Stats LEC 2 - DESCRIBING DISTRIBUTIONS NUMERICALLY 1 Numerical Summaries Describing Data Numerically Central Tendency Mean Median Mode Variation Range

# Stats LEC 2 - DESCRIBING DISTRIBUTIONS NUMERICALLY 1...

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1 DESCRIBING DISTRIBUTIONS NUMERICALLY
2 Numerical Summaries Mean Median Mode Describing Data Numerically Variance Standard Deviation Range Interquartile Range Central Tendency Variation
3 Central Tendency Mean Median Mode n x x n 1 i i ! = = Midpoint of ranked values Most frequently observed value Arithmetic average Measures of center: Mode, Median, Mean
4 Measures of Center: Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 0 1 2 3 4 5 6 No Mode
5 In an ordered list, the median is the “middle” number (50% above, 50% below) Not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 Median = 3 0 1 2 3 4 5 6 7 8 9 10 Median = 3 Measures of Center: Median
6 Measures of Center: Median The Median the point that divides a distribution in half: 50% of the data on the left of the median and the other 50% - on the right. To find the median, arrange the data in ascending (descending) order. If n is odd, the median is the observation with position If n is even, the median is the average of the middle two values with the positions n 1 2 + n n and 1 2 2 +
7 25 Toyota MR2 21 23 Mersedes-Benz SLK230 20 22 Mazda Miata 19 22 Audi TT Roadster 18 20 Mersedes-Benz SLK320 17 20 Honda S 2000 16 20 BMW Z3 Roadster 15 20 Audi TT Quattro 14 19 Porsche Boxter 13 19 BMW Z3 Coupe 12 18 Chrysler Prowler 11 18 Chevrolet Corvette 10 17 Ford Thunderbird 9 17 BMW M Coupe 8 17 Acura NSX 7 16 Mersedes-Benz SL500 6 15 Porsche 911 GT2 5 13 Mersedes-Benz SL600 4 13 BMW Z8 3 11 Ferrari 360 Modena 2 9 Lamborghini Murcielago 1 City mpg Model 50% MEDIAN 0% MIN 100% MAX 25% 1st QUARTILE 75% 3rd QUARTILE
8 9 Min 16 Q1 18 Median 20 Q3 25 Max 5 - Number Summary 7 2 2 5
9 Boxplot is a visual representation of the five number summary. There are four steps to built a boxplot: 1. Interquartile range IQR= Q3 – Q1, shows the spread of the middle 50% of the data 2. Fences lower fence LF = Q1 – 1.5(IQR) upper fence UF = Q3 + 1.5(IQR) 3. Whiskers - extend from Q1 and Q3 to the smallest and largest observations within the fences 4. outliers (if any) - extreme observations that fall outside the fences
10 Boxplot: gas mileage for two-seater cars 0 5 10 15 20 25 30 Collection 1 Box Plot