University:
California State University, Northridge
Course:
MATH 140 | Introductory Statistics
Academic year:

2010
Views:

279

Pages:

26
Author:

ledescasffu

Math 140 Introductory Statistics Chapter 2 Part 2 Quantitative vs. Categorical Data Quantitative: Data about the cases in the form of numbers that can be compared and that can take a large number of values. Categorical: Data where a case either belongs to a category or not. Example (D6) Quantitative variables: Gestation period, average longevity, maximum longevity, speed. Categorical variables: Wild, predator. Different ways to visualize data Quantitative Variables Dot Plots Histograms Stemplots Categorical Variables Bar Graphs Dot Plots Each dot represents the value associated to a case. Dots may have different symbols or colors. Dots may represent more than one case. 5 15 25 35 45 55 65 75 Dot Plots Dot Plots work best when Relatively small number of values to plot Want to keep track of individuals Want to see the shape of the distribution Have one group or a small number of groups that we want to compare Histograms Groups of cases represented as rectangles or bars The vertical axis gives the number of cases (called frequency or count) for a given group of values. By convention borderline values go to the bar on the right. There is no prescribed number for the width of the bars. Relative Frequency Histograms The height of each bar is the proportion of values in that range. (always a number between 0 and 1) The sum of the heights of all the bars equals 1. To change a regular histogram to a relative frequency histogram just divide the frequency of each bar by the total number of values in the data set. This histogram shows the relative frequency distribution of life expectancies for 203 countries around the world. How many countries have a life expectancy of at least 70 but less than 75 years? .30 x 203 = 60.9 What proportion of the countries have a life expectancy of 70 years or more? .30+.19+.07 = .56 = 56 % Histograms (Relative Frequency) Histograms work best when Large number of values to plot Don’t need to see individual values Want to see the general shape of the distribution Have one or a small number of distributions we want to compare We can use a calculator or computer to draw the plots Stemplots Also called stem-and- leaf plots. Numbers on the left are called stems (the first digits of the data value) Numbers on the right are the leaves. (the last digit of the data value) Mammal speeds: 11,12,20,25,30,30,30,32,35, 39,40,40,40,42,45,48,50,70. 1 2 3 4 5 6 7 12 05 000259 000258 0 0 3 | 9 represents 39 miles per hour. Stemplots (split) Each original stem becomes two stems. The unit digits 0,1,2,3,4 are associated with the first stem and they are placed on the first line. The unit digits 5,6,7,8,9 are associated with the second stem and they are placed on the second line from that stem. 1 2 3 4 5 6 7 12 0 5 0002 59 0002 58 0 0 3 | 9 represents 39 miles per hour. Stemplot vs split stemplot Mammal speeds: 11,12,20,25,30,30,30,32,35,39,40,40,40,42,45,48,50,70. 1 2 3 4 5 6 7 12 05 000259 000258 0 0 3 | 9 represents 39 miles per hour. 1 2 3 4 5 6 7 12 0 5 0002 59 0002 58 0 0 3 | 9 represents 39 miles per hour. Stemplots Stemplots work best when Plotting a single quantitative variable Small number of values to plot Want to keep track of individual values (at least approximately) Have two or more groups that we want to compare Bar Graphs One bar for each category. 45 The height of the bar 40 35 tells the frequency. 30 25 Bar graphs have 20 categories in the 15 horizontal axis, as 10 opposed to histograms 5 0 which have measurements. Non-Predator Predator Domestic Wild Both Total 2.3 Measures of Center and Spread Before we used visual methods (estimations) to find out center (e.g. mean) and spread (e.g. SD). Now we will learn how to calculate them exactly. Measures of Center Mean Median Measures of Spread Standard Deviation Inter Quartile Range Measures of Center Mean The average of the data values denoted x. Calculated as: x sum of values ∑ x= = n number of values Example. Data Set: 5,12,34,18,37,11,9,21,30,6 5 + 12 + 34 + 18 + 37 + 11 + 9 + 21 + 30 + 6 = 18.3 x= 10 Measures of Center Median The value that divides the data into equal halves. Denoted median or Q2. Calculated as: List all values in increasing order and find the middle one. If there are n values then the middle one is (n+1)/2 If n is even use the fact that the mid-value between a and b is (a+b)/2 Measures of Center Median (examples) Data set: 5,12,34,18,37,11,9,21,30,6. Ordered data set: 5,6,9,11,12,18,21,30,34,37 12 + 18 = 15 median = 2 2. Data set: 6, 5 , 9, 12, 30, 18, 11, 34, 21. Ordered data set: 5,6,9,11,12,18,21,30,34 Median = 12 Measure of spread around the Mean Most useful measure of Standard Deviation spread when working with random samples. The deviation of a value is how far apart is it from the mean. There are two kinds σn x−x Unfortunately it is easy and σn-1. The default is σn-1. They are calculated as: σn = to see that ∑ ( x − x) = 0 σ n −1 = 2 x x ( − ) ∑ n 2 ( x − x ) ∑ n −1 Measure of spread around the Mean Example. Data: 2,7,8,12,12,19 n = 6, x = (2 + 7 + 8 + 12 + 12 + 19) / 6 = 10 x Sum σn = x − x ( x − x )2 2 -8 64 7 -3 9 8 -2 4 12 2 4 12 2 4 19 9 81 60 0 166 σ n −1 = σn = σ n −1 = ∑ ( x − x) n 2 ∑ ( x − x) n −1 166 6 166 5 2 ≈ 5.2599 ≈ 5.7619 Measure of spread around the Median Q1 = First Quartile or Lower Inter Quartile Range Quartile. Q3 = Third Quartile or Upper Quartile. IQR = The distance between the Lower Quartile and the Upper Quartile. These are calculated as the IQR = Q3 − Q1 medians of each of the two halves determined by the original median. In case n is odd then the original median is removed from each of the two halves. About 50% of the values are between Q1 and Q3. Five Number Summary min = Minimum (value) Q1 = Lower or First Quartile Q2 = Median In addition we also have Range = max – min IQR = Q3 – Q1 Q3 = Upper or Third Quartile max = Maximum (value) Example: Mammal speeds, 11,12,20,25,30,30,30,32,35,39,40,40,40,42,45,48,50,70. min = 11 Q1 = 30 Median = (35+39)/2 = 37 Q3 = 42 max = 70. Range = 70 – 11 = 59 IQR = 42 – 30 = 12 Box Plots A Box Plot is a graphical display of a five-point summary. Example: Mammal speeds, 11,12,20,25,30,30,30,32,35,39,40,40,40,42,45,48,50,70. min = 11 Q1 = 30 Median = (35+39)/2 = 37 Q3 = 42 max = 70. Range = 70 – 11 = 59 IQR = 42 – 30 = 12 min 11 Q1 Q2 Q3 30 37 42 IQR Range max 70 Modified Box Plots A Modified Box Plot also takes into account the outliers. An outlier is a value which is more than 1.5 times the IQR from the nearest quartile. Example: Mammal speeds, 11,12,20,25,30,30,30,32,35,39,40,40,40,42,45,48,50,70. min Q1 Q2 Q3 max 11 30 37 42 70 IQR = 42 – 30 = 12 (1.5)IQR = (1.5)12 = 18 30 – 18 = 12 > 11, so 11 is an outlier. 42 + 18 = 60 <70, so 70 is an outlier. Q1 Q2 Q3 IQR Range 11 30 37 42 70 Box Plots (Modified) Box Plots and Modified Box Plots are useful when plotting a single quantitative variable and: We want to compare shape, center, and spread of two or more distributions. The distribution has a large number of values Individual values do not need to be identified. (Modified) We want to identify outliers. Four different tables Cumulative distributions reflect the total value accumulated from top to bottom (left to right on a plot) on the corresponding table. (More on this p. 78.) Frequency table* Weight 2.99 3.01 3.03 3.05 3.07 3.09 3.11 3.13 3.15 3.17 3.19 3.21 Total Frequency 1 4 4 4 7 17 24 17 13 6 2 1 100 Cumulative frequency table Weight 2.99 3.01 3.03 3.05 3.07 3.09 3.11 3.13 3.15 3.17 3.19 3.21 Frequency 1 5 9 13 20 37 61 78 91 97 99 100 Relative frequency table Cumulative relative frequency table Weight Frequency Weight Frequency 2.99 3.01 3.03 3.05 3.07 3.09 3.11 3.13 3.15 3.17 3.19 3.21 1/100=0.01 4/100=0.04 4/100=0.04 4/100=0.04 7/100=0.07 17/100=0.17 24/100=0.24 17/100=0.17 13/100=0.13 6/100=0.06 2/100=0.02 1/100=0.01 2.99 3.01 3.03 3.05 3.07 3.09 3.11 3.13 3.15 3.17 3.19 3.21 0.01 0.05 0.09 0.13 0.20 0.37 0.61 0.78 0.91 0.97 0.99 1 Total 100/100=1 * This table shows the weights of the pennies in Display 2.3 on page 31.

Introductory Statistics, Chapter 2, Part 2

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