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 2 2 Where d = X - A (Assumed mean) Also s.d = X 2 - X 2 where X= variables N
Example: Blood Serum cholesterol levels of 10 persons are as under: 240288 260272 290263 245277 255251 Calculate the standard deviation with the help Of 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 NN 2 Step Deviation Method: = f d 2 NN 2 X 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 4503 5005 5508 6007 6509 7007 7504 8007 Calculate the standard deviation of the salaries.
Example: Calculate mean and standard deviation of the following Frequency distribution of marks: MarksNo. of Students 0-1005 10-2012 20-3030 30-4045 40-5050 50-6037 60-7021
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 : ExpenditureNo. of students 78-8202 73-7706 68-7207 63-6712 58-6218 53-5713 48-5209 43-4707 38-4204 33-3702 Calculate 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. XY 35108 54107 52105 53105 56106 58107 52104 50103 51104 49101
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 AFactory B No. of employees100150 Average wage32002800 Variance of wage625729 (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 obtainedNo. of Students 10-1402 14-1804 18-2204 22-2608 26-3012 30-3416 36-3810 38-4208 42-4604 46-5006 50-5402 54-5804
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 ness Mean =Median =Mode Zero skew ness Mean =Median =Mode
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: Mean and median are to the right of the mode. Mean>Median>Mode
Negatively Skewed: Mean and Median are to the left of the Mode. Mean<Median<Mode
Karl Pearson’s Coefficient of Skewness: S p = Mean – Mode/ Standard Deviation Bowley’s Coefficient of Skewness S B = Q3 + Q1 – 2 Median/Q3 – Q1