100%(1)1 out of 1 people found this document helpful
This preview shows page 4 - 8 out of 97 pages.
1.2Parameters and StatisticsApopulation parameteris a (typically unknown) numerical constant that describes the populationof interest.Asample statisticis a numerical summary of data that comes from the sample.Typically, we use Greek symbols to denote population parameters and the ”hat” symbol to denotethe sample statistic.For example, the population mean is denoted byμand the sample mean is denoted by ˆμ= ¯x1.3Summary StatisticsLetx1, x2, ..., xndenotenobservations sampled from a population.1.3.1Measures of Location•Themodeof the data is the most frequently encountered value. Note that data can have multiplemodes. Data with two modes are calledbimodaland data with three are calledtrimodal.•Themeanof the data is calculated by taking thearithmetic averageof the data. We use theequation¯x=1nnXi=1xi•Themedianof the data is found by sorting the data in increasing order and then choosing theobservation that divides the data into two equal parts.Ifnis odd, the index of the sorted data that accomplishes this is (n+ 1)/2.Ifnis even, the median is found by taking the average of the numbers in positionsn/2 andn/2 + 1.•Thep-th percentileis the value that divides the sorted data such thatp% of the data are lessthan that value and (1-p)% of the data are greater than it. To find this value, first sort thedata. Then, compute the quantity (p/100)(n+ 1). If this value is an integer, then the data pointin this position is thepth percentile. Otherwise, take the average of the nearest data point tothe left and to the right of this number.Note that the 50th percentile is the median.Other important percentiles are the 25th and75th. Together, the 25th, 50th, and 75th percentiles make up the first, second, and third quar-tiles, respectively.Note:Some statistical packages use different methods of calculating percentiles, such as us-ing a weighted average instead of the simple average used above. Thus, results from computersmay be different than the results you obtain, but they should be close.5
Example 1.3.1.The following values of fracture stress (in megapascals) were measured for a sample of 24 mixtures ofhot mixed asphalt (HMA).30757980801051261381491791791912232322322362402422452472542743844701. Find the mode of the data.2. Find the mean of the data.3. Find the 25th percentile (first quartile) of the data.6
1.3.2Measures of Spread•Thesample varianceis a measure of spread that calculates the sum ofsquared deviationsfromthe center (as measured by the mean).s2=1n-1nXi=1(xi-¯x)2An equivalent formula, which is often easier to use, iss2=1n-1nXi=1x2i-n¯x2!•Thesample standard deviationis simply the square root of the sample variance. That is,s=√s2=vuut1n-1nXi=1(xi-¯x)2Question:Why should we take the square root?