Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Another stem-and-leaf exampleSlide 12Slide 13Slide 14Slide 15Slide 16Slide 17Histogram exampleSlide 19Slide 20Histogram example: symmetric, slightly bimodalSlide 22Slide 23Slide 24Median exampleSlide 26Slight positive skewSlide 28Slide 29Sample variance exampleSlide 31Standard deviation exampleSlide 33Formula for s2: Shortcut exampleSlide 35Slide 36Slide 37Slide 38Slide 39Boxplot exampleAnother boxplot exampleChapter 1Overview and Descriptive Statistics1.1Populations, Samples, and ProcessesPopulations and SamplesA population is a well-defined collection of objects.When information is available for the entire population we have a census. A subset of the population is a sample.Data and ObservationsUnivariate data consists of observations on a single variable (multivariate – more than two variables).Branches of StatisticsDescriptive Statistics – summary and description of collected data.Inferential Statistics – generalizing from a sample to a population.Relationship Between Probability and Inferential StatisticsPopulationSampleProbabilityInferentialStatistics1.2Pictorial and Tabular Methods in Descriptive StatisticsStem-and- Leaf Displays1. Select one or more leading digits for the stem values. The trailing digits become the leaves.2. List stem values in a vertical column.3. Record the leaf for every observation.4. Indicate the units for the stem and leaf on the display.Stem-and-Leaf Example9, 10, 15, 22, 9, 15, 16, 24,11 Observed values:0 9 91 1 0 5 5 62 2 4 Stem: tens digit Leaf: units digitStem-and- Leaf Displays•Identify typical value•Extent of spread about a value•Presence of gaps•Extent of symmetry•Number and location of peaks•Presence of outlying valuesAnother stem-and-leaf example The decimal point is 1 digit to the right of the | 2 | 55 3 | 05888 4 | 03558888888 5 | 00000000333335555577777 6 | 00033333555588 7 | 0003333335555588 8 | 00000335588 9 | 088Dotplots9, 10, 15, 22, 9, 15, 16, 24,11 Observed values:5 10 15 20 25Represent data with dots.Types of VariablesA variable is discrete if its set of possible values constitutes a finite set or an infinite sequence. A variable is continuous if its set of possible values consists of an entire interval on a number line.Histograms Discrete DataDetermine the frequency and relative frequency for each value of x. Then mark possible x values on a horizontal scale. Above each value, draw a rectangle whose height is the relative frequency of that value.Ex. Students from a small college were asked how many charge cards they carry. x is the variable representing the number of cards and the results are below.x #people0 121 422 573 244 95 46 2Rel. Freq0.080.280.380.160.060.030.01Frequency DistributionHistogramsx Rel. Freq.0 0.081 0.282 0.383 0.164 0.065 0.036 0.01Credit card results:ixHistograms Continuous Data: Equal Class WidthsDetermine the frequency and relative frequency for each class. Then mark the class boundaries on a horizontal measurement axis. Above each class interval, draw a rectangle whose height is the relative frequency.Histogram exampleHistogram of e1scorese1scoresFrequency20 40 60 80 1000 5 10 15Histograms Continuous Data: Unequal WidthsAfter determining frequencies and relative frequencies, calculate the height of each rectangle using: The resulting heights are called densities and the vertical scale is the density scale.relative frequency of the classrectangle height = class widthHistogram Shapessymmetric unimodal bimodalpositively skewed negatively skewedHistogram example:symmetric, slightly bimodalHistogram of e1scorese1scoresFrequency20 40 60 80 1000 5 10 151.3Measures of LocationThe MeanThe average (mean) of the n numbers 1 2, ,...,nx x xis wherex1 2...nx x xxn+ + +=1niixn==�Population mean: mThe sample median, is the middle value in a set of data that is arranged in ascending order. For an even number of data points the median is the average of the middle two.Median,x%Population median: m%Median example•In a class of 85 exam scores, the median,is the 43rd number if the scores are listed in ascending order. (Note: In this case there are 42 above the median and 42 below the median.),x% 40 41 42 43 44 45 4657.5 57.5 60.0 60.0 60.0 62.5 62.5Three Different Shapes for a Population Distributionm m=%mm%mm%symmetricpositive skewnegative skewSlight positive skewHistogram of e1scoresMedian=60.0, Mean=61.4Frequency20 40 60 80 1000 5 10 151.4Measures of Variability( )221 1ixxx xSsn n-= =- -�Sample VarianceVariance is a measure of the spread of the data. The sample variance of the sample x1, x2, …xn of n values of X is given byWe refer to s2 as being based on n – 1 degrees of freedom.Sample variance example•First, find sample mean: •Next, add up squared deviations from mean:•Divide by n-1, where n is the number of observations (in this case, 85):61.35x =2 2(62.5 61.35) (90.0 61.35) 21,531.9- + - + =L21,531.9256.384=2s s=The sample standard deviation is the square root of the sample variance:Standard DeviationStandard deviation is a measure of the spread of the data using the same units as the data.Standard deviation example2256.3 16.0s s= = =Formula for s2( )( )222ixx i ixS x x xn= - = -�� �An alternative expression for the numerator of s2 isFormula for s2: Shortcut example•First, sum the scores:•Next, sum the squares: •Numerator of variance equals 21341, 487.5niix==�25215341, 487.5 21,531.985- =15215niix==�Properties of s2Let x1, x2,…,xn be any sample and c be any nonzero constant.2 21 11. If ,..., , then n n y xy x c y x c s s= + = + =2 2 21 12. If ,..., , then ,n n y xy cx y cx s c s= = =where is the sample variance of the x’s and is the sample variance of the y’s.2xs2ysUpper and Lower FourthsAfter the n observations in a data set are ordered from smallest to largest, the lower (upper) fourth is the median of the smallest (largest) half of the data, where the median is included in both halves if n is odd. A measure of the spread that is resistant to outliers is the fourth spread fs = upper fourth – lower fourth.x%Third and first quartilesAfter the n observations in a data set are ordered from smallest to largest, the