1STA 291 - Lecture 11 1STA 291Lecture 11• Describing Quantitative Data– Measures of Central LocationExamples of mean and median– Review of Chapter 5. using the probability rules• You need a Calculator for the exam, but no laptop, no cellphone, no blackberry, no iphone, etc (anything that can transmitting wireless signal is not allowed)• Location: Memorial Hall,• Time: Tuesday 5-7pm.• Talk to me if you have a conflict. STA 291 - Lecture 11 2• A Formula sheet, with probability rules and sample mean etc will be available.• Memorial HallSTA 291 - Lecture 11 32• Feb. 23 5-7pm• Covers up to mean and median of a sample (beginning of chapter 6). But not any measure of spread (i.e. standard deviation, inter-quartile range etc)Chapter 1-5, 6(first 3 sections) + 23(first 5 sections)STA 291 - Lecture 11 4STA 291 - Lecture 11 5Summarizing Data Numerically• Center of the data– Mean (average)– Median– Mode (…will not cover)• Spread of the data– Variance, Standard deviation– Inter-quartile range– RangeSTA 291 - Lecture 11 6Mathematical Notation: Sample Mean• Sample size n• Observations x1, x2,…, xn• Sample Mean “x-bar” --- a statistic SUM=∑121x()/1nniixxxnxn==+++=∑K3STA 291 - Lecture 11 7Mathematical Notation: Population Mean for a finite population of size N• Population size (finite) N• Observations x1, x2,…, xN• Population Mean “mu” --- a Parameter∑=SUM 121()/1NNiixxxNxNµ==+++=∑KInfinite populations• Imagine the population mean for an infinite population. • Also denoted by mu or • Cannot compute it (since infinite population size) but such a number exist in the limit.• Carry the same information.STA 291 - Lecture 11 8µInfinite population• When the population consists of values that can be ordered • Median for a population also make sense: it is the number in the middle….half of the population values will be below, half will be above.STA 291 - Lecture 11 94STA 291 - Lecture 11 10Mean• If the distribution is highly skewed, then the mean is not representative of a typical observation• Example: Monthly income for five persons1,000 2,000 3,000 4,000 100,000• Average monthly income: = 22,000• Not representative of a typical observation.• Median = 3000 STA 291 - Lecture 11 11STA 291 - Lecture 11 12Median• The median is the measurement that falls in the middle of the ordered sample• When the sample size n is odd, there is a middle value• It has the ordered index (n+1)/2• Example: 1.1, 2.3, 4.6, 7.9, 8.1n=5, (n+1)/2=6/2=3, so index = 3,Median = 3rdsmallest observation = 4.65STA 291 - Lecture 11 13Median• When the sample size n is even, average the two middle values• Example: 3, 7, 8, 9, n=4, (n+1)/2=5/2=2.5, index = 2.5Median = midpoint between2ndand 3rdsmallest observation = (7+8)/2 =7.5STA 291 - Lecture 11 14Summary: Measures of LocationMean- Arithmetic Average Mean of a Sample - xMean of a Population - µMedian– Midpoint of the observations when they are arranged in increasing orderMode….Notation: Subscripted variablesn = # of units in the sampleN = # of units in the population x = Variable to be measuredxi = Measurement of the ith unitSTA 291 - Lecture 11 15Mean vs. MedianObservations Median Mean1, 2, 3, 4, 5 3 31, 2, 3, 4, 100 3 223, 3, 3, 3, 3 3 31, 2, 3, 100, 100 3 41.26STA 291 - Lecture 11 16Mean vs. Median• If the distribution is symmetric, then Mean=Median• If the distribution is skewed, then the mean lies more toward the direction of skew• Mean and Median Online AppletExample• the sample consist of 5 numbers, 3.6, 4.4, 5.9, 2.1, and the last number is over 10. (some time we write it as 10+)• Median = 4.4• Can we find the mean here? NoSTA 291 - Lecture 11 17STA 291 - Lecture 11 18Example: Mean and Median• Example: Weights of forty-year old men158, 154, 148, 160, 161, 182, 166, 170, 236, 195, 162• Mean = • Ordered weights: (order a large dataset can take a long time)• 148, 154, 158, 160, 161, 162,166, 170, 182, 195, 236• Median = 1627Eye ball the plot to find mean/medianSTA 291 - Lecture 11 19• Extreme valued observations pulls mean,but not on median.For data with a symmetric histogram, mean=median.STA 291 - Lecture 11 20STA 291 - Lecture 11 218Using probability rule• In a typical week day, a restaurant sells ? Gallons of house soup. • Given that P( sell more than 5 gallon ) = 0.8P( sell less than 10 gallon ) = 0.7 • P( sell between 5 and 10 gallon) = 0.5 STA 291 - Lecture 11 22STA 291 - Lecture 11 23Why not always Median?• Disadvantage: Insensitive to changes within the lower or upper half of the data• Example: 1, 2, 3, 4, 5, 6, 7 vs. 1, 2, 3, 4, 100,100,100• For symmetric, bell shaped distributions, mean is more informative. • Mean is easy to work with. Ordering can take a long time• Sometimes, the mean is more informative even when the distribution is slightly skewedSTA 291 - Lecture 11 24Census Data Lexington Fayette County Kentucky United StatesPopulation 261,545 261,545 4,069,734 281,422,131Area in square miles 306 306 40,131 3,554,141People per sq. mi. 853 853 101 79Median Age 35 34 36 36Median Family Income $42,500 $39,500 $32,101 $40,591 Real Estate Market Data Lexington Fayette County Kentucky United StatesTotal Housing Units 54,587 54,587 806,524 115,904,743Average Home Price $151,776 $151,776 $115,545 $173,585 Median Rental Price $383 $383 $257 $471 Owner Occupied 52% 52% 64% 60%9Given a histogram, find approx mean and medianSTA 291 - Lecture 11 25STA 291 - Lecture 11 26STA 291 - Lecture 11 2710STA 291 - Lecture 11 28Five-Number Summary• Maximum, Upper Quartile, Median, Lower Quartile, Minimum• Statistical Software SAS output (Murder Rate Data) Quantile Estimate 100% Max 20.3075% Q3 10.30 50% Median 6.70 25% Q1 3.900% Min 1.60 STA 291 - Lecture 11 29Five-Number Summary• Maximum, Upper Quartile, Median, Lower Quartile, Minimum• Example: The five-number summary for a data set is min=4, Q1=256, median=530, Q3=1105, max=320,000.• What does this suggest about the shape of the distribution?Box plot • A box plot is a graphic representation of the five number summary --- provided the max is within 1.5 IQR of Q3 (min is within 1.5 IQR of Q1)STA 291 - Lecture 11 3011STA 291 - Lecture 11