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# STATISTICAL METHOD STA 291

UK

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This 40 page Class Notes was uploaded by Helga Torp Sr. on Friday October 23, 2015. The Class Notes belongs to STA 291 at University of Kentucky taught by Jing Xi in Fall. Since its upload, it has received 12 views. For similar materials see /class/228268/sta-291-university-of-kentucky in Statistics at University of Kentucky.

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Date Created: 10/23/15

STA 291 Lecture 8 4 Numerical Descriptive Techniques 41 Measures of Central Location 42 Measures of Variability STA 291 Lecture 8 Midterm exam 1 Next Tuesday 57pm CB 118 No computer No cell phone No PDA No blue tooth devices etc Calculator Yes One formula sheet STA 291 Lecture 8 Makeup exam on the same day Continuous you have 2 hour 730pm930pm the latest STA 291 Lecture 8 in lab Summarizing Data Numerically Center of the data Mean Median Dispersion of the data Standard deviation Variance nterquartie range Q3 Q1 Range Max Min STA 291 Lecture 8 Mean versus Median Mean Numerical data with an approximately symmetric distribution Median numerical or ordinal data The mean is sensitive to outliers the median is not STA 291 Lecture 8 Mean and Median Example For towns with population size 2500 to 4599 in the US Northeast in 1994 the mean salary of chiefs of police was 37527 and the median was 30500 Does this suggest that the distribution of salary was skewed to the left symmetric or skewed to the right STA 291 Lecture 8 7 Mean vs Median Observations Median Mean 1 2 3 4 5 3 3 12 34100 3 22 3 3 3 3 3 3 3 12 3100100 3 412 STA 291 Lecture 8 Census Data Population Area in square miles People per sq mi Median Age Median Family Income Real Estate Market Data Total Housing Units Average Home Price Median Rental Price Owner Occupied Lexington 261 545 306 853 35 42500 Lexington 54587 151 776 383 52 STA 291 Lecture 8 Fayette County 261 545 306 853 34 39500 Fayette County 54587 151 776 383 52 Kentucky 4069734 40131 101 36 32101 Kentucky 806524 1 15545 257 64 United States 281422131 3554141 79 36 40591 United States 1 15904743 173585 471 60 Percentiles The pth percentile is a number such that p of the observations take values below it and 1 OOp take values above it 50th percentile median 25th percentile lower quartile 75th percentile upper quartile 90th percentile of SAT math is 680 STA 291 Lecture 8 10 Quartiles 25th percentile lower quartile approximately median of the observations below the median 75th percentile upper quartile approximately median of the observations above the median STA 291 Lecture 8 FiveNumber Summary Maximum Upper Quartile Median Lower Quartile Minimum Statistical Software SAS output Murder Rate Data Quantile 100 Max 75 Q3 50 Median 25 01 0 Min Estimate 2030 1030 670 390 160 STA 291 Lecture 8 Review Shapes of Distributions Rulzltim quucncy alI39J E Rtla va Frtq Limit us I Lgature FiveNumber Summary Max Upper Quartile Q3 Median Lower Quartile 01 Min Example The fivenumber summary for a data set is minimum4 01256 median530 Q31105 maximum320000 Whatdoes this suggest about the shape of the distribution STA 291 Lecture 8 Box plot Boxwhisker plot Read information off a box plot STA 291 Lecture 8 Measures of Variation Mean and Median only describe the central location but not the spread of the data Two distributions may have the same mean but different variability Statistics that describe variability are called measures of variation STA 291 Lecture 8 17 Measures of Variation Range max min Difference between maximum and minimum value 2 Xi X Variance s2 Z 1 n 2 Standard Deviation s 2 J Elm x n l Interquartile Range Q3 Q1 Difference between upper and lower quartile of the data STA 291 Lecture 8 18 Range Range the Difference between the largest and smallest observation The longest living person now 114 years range 114 O 114 Very much affected by outliers a mis recorded observation may lead to an outlier and affect the range STA 291 Lecture 8 19 Range Example Murder Rate Data with DC Smallest Observation 16 Largest Observation 785 Range 785 16 Murder Rate Data without DC Smallest Observation 16 Largest Observation 203 Range STA 291 Lecture 8 20 sample Deviations The deviation of theith observation Xfrom the sample mean x is the difference between them x x The sum of all deviations is zero because the sample mean is the center of gravity of the data negative deviation cancel with positive deviation Therefore we use the sum of the squared deviations as a measure of variation STA 291 Lecture 8 21 Deviations Example Data 1 7 4 3 1O Mean 1 743105 2555 data Deviation Dev square 1 1 5 4 16 3 3 5 2 4 4 4 5 1 1 7 7 5 2 4 1O 1O 5 5 25 Sum25 Sum 0 sum 50 STA 291 Lecture 8 22 Sample Variance 2 Elba ff S n l The variance ofn observations is the sum of the squared deviations divided by n1 STA 291 Lecture 8 23 Variance Example Observation Mean Deviation Squared Deviation 1 5 16 3 5 4 4 5 1 7 5 4 1O 5 25 Sum of the Squared Deviations 50 n1 514 Sum of the Squared Deviations n1 504125 STA 291 Lecture 8 24 Variance Interpretation The variance is average squared distance from the mean Unit square of the unit of the original data Average money spent on Spring Break 350 with variance 2500 square what is square I Difficult to interpret Solution Take the square root of the variance and the unit is the same as for the original data STA 291 Lecture 8 25 Sample Standard Deviation SD The standard deviation 8 is the positive square root of the variance S 2004 17 1 STA 291 Lecture 8 26 Example cont Sample SD S Interpretation A typical deviation is around 35 positive or negative Or a typical observation can be 35 units above or below the mean STA 291 Lecture 8 27 Standard Deviation Properties S 2 0 always s0 only when all observations are the same 8 is sensitive to outliers like mean Same can be said for the variance STA 291 Lecture 8 28 Standard Deviation Interpretation Empirical Rule Ifthe histogram of the data is approximately symmetric and bellshaped then About 68 of the data are within one standard deviation from the mean About 95 of the data are within two standard deviations from the mean About 997 of the data are within three standard deviations from the mean STA 291 Lecture 8 29 STA 291 Lecture 8 30 EHH EEumuz wax II39I 39 quotit Ell ail sq 4 3936 J M1 STA ngture example guess SD from a good looking histogram STA 291 Lecture 8 32 Another Example Distribution of SAT score is approximately bellshaped with mean 500 and standard deviation 100 About 68 of the scores are between and About 95 are between and STA 291 Lecture 8 33 Example Distribution of SAT score is approximately be shaped with mean 500 and standard deviation 100 About 68 of the scores are between 500 100 ie between 400 and 600 About 95 are between 500 200 Le between 300 and 700 STA 291 Lecture 9 34 If you have a score of 700 you are in the top STA 291 Lecture 8 35 Summary Measures of Location Central Tendency Where is the data located Where is the middle of the data Mean Median Measures of Variation How variable are the data How spread out about the middle are the data Range Standard Deviation Variance Interquartile Range STA 291 Lecture 8 36 Homework 4 Due Tuesday next week Feb 12 5 PM Please read Chapter 3 about the Art amp Science ofgraphical presentations Suggested problems from the textbook 38 42 43 44 STA 291 Lecture 8 37 Attendance Survey Question 8 On a 4 x6 index card Pease write down your name and section number Today s Question Which topic you want to review next Monday STA 291 Lecture 8 38 Interquartile Range The Interquartile Range IQR is the difference between upper and lower quartile IQR Q3 Q1 IQR Range of values that contains the middle 50 of the data IQR increases as variability increases Example Murder Rate Data Q1 39 Q3 103 IQR STA 291 Lecture 8 39 Interquartile Range The Interquartile Range IQR is the difference between upper and lower quartile IQR Q3 Q1 IQR Range of values that contains the middle 50 of the data IQR increases as variability increases Example Murder Rate Data Q1 39 Q3 103 IQR STA 291 Lecture 8 40

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