Measure of Dispersion, Range, Mean and Standard Deviation, Correlation and Regression Analysis
- 1. Measure of dispersion, Range, Mean and Standard Deviation, Correlation and Regression Analysis Presented by: Parth Chauhan M.Sc. Forensic Science LNJN NICFS, MHA FS-105
- 2. WHAT IS DISPERSION ? Dispersion measures the extent to which the items vary from some central value. central value = Mean / Mode / Median Example : 2 , 4 , 9 Arithmetic Mean = �𝑥𝑥𝑖𝑖 𝑛𝑛 = 5 ⇒ 2 to 5 has deviation of 3 ⇒ 4 to 5 has deviation of 1 ⇒ 9 to 5 has deviation of 4 Difference of the deviation of overall series from the central value is Dispersion. Also known as scatter value, spread value or variation.
- 3. MEASUREMENTS OF DISPERSION • Range • Quartile Deviation • Mean Deviation • Standard Deviation • Variance • Coeff. of Range • Coeff. of Quartile Deviation • Coeff. of Mean Deviation • Coeff. of Standard Deviation • Coeff. of Variance ABSOLUTE MEASURE RELATIVE MEASURE (UNIT SAME) (UNIT FREE)
- 4. ⇒ R A N G E ⇒ RANGE = L – S Where, L = Largest Observation S = Smallest Observation In Individual Series,
- 5. In Discrete Series, In Continuous Series ⇒ Inclusive series ,
- 6. In Continuous Series ⇒ Exclusive series , ⇒
- 7. ⇒ M E A N D E V I A T I O N Mean Deviation (MD) Mean (by default) Mode Median In Individual series, Step 1. Find Mean, � 𝑥𝑥 = �𝑥𝑥𝑖𝑖 𝑛𝑛 Step 2. Find absolute deviation, ( 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 ) Step 3. ∑ 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 Step 4. MD = � 𝑥𝑥𝑖𝑖− ̅ 𝑥𝑥 𝑛𝑛
- 8. In Discrete series, Step 1. Find Mean, � 𝑥𝑥 = � 𝑓𝑓𝑖𝑖𝑋𝑋𝑖𝑖 � 𝑓𝑓𝑖𝑖 Step 2. Find absolute deviation, ( 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 ) Step 3. Find, 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 Step 4. MD = � 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖− ̅ 𝑥𝑥 � 𝑓𝑓𝑖𝑖
- 9. In Continuous series, Step 1. Find mid point. Step 2. Find Mean, � 𝑥𝑥 = � 𝑓𝑓𝑖𝑖𝑋𝑋𝑖𝑖 � 𝑓𝑓𝑖𝑖 Step 3. Find absolute deviation, ( 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 ) Step 4. Find, 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 Step 5. MD = � 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖− ̅ 𝑥𝑥 � 𝑓𝑓𝑖𝑖
- 10. ⇒ S T A N D A R D D E V I A T I O N 𝜎𝜎 = � 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖− ̅ 𝑥𝑥 2 𝑁𝑁 ̅ 𝑥𝑥 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑁𝑁 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑖𝑖𝑖𝑖 𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠 𝜎𝜎 = ∑ 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 2 𝑁𝑁 − 1 ̅ 𝑥𝑥 = Sample mean N = Total no of set in a sample In Individual series, Step 1. Calculate Mean, � 𝑥𝑥 = � 𝑋𝑋𝑖𝑖 𝑛𝑛 Step 2. Calculate (𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥) Step 3. Calculate 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 2 Step 4. 𝜎𝜎 = � 𝑥𝑥𝑖𝑖− ̅ 𝑥𝑥 2 𝑁𝑁
- 11. In Discrete series, Step 1. Calculate Mean, � 𝑥𝑥 = ∑ 𝑓𝑓𝑓𝑓 ∑ 𝑓𝑓 Step 2. Calculate (𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥) Step 3. Calculate 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 2 Step 4. Calculate 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 2 Step 5. 𝜎𝜎 = � 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖− ̅ 𝑥𝑥 2 𝑁𝑁
- 12. In Continuous series, Step 1. Calculate mid point (x) Step 2. Calculate Mean, � 𝑥𝑥 = ∑ 𝑓𝑓𝑓𝑓 ∑ 𝑓𝑓 Step 3. Calculate (𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥) Step 4. Calculate 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 2 Step 5. Calculate 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖 − ̅ 𝑥𝑥 2 Step 6. 𝜎𝜎 = � 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖− ̅ 𝑥𝑥 2 𝑁𝑁
- 13. WHAT IS CORRELATION ANALYSIS ? Correlation analysis deals with association between two or more variables. The degree of relationship between the variables under consideration is measured through the correlation analysis. The measure of correlation called the “Correlation Coefficient” or “Correlation index” summarizes in one figure the direction & correlation.
- 14. • Direct correlation • If both the variables varies in same direction. • X and Y both increase or decrease • value lies between 0 to 1 Positive Correlation Negative Correlation • Inverse correlation • If varies in opposite direction. • X increase and Y decrease or vice versa • Value lies between -1 to 0 Correlation Coefficient (r) ranges between -1 to 1 • If r = 1 i.e. perfect positive correlation • If r = -1 i.e. perfect negative correlation • If r = 0 i.e. No correlation
- 15. ⇒ 𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐧𝐧′𝐬𝐬 𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏 𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌 𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌 Cov (x,y) = ∑ 𝑥𝑥− ̅ 𝑥𝑥 𝑦𝑦− � 𝑦𝑦 𝑛𝑛 ⇒ ⇒ 𝜎𝜎𝑥𝑥 = ∑ 𝑥𝑥 − ̅ 𝑥𝑥 2 𝑛𝑛 𝜎𝜎𝑦𝑦 = ∑ 𝑦𝑦 − � 𝑦𝑦 2 𝑛𝑛 ⇒ ⇒ r = To find the value of r , we require ∑ 𝑥𝑥 , ∑ 𝑦𝑦 , � 𝑥𝑥2 ,� 𝑦𝑦2 & ∑ 𝑥𝑥𝑥𝑥 from the dataset. Where, n is the no of pair of observations ∑ 𝑋𝑋𝑋𝑋 − ∑ 𝑥𝑥𝑥𝑥𝑥𝑥 ∑ 𝑥𝑥2 − ∑ 𝑥𝑥 2 ⋅ ∑ 𝑦𝑦2 − ∑ 𝑦𝑦 2
- 16. Step 1. Calculate 𝑥𝑥2 , 𝑦𝑦2 & 𝑥𝑥𝑥𝑥 Step 2. Calculate � 𝑥𝑥2 ,� 𝑦𝑦2 & ∑ 𝑥𝑥𝑥𝑥 Step 3. Put values in the correlation formula. Step 4. Simplify and find the value.
- 17. WHAT IS REGRESSION ANALYSIS ? Regression Analysis is the technique for measuring or estimating the relationship among variables. Regression Analysis provides estimates of values of the dependent variable from the values of the independent variables. The regression lines describes the average relationship existing between x and y.
- 18. REGRESSION LINES Regression line of y on x • 𝑦𝑦 − � 𝑦𝑦 = 𝑏𝑏𝑦𝑦𝑦𝑦 𝑥𝑥 − ̅ 𝑥𝑥 • 𝑏𝑏𝑦𝑦𝑦𝑦 = 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛−∑ 𝑥𝑥𝑥𝑥𝑥𝑥 𝑛𝑛𝑛𝑛𝑥𝑥2− ∑ 𝑥𝑥 2 • x = Independent variable • y = dependent variable • Used to calculate y for given x Regression line of x on y • 𝑥𝑥 − ̅ 𝑥𝑥 = 𝑏𝑏𝑥𝑥𝑥𝑥 𝑦𝑦 − � 𝑦𝑦 • 𝑏𝑏𝑥𝑥𝑥𝑥 = 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛−∑ 𝑥𝑥𝑥𝑥𝑥𝑥 𝑛𝑛𝑛𝑛𝑦𝑦2− ∑ 𝑦𝑦 2 • x = dependent variable • y = Independent variable • Used to calculate x for given y
- 19. Step 1. Calculate 𝑥𝑥2, 𝑦𝑦2 & 𝑥𝑥𝑥𝑥 Step 2. Calculate � 𝑥𝑥2 ,� 𝑦𝑦2 & ∑ 𝑥𝑥𝑥𝑥 Step 3. Put values in the regression coefficient formula. Step 4. Simplify and find the value of 𝑏𝑏𝑦𝑦𝑦𝑦 and 𝑏𝑏𝑥𝑥𝑥𝑥. Step 5. put values in regression lines equation to find out exact equations : 𝑦𝑦 − � 𝑦𝑦 = 𝑏𝑏𝑦𝑦𝑦𝑦 𝑥𝑥 − ̅ 𝑥𝑥 𝑥𝑥 − ̅ 𝑥𝑥 = 𝑏𝑏𝑥𝑥𝑥𝑥 𝑦𝑦 − � 𝑦𝑦
- 20. R E F E R E N C E S • Craig Adam; “Essential Mathematics and Statistics for Forensic Science”, Wiley Blackwell, 2010 • N.M Shah; “Statistics and Economics”, APC Publications, 2019
- 21. T H A N K Y O U