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Chapter 1 Overview and Descriptive Statistics Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 1.1 Populations, Samples, and Processes Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Populations and Samples A 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. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Data and Observations Univariate data consists of observations on a single variable (multivariate – more than two variables). Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Branches of Statistics Descriptive Statistics – summary and description of collected data. Inferential Statistics – generalizing from a sample to a population. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Relationship Between Probability and Inferential Statistics Probability Population Sample Inferential Statistics Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 1.2 Pictorial and Tabular Methods in Descriptive Statistics Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Stem-and- Leaf Displays 1. 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. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Stem-and-Leaf Example Observed values: 9, 10, 15, 22, 9, 15, 16, 24,11 0 99 1 10556 2 24 Stem: tens digit Leaf: units digit Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Stem-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 values Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Another 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 | 088 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Dotplots Represent data with dots. Observed values: 9, 10, 15, 22, 9, 15, 16, 24,11 5 10 15 20 25 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Types of Variables A 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. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Histograms Discrete Data Determine 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. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 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 #people Rel. Freq 0 12 0.08 1 2 3 4 42 57 24 9 0.28 0.38 0.16 0.06 5 6 4 2 0.03 0.01 Frequency Distribution Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Histograms Credit card results: Relative Frequency x Rel. Freq. 0 1 0.08 0.28 0.4 2 3 4 5 0.38 0.16 0.06 0.03 0.2 6 0.01 0.3 xi 0.1 0 0 1 2 3 4 5 6 Number of Cards Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Histograms Continuous Data: Equal Class Widths Determine 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. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Histogram example 10 5 0 Frequency 15 Histogram of e1scores 20 40 60 80 100 e1scores Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Histograms Continuous Data: Unequal Widths After determining frequencies and relative frequencies, calculate the height of each rectangle using: relative frequency of the class rectangle height = class width The resulting heights are called densities and the vertical scale is the density scale. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Histogram Shapes symmetric unimodal positively skewed bimodal negatively skewed Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Histogram example: symmetric, slightly bimodal 10 5 0 Frequency 15 Histogram of e1scores 20 40 60 80 100 e1scores Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 1.3 Measures of Location Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Mean The average (mean) of the n numbers x1, x2 ,..., xn is x where n x1 x2 ... xn x n xi i 1 n Population mean: Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Median The sample median, x, 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. Population median: Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Median example • In a class of 85 exam scores, the median, x, 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.) 40 41 42 43 44 45 46 57.5 57.5 60.0 60.0 60.0 62.5 62.5 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Three Different Shapes for a Population Distribution symmetric negative skew positive skew Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Slight positive skew 10 5 0 Frequency 15 Histogram of e1scores 20 40 60 80 100 Median=60.0, Mean=61.4 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 1.4 Measures of Variability Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sample Variance Variance 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 by x x i 2 s n 1 2 S xx n 1 We refer to s2 as being based on n – 1 degrees of freedom. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sample variance example • First, find sample mean: x 61.35 • Next, add up squared deviations from mean: (62.5 61.35)2 (90.0 61.35)2 21,531.9 • Divide by n-1, where n is the number of observations (in this case, 85): 21,531.9 256.3 84 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Standard Deviation Standard deviation is a measure of the spread of the data using the same units as the data. The sample standard deviation is the square root of the sample variance: s s 2 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Standard deviation example 2 s s 256.3 16.0 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Formula for s2 An alternative expression for the numerator of s2 is S xx xi x 2 xi2 xi 2 n Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Formula for s2: Shortcut example • First, sum the scores: • Next, sum the squares: • Numerator of variance equals n x i 1 5215 i n x i 1 2 i 341, 487.5 2 5215 341, 487.5 21,531.9 85 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Properties of s2 Let x1, x2,…,xn be any sample and c be any nonzero constant. 2 2 1. If y1 x1 c,..., yn xn c, then s y sx 2 2 2 2. If y1 cx1,..., yn cxn , then s y c s x , 2 where s x is the sample variance of the x’s 2 and s y is the sample variance of the y’s. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Upper and Lower Fourths After 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 x 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. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Third and first quartiles After the n observations in a data set are ordered from smallest to largest, the first (third) quartile is the median of the smallest (largest) half of the data, where the median x is included in both halves if n is odd. A measure of the spread that is resistant to outliers is the interquartile range or IQR fs = 3rd quartile – 1st quartile. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Outliers Any observation farther than 1.5fs from the closest fourth is an outlier. An outlier is extreme if it is more than 3fs from the nearest fourth, and it is mild otherwise. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Boxplots lower fourth extreme outliers mild outliers upper fourth median Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 60 40 e1scores 80 100 Boxplot example Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 0.65 0.60 0.55 0.50 0.45 e1questions 0.70 0.75 Another boxplot example Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.