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# Intro to Statistics Chapter 2 sta2023

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This 10 page Class Notes was uploaded by Danielle Adams on Wednesday May 11, 2016. The Class Notes belongs to sta2023 at University of Florida taught by Maria Ripoli in Spring 2016. Since its upload, it has received 12 views.

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Date Created: 05/11/16

Chapter 2 Exploring Data with Graphs and Numerical Summaries 2.1 Different Types of Data Categorical Variables: These are variables that you can put into different groups, generally non-numerical answers. Things like favorite colors, political party afﬁliation, or letter grades are categorical variables. Quantitative Variables: These are variables that are represented by numerical values. An amount of something. There are two types are quantitative variables: Discrete: these are characterized as having a FINITE amount of possible outcomes. Continuous: these, on the other hand, have an inﬁnite list of possible outcomes and can generally be measured. Note: For quantitative data: there are two key features. Example: The class list below has all the information for each student in a class at the end of the semester, including their year in school, major, exam grades, project grades, number of absences, their average in the class and their ﬁnal letter grade in the class. Studen Name Year Major Exam Exam Project Project Abs Avg Grade t ID # 1 2 1 2 46895382Aiken, 1 Psych 78 82 20 24 2 81.6 B John 21657845Bailey, 2 PolSci 62 74 15 19 10 68.0 D Kim 13695544Carr, 2 BusAd 95 95 92 25 24 94.4 A May m Which of the previous variables are: Categorical Discrete Quantitative Continuous Quantitative Major Absent Average Grade Exam, Project Year Year Note: You need to be able to ﬁnd the average for quantitative data. 2.2 Graphical Summaries of Data The type of graph used depends on the type of variable. Most graphs are done with a computer, particularly for large data sets. Graphs for Categorical Variables: Bar Charts and Pie Charts Example: Year in school for students in classroom Frequency (Count) Proportion Percentage Freshman 4 0.055 5.5% Sophomore 28 0.384 38.4% Junior 30 0.411 41.1% Senior 11 0.151 15.1% Total 73 1.001 100.1% Bar Chart Pie Chart Frequency Freshman Sophomore Junior Senior 80 Total 60 3% 19% 40 50% 20 21% 8% 0 Freshman Junior Total Graphs for Quantitative Variables: Dotplots, Stem-and-Leaf Plots, and Histograms Example: Grades on an exam for a small class: 82, 76, 65, 94, 72, 80, 91, 45, 72, 86, 89 Dotplot 40 50 60 70 80 90 100 Stemplot 4 5 5 6 5 7 6 2 2 8 2 0 6 9 9 4 1 Histogram Region 1 4 3 2 1 0 30-40 40-50 50-60 60-70 70-80 80-90 90-100 For this data set, what can you say about: a) The center of distribution? what would be an average estimate? Average grade is around the high 70s b) The spread of the distribution? the values go from what to about what? Most of the grades are between 40 and 100 c) The shape of distribution? bell shaped with an outlier d) Unusual observations? Outliers? 45 Some Common Shapes: Mound or Bell-Shape - Symmetric - One Peak - Example: Weights - One of the most common curves in this class Uniform or Rectangular - Equally Likely to Occur - Example: Rolling a dice Bimodal - Two Peaks - Several Bars, Two Groups - Example: MPG for Standard and Hybrid Cars Skewed Left - Grades on an Exam - Tail to the Lower Left - Example: Grades on an Easy Exam Skewed Right - Tail to the Right - Example: Income per hour 2.3 Measuring the Center of Quantitative Data Measures of Center - Mean: - Median: - Mode: Mathematical Deﬁnitions: Notation: n = # of observations in the data set x1, x2, x3, ….., xn = ﬁrst, second, third, …., last observation Summation Notation (Capital Sigma) Formulas: Mean: (x1 + x2 + …. + Xn)/n Median: - Put the observations in order - Find the middlemost value. Position of median Pos((n+1)/2) Example: Number of friends in Facebook, for a sample of 9 female Facebook Members 288 254 476 329 191 121 404 184 505 a) Find the mean: (288+254+476+329+191+121+404+184+505)/9 = 305.18 b) Find the median: 121 184 191 254 288 329 404 476 505 Example: Number of friends in Facebook, for a sample of 10 male Facebook members 63 342 345 172 46 458 106 153 244 810 a) Find the mean: add all of the samples together and divide by the total number of samples = 274.1 b) Find the median: 208 POS(10+1)/2 = 5.5 between 172 and 244 which is 208 Example: The following is a templet done in Minitab of self-reported college GPAs of students enrolled in a large introductory statistics course. Find the median of this set of data. Notes: - First column represents the cumulative counts from top and bottom. - The line with the parentheses contains the median. - Second column contains the stems. - Rest of the columns contain the leaves. - Leaf Unit: 0.10 decimal between stem and leaf Example from the ﬁrst row: 2 01 means 2.0 and 2.1 1.00 no decimal 10.0 add a zero after the leaf 0.01 move decimal to the left of stem 2.4 Measuring the Variability of Quantitative Data Measures of Variability Range = maximum - minimum Example: Find the range of the distributions of the number of friends in Facebook for the samples of female and male Facebook members. Data appears below in order. Female: 121 184 191 254 288 329 404 476 505 505 - 121 = 384 In order to ﬁnd the range, we’re going to take the highest value and subtract the lowest value from it. Male: 46 65 106 153 172 244 342 345 458 810 810- 46 = 764 Variance and Standard Deviation Measures of spread around the mean, particularly useful for bell-shaped and symmetric distributions. Variance - averaged squared deviation from the mean: s 2 - its units of measurement are those of the original data squared - we need to take the square root before we interpret Standard Deviation - square root of the variance: s - its units of measurement are the same as those of the original data - typical distance from the points to the mean Mathematical Deﬁnitions: Formulas: V2riance: 2 s = ∑(x-(mean)) / (n-1) Standard Deviation: s = √s2 Why n-1 in the denominator? The denominator in the formula for the variance, n-1, represents the number of degrees of freedom. This is the number of independent quantities we are adding up in the numerator. Since the mean is computed from those n observations, only n-1 of the distances from the observations to the mean are independent of each other. Meaning that the mean is one of the n values and thus has to be removed from the number of values we divide by. Example: Suppose your whole grade in a class is based on four exams, worth 100 points each. Here are the grades you know: Exam 1 Exam 2 Exam 3 Exam 4 Average 82 76 85 ????? 80.5 How many possible values of exam 4 grade are there? ONLY ONE Examples: Two very simple data sets. For each one, make a quick plot of their distribution, ﬁnd the mean, median, and range. Compare the two distributions. Then ﬁnd the standard deviation using the formula, and also using the statistical function on your calculator. Interpreting the Standard Deviation, s: - The larger the standard deviation, s, the more spread out the data is. - s can never be negative - s can only be zero if there is no variability in the data, meaning all the points have the exact same value - - s is very much affected by outliers s works best for bell-shaped and symmetric distributions Empirical Rule: in any bell-shaped and symmetric distribution you ﬁnd will ﬁnd approximately: -68% of the observations within one standard deviation of the mean. 95% of the observations within two standard deviations of the mean. 99.7% of the observations within three standard deviations of the mean. Example: One of the many different scales for IQ scores states that the IQ values for the whole population follow a bell-shaped and symmetric distribution with a mean 100 points and standard deviation of 16 points. 2.5 Describing Distributions Using Percentiles and Boxplots Quartiles divide the data set into four quarters: - Q1 = Lower Quartile - 25% of observations below it = 25th percentile - Q2 = Median - 50% of observations below it = 50th percentile - Q3 = Upper Quartile - 75% of observations below it = 75th percentile There are many ways to ﬁnd the quartiles, each resulting in slightly different numbers. For our purposes, - Q1 will be the median of the lower half of the data - Q3 will be the median of the upper half of the data Example: Find the quartiles of the distributions of the number of friends on Facebook for the samples of female and male Facebook members. Data appears in order below. Female: 121 184 191 254 288 329 404 476 505 Male: 46 65 106 153 172 244 342 345 458 810 InterQuartile Range - measures the spread of the central 50% of the data. IQR = Q3 - Q1 Five number summary of positions: Minimum, Q1, Median, Q3, Maximum Boxplots: graphs based on the ﬁve number summary. - Box contains the central 50% of the data (going from Q1 to Q3). Line crossing the box represents the median. Whiskers extended to the minimum and maximum observations. - In Minitab the whiskers only extend to the smallest and largest observations that are not considered outliers. The outliers are plotted individually, with an asterisk if it’s a “mild” outlier, and an open circle for an extreme outlier.

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