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2: CHAPTER DISPLAYING AND DESCRIBING DATA 2.1 Variables Consider a class of 30 students: ............ ............ Gender (male, female) Hair color (blond, brown, black,..) Height (in feet, inches) Weight (in pounds) Age (in years) Variable any characteristic of a person or thing that can be expressed as a number or a label. Value the actual number or label assigned to a particular variable. VARIABLE Gender Hair color Height Weight POSSIBLE VALUES male, female blond, brown, black,... 5' 10'', 6'1" 160, 176, 180,... 1 Variables Categorical (values are labels) Quantitative (values are numbers) Categorical variables: gender, hair color, marital status Quantitative variables: weight, height, age, income Describing Variables: Categorical: Specify the number (or percentage) of observations in each category. Quantitative: Determine the center (typical value) and spread (variability) of the data. Quantitative Variables Discrete (isolated points on the number line) Continuous (possible values form an interval ) 2 Example: Discrete variables: number of children in a household Continuous variables: weight, height, income. Frequency Tables Frequency Table for Categorical Variables lists all possible values for a variable with the corresponding number of observations (and relative frequency) for each value. Gender Males Females Frequency 18 12 Relative Frequency 0.60 0.40 Frequency table for Quantitative Variable- based on counts of observations in each of several non-overlapping intervals of equal width covering the whole range of values. Weight (in pounds) 140 or higher but smaller than 150 150 or higher but smaller than 160 ... Frequency 3 6 ... Relative Frequency 0.10 0.20 ... 3 2.2 Displaying Categorical Variables Consider a class of 30 with 18 males and 12 females. (a) Bar Graph A vertical bar erected over each category; the height of the bar is the frequency or the percentage of observations in the category. Percent 60 40 Females Males (b) Pie Chart Females Males Slices represent categories; size of each slice corresponds to the percentage for the category 4 2.3 Displaying Quantitative Variables Example: 30 examination scores: 75 61 62 66 86 79 77 61 83 66 58 54 64 70 52 73 77 45 91 80 82 65 58 48 59 94 67 86 78 55 Conclusions: The scores vary, The lowest score=45, the highest score=94 The scores are not distributed evenly between 45 and 94 (16 scores between 60 and 80) (a) Dot Plots 45 48 58 91 94 Place a dot above its value, multiple values produce stack of points placed above the value. (b) Stem-and-Leaf Plots 1. Split each observation into two parts: a stem and a leaf. Stems may have as many digits as required; each leaf contains only a single digit. 5 82 312 stem 2. 3. leaf stem leaf List the stems vertically in increasing order from top to bottom, and draw a vertical line to the right of the stems. Add leaves to the right of the line in increasing order. Example: Stem-and-leaf plot for the 30 exam scores: 4 5 6 7 8 9 (c) Histograms 58 245889 11245667 0357789 02366 14 1. Divide the range of the data into non-overlapping classes of equal width. ( )( )( )( )( 6 Convention: Left-hand limit of each class is included, righthand limit is excluded. 2. 3. Count the number of observations (frequency) in each class. Erect over each class a rectangle whose height equals to the frequency or percentage of that class. Frequency Table: Class 40-50 50-60 60-70 70-80 80-90 90-100 Frequency 2 6 8 7 5 2 8 6 2 40 50 60 70 80 90 100 Using StatCrunch to obtain histograms: Graphics Histogram 7 Graphs describe the distribution of the data. Describing distributions Center Spread Shape Unimodal distributions- a single mound (IQs scores) Bimodal distributions- two distinct mounds (heights of males and females in a population) Shapes of distributions Symmetric Skewed Skewed to the right Skewed to the left (the right tail longer than the left tail) (the left tail longer than the right tail) 8 4 3 Frequency 2 1 0 Symmetric 5 4 Frequency 3 2 1 0 Skewed Right 5 4 Frequency 3 2 1 0 Skewed Left 9 2.4 Time Plots Variable | | | | | Equally-spaced time intervals Trend: persistent long-term rise or fall Cycle: up-and-down movements Example: Variable = Gasoline sales (in thousands of liters) Year 1 Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Sales 39 37 61 58 18 56 82 27 41 69 49 66 54 42 90 66 2 3 4 10 Time Plot of Sales 90 70 80 sales 60 50 40 30 20 Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Line shows the increasing trend Variation: Large amount of random variation. Trend: Clear trend of increasing sales. Cycle: Seasonal pattern that exhibits peaks in the third quarter of each year and valleys in the first quarter of the year. 11 2.5 Measures of Center and Spread Center Small spread Large spread (a) Measures of Center The Mean The mean x of n observations x1, x2, ..., xn is defined as x= x1 + x2 + ... + xn n In compact notation: x = x . i n Example: 30 30 40 50 50 60 40 12 x = 300 / 7 = 42.857 30 30 40 50 50 60 340 x = 600 / 7 = 85.714 Conclusion: The mean is not resistant measure of center (very sensitive to outliers). The Median To compute the median: (i) (ii) The single middle value if n is odd, Median = The average of the two middle values if n is even. Example: Data set 1: 30 60 40 30 50 40 50 Ordered list: 30 30 40 40 50 50 60 Median = Data set 2: 30 60 40 30 50 48 50 40 Ordered list: 30 30 40 40 48 50 50 60 Median = Arrange all observations in order, from smallest to largest 13 60 in data set 2 replaced by 600 Data set 3: 30 600 40 30 50 48 50 40 Ordered list: 30 30 40 40 48 50 50 600 Median = Conclusion: The median is resistant measure of center. The Mode The value that occurs most frequently. (b) Measures of Spread Variance and Standard Deviation x x1 x1 - x Observations: x1, x2, ..., xn Variance s2 is defined as ( x1 - x ) 2 + ( x2 - x ) 2 + ...( xn - x ) 2 s = . n -1 2 14 Compact notation: s 2 (x - x ) = i 2 n -1 . 2 Standard deviation s: s = s Properties of s: 1. 2. 3. Measures the spread of observations about the mean. s is not resistant to outliers. s=0 only when there is no spread (all observations equal). Example: Compute the variance and standard deviation of the observations: 20, 40, 50, 30, 60, 70 Solution: 15 The Empirical Rule In the distribution is bell-shaped, then approximately 68% of the observations fall within 1 standard deviation of the mean 95% of the observations fall within 2 standard deviations of the mean All or nearly all observations fall within 3 standard deviations of the mean. Bell-shaped distribution x s s 68% 2s 95% 2s All or nearly all observations Example: IQ scores for the 20 to 34 age group follow approximately a bell-shaped distribution with the mean = 110 and standard deviation= 25. About what percent of people in this age group have scores above 110? Above 160? Between 135 and 160? 16 Quartiles M=median LOWER HALF UPPER HALF Q1 = Median of the Lower Half, Q2 = Overall Median, Q3 = Median of the Upper Half. Q1 Q2 Q3 Interquartile range IQR: IQR = Q3 Q1. IQR is a measure of spread in the data. The Five-Number Summary: Minimum Q1 Q2 (median) Q3 Maximum 17 Example: Refer to the 30 exam scores. Find all quartiles, IQR, and the five-number summary for the data. 2.6 Outliers Outliers- observations separated from the main body of data outlier Not an outlier Detecting Potential Outliers (a) 1.5IQR Rule Potential outlier an observation 1.5*IQR below Q1 or 1.5*IQR above Q3 18 Q1 1.5 IQR Q2 Q3 1.5 IQR Example: Are there are any outliers in the above example? Solution: (b) Z-Scores For a bell-shaped distribution, it is very unlikely for an observation to fall more than three standard deviations more from the mean. z - score = observation - mean st.deviation Potential outlier in a bell-shaped distribution an observation that falls more than three standard deviations from the mean 19 Example: Calculate the z-score for the smallest and the largest observations in the exam scores data. Are they outliers based on their z-scores? Solution: 2.7 Boxplots Outlier (more than 1.5 IQR above Q3) The largest observation within 1.5 IQR from Q3 Q3 IQR Q2 Q1 The smallest observation within 1.5 IQR from Q1 Outlier (more than 1.5 IQR above Q3) 20 Side-BySide box plots Skewed right Skewed left Symmetric Example: Obtain the box plot for the 30 exam scores. 21 2.8 Appropriate Measures of Center and Spread Measures of Center Spread x Distribution Approximately symmetric (no outliers) Skewed median s IQR Henryk Kolacz, September 2006 Revised, September 2007 22
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