviation is useful in judging the effectiveness of the mean.

Standard Deviation Individual Series:

= x 2 N

Where x = X - X

If Assumed mean is taken

=

d - d

N N

22

Where d = X - A (Assumed mean)

Also

s.d = X2 - X 2 where X= variables

N N

Example:

Blood Serum cholesterol levels of 10 persons are as under:

240 288260 272290 263245 277255 251

Calculate the standard deviation with the helpOf assumed mean.

Standard Deviation (Discrete series)/ Continuous Series

• Actual mean method• Assumed mean method• Step Deviation method

Actual mean method:

= fx

2

N

Where x = X – (mean)

Assumed Mean Method:

= f d 2 f d

NN

2

Step Deviation Method:

= f d 2 f d

NN

2X i

Here d = X – A / i

Example:

The annual salaries of a group of employees are given in the following table:

Salaries(in 000) Number of persons 45 03

50 05 55 08 60 07 65 09 70 07 75 04 80 07

Calculate the standard deviation of the salaries.

Example:

Calculate mean and standard deviation of the followingFrequency distribution of marks:

Marks No. of Students0-10 0510-20 1220-30 3030-40 4540-50 5050-60 3760-70 21

Coefficient of Variations:

C.V = / X x 100

Example: The following table shows that monthly expenditures of 80 students of a university on morning breakfast :

Expenditure No. of students78-82 0273-77 0668-72 0763-67 1258-62 1853-57 1348-52 0943-47 0738-42 0433-37 02Calculate standard deviation and coefficient of variation of above data

Example:

From the prices of shares of X and Y below find out which is more stable in value.

X Y35 10854 10752 10553 10556 10658 10752 10450 10351 10449 101

Variance: It is the square of the standard deviation

i.e..Variance = 2

Example:The number of employees, wages per employee and the variance of the wages per employee for two factories is given below

Factory A Factory BNo. of employees 100 150Average wage 3200 2800Variance of wage 625 729(a) In which factory is there a greater variation in the distribution of wages per employee.(b) Suppose in factory B, the wages of an employee were wrongly noted as Rs. 3050 instead of Rs. 3650, what would be the correct variance for factory B.

Example:The mean of 5 observations is 4.4 and the variance is 8.24. If the three of the five observations are 1, 2 and 6, find the other two.

Example:The following table gives the marks obtained by a group of 80 students in an examination. Calculate the variance.Marks obtained No. of Students10-14 0214-18 0418-22 0422-26 0826-30 1230-34 1636-38 1038-42 0842-46 0446-50 0650-54 0254-58 04

Skew ness:

 “It refers to the asymmetry or lack of symmetry in the shape of a frequency distribution.”

As far the study of central tendency the statistical average is calculated and for scatter of values, dispersion is measured. In the same way to study the symmetrical or asymmetrical nature of series, skew ness is calculated

  Types of frequency distribution:

  1. Normal frequency distribution

2. Asymmetrical Distribution

1. Normal frequency distribution

One main feature of normal distribution is that mean, median and mode are found equal .in such a distribution the frequencies gradually increase, they are maximum in the center and then decrease. When this distribution is plotted on a graph it will be a bell-shaped graph. It is also called normal curve.

Zero skew nessZero skew ness Mean =Median =ModeMean =Median =Mode

M o d e

M ed ia n

M ea n

2. Asymmetrical Distribution:- In a asymmetrical distribution the rate of increase or decrease of frequencies is not same. Mean, median and mode are not equal such a distribution is called asymmetrical distribution. It is of two types:

(i) Positive Skew ness: When in a series mean is more than median and median is more than mode then skew ness is positive i.e curve is seen more towards left.

(ii) Negative Skew ness:- When in a series mean is less than median and median is less than mode then skew ness is negative i.e curve is seen more towards right.

Positively skewed:Positively skewed: Mean and median are to the right of the mode. Mean and median are to the right of the mode.

Mean>Median>Mode

M o d e

M ed ia n

M ea n

Negatively SkewedNegatively Skewed:: Mean and Median are to the left of the Mode. Mean and Median are to the left of the Mode.

Mean<Median<ModeMean<Median<Mode

M o d eM ea n

M ed ia n

Karl Pearson’s Coefficient of Skewness:

Sp = Mean – Mode/ Standard Deviation

Bowley’s Coefficient of Skewness

SB = Q3 + Q1 – 2 Median/Q3 – Q1