FM217_IntroStats_Part2.pdf

Introduction to Data Science Introductory Statistics / Probability – Part II September 6, 2024  Percentiles Position of nth percentile = (Population Size) * n/100 Akin to dividing the sorted data into 100 buckets and picking the n th bucket. ...

Introduction to Data Science Introductory Statistics / Probability – Part II September 6, 2024  Percentiles Position of nth percentile = (Population Size) * n/100 Akin to dividing the sorted data into 100 buckets and picking the n th bucket.  Quantiles A q-quantile splits the data into q equal parts. Akin to dividing the sorted data into q buckets and picking the nth bucket. Position of nth q-quantile = ?  Quantiles A q-quantile splits the data into q equal parts. Akin to dividing the sorted data into q buckets and picking the nth bucket. Position of nth q-quantile = (Population Size) * n/q  Quantiles - Example A q-quantile splits the data into q equal parts. X = {1,1,2,2,3,3,4,4,5,5} q= 5 : { 1,1, | 2,2 | 3,3 | 4,4 | 5,5 } q =2: { 1,1,2,2,3 | 3,4,4,5,5 } q=10 : { 1, | 1, | 2, | 2, | 3, | 3, | 4, | 4, | 5, | 5 }  Quantiles Quantiles help get an idea of the spread of data at a required granularity.  Quantiles Quantiles help get an idea of the spread of data at a required granularity. When q = 100, 100-quantiles = percentiles When q = 10, 10-quantiles = deciles When q = 4, 4-quantiles = quartiles : :  Quantiles Quantiles help get an idea of the spread of data at a required granularity. When q = 100, 100-quantiles = percentiles When q = 10, 10-quantiles = deciles When q = 4, 4-quantiles = quartiles : : Q) What are the bounds on the values that q can take?  Quartiles Quantiles with q = 4 Give a good idea about the centre as well as the spread of the data. X = [0,….Q1,….Q2,….Q3,….n ] Q1 = 1st quartile Q2 = 2nd quartile Q3 = 3rd quartile  Quartiles Quantiles with q = 4 Give a good idea about the centre as well as the spread of the data. X = [0,….Q1,….Q2,….Q3,….n ] Q1 = 1st quartile = Lower quartile Q2 = 2nd quartile = Middle quartile Q3 = 3rd quartile = Upper quartile  Quartiles Quantiles with q = 4 Give a good idea about the centre as well as the spread of the data. X = [0,….Q1,….Q2,….Q3,….n ] Q1 = 1st quartile = Lower quartile = 25th Percentile Q2 = 2nd quartile = Middle quartile = 50th Percentile Q3 = 3rd quartile = Upper quartile = 75th Percentile  Quartiles Quantiles with q = 4 Give a good idea about the centre as well as the spread of the data. X = [0,….Q1,….Q2,….Q3,….n ] Q1 = 1st quartile = Lower quartile = 25th Percentile = Median of lower half of data Q2 = 2nd quartile = Middle quartile = 50th Percentile = Median Q3 = 3rd quartile = Upper quartile = 75th Percentile = Median of upper half of data Boxplots  Boxplots Whiskers could end at: 1. Minimum and Maximum 2. 2nd Percentile and 98th Percentile 3. (Q1 - 1.5 IQR) and (Q3 + 1.5 IQR) 4. (Q1 - XYZ) and (Q3 + XYZ) 5. … Boxplots Boxplots  Boxplots Box-and-whisker plots or Box plots display the centrality, dispersion and skew of data. Can also be used to highlight the outliers. Helpful in visually comparing (multiple aspects of) different datasets or distributions.  Standardized Variable / Dataset Scaled version of the original variable/dataset such that Mean = 0 Standard Deviation = 1  Standardized Variable / Dataset Scaled version of the original variable/dataset such that Mean = 0  Standardized Variable / Dataset Scaled version of the original variable/dataset such that Mean = 0 Standard Deviation = 1  Standardized Variable / Dataset Scaled version of the original variable/dataset such that Mean = 0 Standard Deviation = 1 X = {10, 20, 30} After Standardization : {-SQRT(3), 0, SQRT(3)}  Central Moments The ith Central Moment is the Expected value of the ith power of deviation from the mean.  Central Moments The deviation from the mean.  Central Moments The ith power of deviation from the mean.  Central Moments The ith Central Moment is the Expected value of the ith power of deviation from the mean. Central Moments Standardized Central Moments Standardized Central Moments Third Standardized Central Moment Skewness  Third Standardized Central Moment Skewness Right skewed or Left skewed or Positive skewed Negative skewed Fourth Standardized Central Moments Kurtosis Fourth Standardized Central Moments Kurtosis  Covariance Cov(X,Y) = E[ ( X – E[X] ) ( Y – E[Y] ) ] = E[ XY – X E[Y] – Y E[X] + E[X] E[Y] ] = E[ XY ] – E[X] E[Y] – E[X] E[Y] + E[X] E[Y] = E[ XY ] – E[X] E[Y] Cov(X,Y) < 0 Cov(X,Y) = 0 Cov(X,Y) > 0  Correlation Coefficient The Correlation Coefficient is generally a scale independent version of the covariance. Hence, it is a better indicator of the correlation of the two variables or datasets. Pearson's Correlation Coefficient: ρX,Y = cov(X,Y) / (σX σY) ρX,Y = - 0.8 ρX,Y = 0 ρX,Y = 0.8