statistics topic-2 Measures of Dispersion & skewness 1.Measures of Dispersion
intro--
Dispersion is a measure of the variation of the items. If value of items are same then there is no variance and dispersion will be zero.
More variations more will be dispersion. MethoDs of Measuring Dispersion
1.Range 3.Means 5.Coefficient deviation of variation
2.Interquartile range 4.Standard 6.Lorenze & deviation. Quartile deviation curve. 1.Range
It is simplest measure of dispersion.
It is define the difference b/w largest and smallest value.
R=L - S
15,22,17,32,28
Largest value = 32 smallest value = 15
R=L - S
R= 32-15= 17. Coefficient of Range
L-S 32-15 17 = = 0.36 L+S 32+15 47 2.Interquartile range & Quartile deviation
A. Interquartile range.
It is the difference b/w upper upper quartile (Q3) and lower quartile(Q3)
IQ.Range = Q3- Q1
B. Quartile deviation.
It also called as semi-interquartile. Q3-Q1 Q.D = 2 C. Coefficient of Quartile deviation.
Q3-Q1 Coeff. Of Q.D = Q3+Q1 3.Means deviation
It is also known as average deviation.
It is the difference b/w upper upper quartile (Q3) and lower quartile(Q3) Mean deviation can be can be computed either from the mean or median. ∑(X-M) M.D from median = N
∑(X-x)̄ M.D from Mean = N 4.Standard deviation
Standard deviation is most widely used measure of dispersion.
S.D first used by Karl Pearson. In 1893.
S.D is also called as root mean square deviation.
S.D is denoted as - σ
2 σ = ∑(X-x̄̄) N S.D is widely used in sampling theory and test of significant. Coefficient of S.D = σ x̄ Difference between mean deviation and S.D
1.In M.D sign + and – ignored. In S.D +,- sign not ignored.
2.M.D can be calculated either from mean ,median and mode. S.D is always computed from mean only. Variance
It is also a measure of dispersion.
The term variance is first used by R.A.Fisher in 1918.
Variance is the square of the standard deviation.
Variance= σ2
∑f(X-x̄̄)2 Variance N = 5.Coefficient of variance.
Coefficient of variation is an important relative measure of dispersion.
It was developed by Karl Pearson. It is widely used in comparing the variability of two or more series.
It is denoted as C.V.
σ C.V. *100 x̄ = σ C.V. *100 x̄ = Variance= 324
324 = 18
18 * 100 180 1800 =10 180 6.Lorenze curve
It is graphical method of dispersion.
It studies the degree of inequality in the distribution of income and wealth b/w countries. 1.Measures of skewness 1.Measures of skewness
Skewness means lack of symmetry in frequency distribution.
It gives us idea about the shape of the frequency curve. When a distribution is not symmetrical it is called a skewed distribution.
Skewness tells us about the asymmetry of the frequency distribution. syMMetrical anD skeweD Distribution.
1.Symmetrical distribution-
In a symmetrical dist. Skewness is not present. In this situation mean,median and mode are equal.
x̄=M=Z 1.Perfectly symmetrical distribution.
In this case x̄,M, and Z. are equal.
In this case normal distribution is bell shaped.
-VE +VE x=M=Z̄ In normal distribution mean is zero and variance is 1. 2.skewed distribution-
a. positively skewed b. Negatively distribution. skewed distribution. a. positively skewed.
In this case x̄,>M> Z. .
In this case normal distribution is +vly skewed..
-VE Z M x̄ +VE Tail right side. b. negatively skewed.
In this case x̄, In this case normal distribution is -vly skewed.. -VE x̄ M Z +VE Tail left side. Measures of skewness 1.Karl Pearson 3.kelly’s method. method 2.Bowley Method 1.Karl Pearson method Carl Pearson method based on 4 components. Mean,median,mode,and S.D. Absolute measure Relative measure/coef. Of skewness x̄-z Sk= x̄-z Coeff. Of Sk= σ When mode is ill(eliminate) When mode is ill(eliminate) Sk= 3(x̄-M) 3(x̄-z) Coeff. Of Sk= σ In Karl Pearson method , coeff. Of skewness usually lies bw +1 or -1. If mode(z) is ill then value of skewness lies b/w +3 or -3. 2.Bowley Method Bowley method is based upon median(M). Bowley used first quartile (Q1) and third quartile(Q3). Bowley method is also called quartile method of measuring skewness . Absolute measure Relative measure/coef. Of skewness Q3+Q1-2M Coeff. Of Sk= Sk= Q3+Q1-2M sQ3-Q1 3.kelly’s method. Kellys’s method is based on percentiles (%) and deciles(.) Absolute measure Relative measure/coef. Of skewness P90+P10-2M Coeff. Of Sk= Sk= P90+P10-2M P90-P10 This method is not popular . It is suitable when the skewness is based on percentiles or deciles. statistics topic-3 Measures of kurtosis & correlation