STATS 250 1st Edition Lecture 2 Outline of Last Lecture I. Syllabus InformationII. Definition of StatisticsIII. Raw DataIV. Types of VariablesV. Summarizing One or Two Categorical VariablesVI. Exploring Features of Quantitative Data with PicturesOutline of Current Lecture I. Numerical Summaries of Quantitative VariablesII. How to Handle OutliersIII. Features of Bell-Shaped DistributionsIV. Population vs. SampleCurrent LectureI. Numerical Summaries of Quantitative Variablesa. Notation for a generic raw set of data: x1, x2, x3… xn, where n = # of items in the data set or sample sizeb. Describing the location or center of a data seti. There are two basic measure of location or center1. Mean – the numerical average value2. Median – the middle value when data arranged from smallest to largestii. Graph the data to determine if the data is symmetric or skewediii. Compute the mean and median1. Mean is affected by outliersThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.a. If the data is positively skewed, the mean will be larger than the medianb. If the data is negatively skewed, the mean will be smaller than the medianc. If the data is symmetrical, median and mean will be approximately the same2. Median is resistant to outliersc. Describing Spreadi. Range: maximum minus minimumii. Percentiles: the pth percentile is the value such that p% of the observations fall at or below that value1. Median (M): 50th percentile2. First quartile (Q1): 25th percentile3. Third quartile (Q3): 75th percentileiii. Interquartile Range (IQR): Measures the spread over the middle 50% of the data.1. IQR = Q3 – Q1d. Boxplotsi. Graphical representation of the five-number summary1. Five-number summary uses maximum, minimum, Q1, Q3, and medianii. Label an axis covering max and miniii. Make a box with ends at Q1 and Q3iv. Draw a line in the box at Mv. Check for possible outliers using the 1.5*IQR rule and label them as dots1. Outliers are observations that are more than 1.5*IQR outside of the quartiles2. Q1-(1.5*IQR) or Q3+(1.5*IQR)vi. Extend lines from end of box to max and min that are not outliersII. How to Handle Outliersa. The outlier is a legitimate data value and represents natural variability for the group and variables measured: do not discard in this case, they provide important information about location and spreadb. A mistake was made while taking a measurement or entering it into the computer: if this can be verified, the values should be discarded or correctedc. The individual in question belongs to a different group than the bulk of individuals measured: values may be discardedIII. Features of Bell-Shaped Distributionsa. AKA normal distribution or normal curveb. Standard deviation (s)i. A measure of the spread of the observations from the meanii. s2 is called the variance, and is also in units squarediii. If s=0, there is no variabilityiv. s is affected by outliersc. The Empirical Ruei. For bell-shaped curves, approximately…1. 68% of values fall within 1 standard deviation of the mean2. 95% of the values fall within 2 standard deviations of the mean3. 99.7% of the values fall within 3 standard deviations of the mean4. In either directionIV. Population vs. Samplea. Symbols are different!b. Population: entire set of items/people in a given situationi. Numerical summary of a population is referred to as a parameterii. Population size: Niii. Population mean: iv. Population standard deviation: c. Sample: part of a larger population used to collect datai. Numerical summary based on a sample is referred to as a statisticii. Sample size: niii. Sample mean:iv. Sample standard