1 Chapter 1 Exploring Data Introduction: Data Analysis: Making Sense of Data 1.1 Analyzing Categorical Data 1.2 Displaying Quantitative Data with Graphs 1.3 Describing Quantitative Data with Numbers
2 Introduction Data Analysis: Making Sense of Data Learning Objectives After this section, you should be able to DEFINE Individuals and Variables DISTINGUISH between Categorical and Quantitative variables DEFINE Distribution DESCRIBE the idea behind Inference
3 Statistics is the science of data. Data Analysis is the process of organizing, displaying, summarizing, and asking questions about data. Definitions: Individuals objects (people, animals, things) described by a set of data Variable - any characteristic of an individual Data Analysis Categorical Variable places an individual into one of several groups or categories. Quantitative Variable takes numerical values for which it makes sense to find an average.
4 Read Example : Census at school ( P-3) Now do Exercise 3 on P- 7 Answers: a) Individuals: AP Stat students who completed a questionnaire on the first day of class. b) Categorical: gender, handedness, favorite type of music Quantitative: height( inches), amount of time the students expects to spend on HW (mins), the total value of coin(cents) c) Female, right-handed, 58 inches tall, spends 60 mins on HW, prefers alternative music, had 76 cents in her pocket.
5 Do Activity : Hiring Discrimination It just won t fly Follow the directions on Page 5 Perform 5 repetitions of your simulation. Turn in your results to your teacher. (Make Groups of 4)
6 A variable generally takes on many different values. In data analysis, we are interested in how often a variable takes on each value. Definition: Distribution tells us what values a variable takes and how often it takes those values 2009 Fuel Economy Guide Example MODEL MPG Acura RL 922 Audi A6 Quattro 1023 Bentley Arnage 1114 BMW Buick Lacrosse 1328 Cadillac CTS 1425 Chevrolet Malibu 1533 Chrysler Sebring 1630 Dodge Avenger 1730 Variable of Interest: MPG 2009 Fuel Economy Guide MODEL MPG <new> MODEL MPG <new> Dodge Avenger 1630 Hyundai Elantra 1733 Jaguar XF 1825 Kia Optima 1932 Lexus GS Lincolon MKZ 2128 Mazda Mercedes-Benz E Mercury Milan Fuel Economy Guide Mercedes-Benz E Mercury Milan 29 Mitsubishi Galant 27 Nissan Maxima 26 Rolls Royce Phantom 18 Saturn Aura 33 Toyota Camry 31 Volkswagen Passat 29 Volvo S80 25 Dotplot of MPG Distribution
7 Examine each variable by itself. Then study relationships among the variables. How to Explore Data 2009 Fuel Economy Guide MODEL 2009 Fuel Economy Guide MPG 2009 Fuel Economy Guide MODEL MPG <new> MODEL MPG <new> Acura RL 922 Dodge Avenger 1630 Mercedes-Benz E Audi A6 Quattro 1023 Hyundai Elantra 1733 Mercury Milan 29 Bentley Arnage 1114 Jaguar XF 1825 Mitsubishi Galant 27 BMW Kia Optima 1932 Nissan Maxima 26 Buick Lacrosse 1328 Lexus GS Rolls Royce Phantom 18 Cadillac CTS 1425 Lincolon MKZ 2128 Saturn Aura 33 Chevrolet Malibu 1533 Mazda Toyota Camry 31 Chrysler Sebring 1630 Mercedes-Benz E Volkswagen Passat 29 Dodge Avenger 1730 Mercury Milan 2429 Volvo S80 25 Start with a graph or graphs Data Analysis Add numerical summaries
8 From Data Analysis to Inference Population Sample Collect data from a representative Sample... Data Analysis Make an Inference about the Population. Perform Data Analysis, keeping probability in mind
9 Inference: Inference is the process of making a conclusion about a population based on a sample set of data.
10 Introduction Data Analysis: Making Sense of Data Summary In this section, we learned that A dataset contains information on individuals. For each individual, data give values for one or more variables. Variables can be categorical or quantitative. The distribution of a variable describes what values it takes and how often it takes them. Inference is the process of making a conclusion about a population based on a sample set of data.
11 Do Notes worksheet : First page
12 Looking Ahead In the next Section We ll learn how to analyze categorical data. Bar Graphs Pie Charts Two-Way Tables Conditional Distributions We ll also learn how to organize a statistical problem.
13 Section 1.1 Analyzing Categorical Data Learning Objectives After this section, you should be able to CONSTRUCT and INTERPRET bar graphs and pie charts RECOGNIZE good and bad graphs CONSTRUCT and INTERPRET two-way tables DESCRIBE relationships between two categorical variables ORGANIZE statistical problems
14 Categorical Variables place individuals into one of several groups or categories The values of a categorical variable are labels for the different categories The distribution of a categorical variable lists the count or percent of individuals who fall into each category. Example, page 8 Variable Format Frequency Table Count of Stations Adult Contemporary 1556 Adult Standards 1196 Contemporary Hit 569 Country 2066 News/Talk 2179 Format Relative Frequency Table Percent of Stations Adult Contemporary 11.2 Adult Standards 8.6 Contemporary Hit 4.1 Country 14.9 News/Talk 15.7 Analyzing Categorical Data Values Oldies 1060 Religious 2014 Rock 869 Spanish Language 750 Other Formats 1579 Total Oldies 7.7 Religious 14.6 Rock 6.3 Count Spanish Language 5.4 Percent Other Formats 11.4 Total 99.9
15 Displaying categorical data Frequency tables can be difficult to read. Sometimes is is easier to analyze a distribution by displaying it with a bar graph or pie chart. Format Frequency Table Count of Stations Adult Contemporary 1556 Adult Standards 1196 Contemporary Hit 569 Country 2066 News/Talk 2179 Oldies 1060 Format Relative Frequency Table Percent of Stations Adult Contemporary 11.2 Adult Standards 8.6 Contemporary Hit 4.1 Country 14.9 News/Talk 15.7 Oldies 7.7 Analyzing Categorical Data Religious 2014 Rock 869 Spanish Language 750 Other Formats 1579 Total Religious 14.6 Rock 6.3 Spanish Language 5.4 Other Formats 11.4 Total 99.9
16 Graphs: Good and Bad Bar graphs compare several quantities by comparing the heights of bars that represent those quantities. Our eyes react to the area of the bars as well as height. Be sure to make your bars equally wide. Avoid the temptation to replace the bars with pictures for greater appeal this can be misleading! Analyzing Categorical Data Alternate Example This ad for DIRECTV has multiple problems. How many can you point out?
17 Example: Bar graph What personal media do you own? Here are the percents of year olds who own the following personal media devices. Device % who won Cell Phone 85 MP3 player 83 Handheld Video game player 41 Laptop 38 Portable CD/tape player 20 a) Make a well-labeled bar graph. What do you see? b) Would it be appropriate to make a pie chart for these data? Why or why not?
18 Two-Way Tables and Marginal Distributions When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables. Definition: Two-way Table describes two categorical variables, organizing counts according to a row variable and a column variable. Example, p. 12 Young adults by gender and chance of getting rich Female Male Total Almost no chance Some chance, but probably not A chance A good chance Almost certain Total What are the variables described by this twoway table? How many young adults were surveyed? Analyzing Categorical Data
19 Two-Way Tables and Marginal Distributions Definition: The Marginal Distribution of one of the categorical variables in a two-way table of counts is the distribution of values of that variable among all individuals described by the table. Note: Percents are often more informative than counts, especially when comparing groups of different sizes. Analyzing Categorical Data To examine a marginal distribution, 1)Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. 2)Make a graph to display the marginal distribution.
20 Percent + Two-Way Tables and Marginal Distributions Example, p. 13 Young adults by gender and chance of getting rich Female Male Total Almost no chance Some chance, but probably not A chance A good chance Almost certain Total Response Percent Almost no chance 194/4826 = 4.0% Some chance 712/4826 = 14.8% A chance 1416/4826 = 29.3% A good chance 1421/4826 = 29.4% Almost certain 1083/4826 = 22.4% Examine the marginal distribution of chance of getting rich. Chance of being wealthy by age 30 Almost none Some chance chance Good chance Survey Response Almost certain Analyzing Categorical Data
21 Relationships Between Categorical Variables Marginal distributions tell us nothing about the relationship between two variables. Definition: A Conditional Distribution of a variable describes the values of that variable among individuals who have a specific value of another variable. To examine or compare conditional distributions, 1)Select the row(s) or column(s) of interest. 2)Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s). 3)Make a graph to display the conditional distribution. Use a side-by-side bar graph or segmented bar graph to compare distributions.
22 Percent Percent + Two-Way Tables and Conditional Distributions Example, p. 15 Young adults by gender and chance of getting rich Female Male Total 100% Almost no chance % Some chance, but probably not % A chance % % A good chance % Almost certain % Total % Response Male Almost no chance 98/2459 = 4.0% Some chance 286/2459 = 11.6% A chance 720/2459 = 29.3% A good chance 758/2459 = 30.8% Almost certain 597/2459 = 24.3% Female 96/2367 = 4.1% 426/2367 = 18.0% 696/2367 = 29.4% 663/2367 = 28.0% 486/2367 = 20.5% % 10% 0% Chance of being wealthy by age 30 Calculate the conditional distribution of opinion among males. Examine the relationship between gender and opinion. Chance of of being wealthy by by age age Males Opinion Females Almost no Some Some chance chancegood Good chance chance chance chance chance Almost certain Almost certain Good chance chance Some chance Almost no chance Males Males Females
23 Organizing a Statistical Problem As you learn more about statistics, you will be asked to solve more complex problems. Here is a four-step process you can follow. How to Organize a Statistical Problem: A Four-Step Process State: What s the question that you re trying to answer? Plan: How will you go about answering the question? What statistical techniques does this problem call for? Analyzing Categorical Data Do: Make graphs and carry out needed calculations. Conclude: Give your practical conclusion in the setting of the real-world problem.
24 Do : P- 25 # 26. State: Do the data support the idea that people who get angry easily tend to have more heart disease? Plan: We suspect that people with different anger levels will have different rates of CHD, so we will compare the conditional distributions of CHD for each anger level. Do: We will display the conditional distributions in a table to compare the rate of CHD occurrence for each of the 3 anger levels Conclude: The conditional distributions show that while CHD occurrence is quite small overall, the percent of the population with CHD does increase as eth anger level increases.
25 Section 1.1 Analyzing Categorical Data Summary In this section, we learned that The distribution of a categorical variable lists the categories and gives the count or percent of individuals that fall into each category. Pie charts and bar graphs display the distribution of a categorical variable. A two-way table of counts organizes data about two categorical variables. The row-totals and column-totals in a two-way table give the marginal distributions of the two individual variables. There are two sets of conditional distributions for a two-way table.
26 Section 1.1 Analyzing Categorical Data Summary, continued In this section, we learned that We can use a side-by-side bar graph or a segmented bar graph to display conditional distributions. To describe the association between the row and column variables, compare an appropriate set of conditional distributions. Even a strong association between two categorical variables can be influenced by other variables lurking in the background. You can organize many problems using the four steps state, plan, do, and conclude.
27 Looking Ahead In the next Section We ll learn how to display quantitative data. Dotplots Stemplots Histograms We ll also learn how to describe and compare distributions of quantitative data.
28 Section 1.2 Displaying Quantitative Data with Graphs Learning Objectives After this section, you should be able to CONSTRUCT and INTERPRET dotplots, stemplots, and histograms DESCRIBE the shape of a distribution COMPARE distributions USE histograms wisely
29 Dotplots One of the simplest graphs to construct and interpret is a dotplot. Each data value is shown as a dot above its location on a number line. How to Make a Dotplot 1)Draw a horizontal axis (a number line) and label it with the variable name. 2)Scale the axis from the minimum to the maximum value. 3)Mark a dot above the location on the horizontal axis corresponding to each data value. Number of Goals Scored Per Game by the 2004 US Women s Soccer Team Displaying Quantitative Data
30 Examining the Distribution of a Quantitative Variable The purpose of a graph is to help us understand the data. After you make a graph, always ask, What do I see? How to Examine the Distribution of a Quantitative Variable In any graph, look for the overall pattern and for striking departures from that pattern. Describe the overall pattern of a distribution by its: Shape Center Spread Outliers Don t forget your SOCS! Note individual values that fall outside the overall pattern. These departures are called outliers. Displaying Quantitative Data
31 Examine this data The table and dotplot below displays the Environmental Protection Agency s estimates of highway gas mileage in miles per gallon (MPG) for a sample of 24 model year 2009 midsize cars Fuel Economy Guide Example, page 28 MODEL MPG Acura RL 922 Audi A6 Quattro 1023 Bentley Arnage 1114 BMW Buick Lacrosse 1328 Cadillac CTS 1425 Chevrolet Malibu 1533 Chrysler Sebring 1630 Dodge Avenger Fuel Economy Guide MODEL MPG <new> MODEL MPG <new> Dodge Avenger 1630 Hyundai Elantra 1733 Jaguar XF 1825 Kia Optima 1932 Lexus GS Lincolon MKZ 2128 Mazda Mercedes-Benz E Mercury Milan Fuel Economy Guide Mercedes-Benz E Mercury Milan 29 Mitsubishi Galant 27 Nissan Maxima 26 Rolls Royce Phantom 18 Saturn Aura 33 Toyota Camry 31 Volkswagen Passat 29 Volvo S80 25 Displaying Quantitative Data Describe the shape, center, and spread of the distribution. Are there any outliers?
32 Describing Shape When you describe a distribution s shape, concentrate on the main features. Look for rough symmetry or clear skewness. Definitions: A distribution is roughly symmetric if the right and left sides of the graph are approximately mirror images of each other. A distribution is skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side. It is skewed to the left (left-skewed) if the left side of the graph is much longer than the right side. Symmetric Skewed-left Skewed-right Displaying Quantitative Data
33 You Do: Smart Phone Battery Life Here is the estimated battery life for each of 9 different smart phones (in minutes) according to Smart Phone Battery Life (minutes) Make a dot plot. Apple iphone 300 * Then describe the shape, center, and spread. Motorola Droid 385 Palm Pre 300 Blackberry Bold Blackberry Storm Motorola Cliq 360 Samsung Moment Blackberry Tour HTC Droid 460 of the distribution. Are there any outliers?
34 Solution: Collection 1 Dot Plot BatteryLife (minutes) Shape: There is a peak at 300 and the distribution has a long tail to the right (skewed to the right). Center: The middle value is 330 minutes. Spread: The range is = 160 minutes. Outliers: There is one phone with an unusually long battery life, the HTC Droid at 460 minutes.
35 Place U.K South Africa + Comparing Distributions Some of the most interesting statistics questions involve comparing two or more groups. Always discuss shape, center, spread, and possible outliers whenever you compare distributions of a quantitative variable. Example, page 32 Compare the distributions of household size for these two countries. Don t forget your SOCS! ( Compare each characteristic)
36 Stemplots (Stem-and-Leaf Plots) Another simple graphical display for small data sets is a stemplot. Stemplots give us a quick picture of the distribution while including the actual numerical values. How to Make a Stemplot 1)Separate each observation into a stem (all but the final digit) and a leaf (the final digit). 2)Write all possible stems from the smallest to the largest in a vertical column and draw a vertical line to the right of the column. 3)Write each leaf in the row to the right of its stem. 4)Arrange the leaves in increasing order out from the stem. Displaying Quantitative Data 5)Provide a key that explains in context what the stems and leaves represent.
37 Stemplots (Stem-and-Leaf Plots) These data represent the responses of 20 female AP Statistics students to the question, How many pairs of shoes do you have? Construct a stemplot Key: 4 9 represents a female student who reported having 49 pairs of shoes. Displaying Quantitative Data Stems Add leaves Order leaves Add a key
38 Splitting Stems and Back-to-Back Stemplots When data values are bunched up, we can get a better picture of the distribution by splitting stems. Two distributions of the same quantitative variable can be compared using a back-to-back stemplot with common stems. Females Males split stems Females Males Key: 4 9 represents a student who reported having 49 pairs of shoes.
39 From both sides, compare and write about the center, spread, shape and possible outliers. Do CYU : P- 34. Check Your Understanding, page 34: 1. In general, it appears that females have more pairs of shoes than males. The median report for the males was 9 pairs while the female median was 26 pairs. The females also have a larger range of 57-13=44 in comparison to the range of 38-4=34 for the males. Finally, both males and females have distributions that are skewed to the right, though the distribution for the males is more heavily skewed as evidenced by the three likely outliers at 22, 35 and 38. The females do not have any likely outliers. 2. b 3. b 4. b
40 Histograms Quantitative variables often take many values. A graph of the distribution may be clearer if nearby values are grouped together. The most common graph of the distribution of one quantitative variable is a histogram. How to Make a Histogram 1)Divide the range of data into classes of equal width. 2)Find the count (frequency) or percent (relative frequency) of individuals in each class. 3)Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals.
41 Number of States + Example, page 35 Making a Histogram The table on page 35 presents data on the percent of residents from each state who were born outside of the U.S. Frequency Table Class 0 to < to < to < to < to < to <30 1 Total 50 Count Percent of foreign-born residents Displaying Quantitative Data
42 Using Histograms Wisely Here are several cautions based on common mistakes students make when using histograms. Cautions 1)Don t confuse histograms and bar graphs. 2)Use percents instead of counts on the vertical axis when comparing distributions with different numbers of observations. 3)Just because a graph looks nice, it s not necessarily a meaningful display of data. ( P- 40: Read the example)
43 Try Check your understanding: P 39 Many people.. (Also put these #s in the calculator.) Solution: 2. The distribution is roughly symmetric and bell-shaped. The median IQ appears to be between 110 and 120 and the IQ s vary from 80 to 150. There do not appear to be any outliers.
44 Section 1.2 Displaying Quantitative Data with Graphs Summary In this section, we learned that You can use a dotplot, stemplot, or histogram to show the distribution of a quantitative variable. When examining any graph, look for an overall pattern and for notable departures from that pattern. Describe the shape, center, spread, and any outliers. Don t forget your SOCS! Some distributions have simple shapes, such as symmetric or skewed. The number of modes (major peaks) is another aspect of overall shape. When comparing distributions, be sure to discuss shape, center, spread, and possible outliers. Histograms are for quantitative data, bar graphs are for categorical data. Use relative frequency histograms when comparing data sets of different sizes.
45 Do # 56 P- 46
46 Looking Ahead In the next Section We ll learn how to describe quantitative data with numbers. Mean and Standard Deviation Median and Interquartile Range Five-number Summary and Boxplots Identifying Outliers We ll also learn how to calculate numerical summaries with technology and how to choose appropriate measures of center and spread.
47 Section 1.3 Describing Quantitative Data with Numbers Learning Objectives After this section, you should be able to MEASURE center with the mean and median MEASURE spread with standard deviation and interquartile range IDENTIFY outliers CONSTRUCT a boxplot using the five-number summary CALCULATE numerical summaries with technology
48 Measuring Center: The Mean The most common measure of center is the ordinary arithmetic average, or mean. Definition: To find the mean (pronounced x-bar ) of a set of observations, add their values and divide by the number of observations. If the n observations are x 1, x 2, x 3,, x n, their mean is: x x sum of observations x 1 x 2... x n n n In mathematics, the capital Greek letter Σis short for add them all up. Therefore, the formula for the mean can be written in more compact notation: Describing Quantitative Data x x i n
49 Measuring Center: The Median Another common measure of center is the median. In section 1.2, we learned that the median describes the midpoint of a distribution. Definition: The median M is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger. To find the median of a distribution: 1)Arrange all observations from smallest to largest. 2)If the number of observations n is odd, the median M is the center observation in the ordered list. 3)If the number of observations n is even, the median M is the average of the two center observations in the ordered list. Describing Quantitative Data
50 Measuring Center Use the data below to calculate the mean and median of the commuting times (in minutes) of 20 randomly selected New York workers. Example, page x Key: 4 5 represents a New York worker who reported a 45- minute travel time to work. M minutes 22.5 minutes Describing Quantitative Data
51 Comparing the Mean and the Median The mean and median measure center in different ways, and both are useful. Don t confuse the average value of a variable (the mean) with its typical value, which we might describe by the median. Comparing the Mean and the Median The mean and median of a roughly symmetric distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is usually farther out in the long tail than is the median. Describing Quantitative Data
52 Measuring Spread: The Interquartile Range (IQR) A measure of center alone can be misleading. A useful numerical description of a distribution requires both a measure of center and a measure of spread. How to Calculate the Quartiles and the Interquartile Range To calculate the quartiles: 1)Arrange the observations in increasing order and locate the median M. 2)The first quartile Q 1 is the median of the observations located to the left of the median in the ordered list. 3)The third quartile Q 3 is the median of the observations located to the right of the median in the ordered list. The interquartile range (IQR) is defined as: Describing Quantitative Data IQR = Q 3 Q 1
53 Do CYU P- 55
54 Find and Interpret the IQR Example, page 57 Travel times to work for 20 randomly selected New Yorkers Q 1 = 15 M = 22.5 Q 3 = 42.5 IQR = Q 3 Q 1 = = 27.5 minutes Describing Quantitative Data Interpretation: The range of the middle half of travel times for the New Yorkers in the sample is 27.5 minutes.
55 Identifying Outliers In addition to serving as a measure of spread, the interquartile range (IQR) is used as part of a rule of thumb for identifying outliers. Definition: The 1.5 x IQR Rule for Outliers Call an observation an outlier if it falls more than 1.5 x IQR above the third quartile or below the first quartile. Example, page 57 In the New York travel time data, we found Q 1 =15 minutes, Q 3 =42.5 minutes, and IQR=27.5 minutes. For these data, 1.5 x IQR = 1.5(27.5) = Q x IQR = = Q x IQR = = Any travel time shorter than minutes or longer than minutes is considered an outlier Describing Quantitative Data
56 The Five-Number Summary The minimum and maximum values alone tell us little about the distribution as a whole. Likewise, the median and quartiles tell us little about the tails of a distribution. To get a quick summary of both center and spread, combine all five numbers. Definition: The five-number summary of a distribution consists of the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written in order from smallest to largest. Minimum Q 1 M Q 3 Maximum Describing Quantitative Data
57 Calculator: Command to get 5 number summary: Put all #s in the list. Stat -> Edit-> List L1-> Put the numbers Now, for the command for 5 number summary, Stat -> Calc -> choose 1:1-VarStats L1 Enter
58 Boxplots (Box-and-Whisker Plots) The five-number summary divides the distribution roughly into quarters. This leads to a new way to display quantitative data, the boxplot. How to Make a Boxplot Draw and label a number line that includes the range of the distribution. Draw a central box from Q 1 to Q 3. Note the median M inside the box. Describing Quantitative Data Extend lines (whiskers) from the box out to the minimum and maximum values that are not outliers.
59 Construct a Boxplot Example Consider our NY travel times data. Construct a boxplot Min=5 Q 1 = 15 M = 22.5 Q 3 = 42.5 Max=85 Recall, this is an outlier by the 1.5 x IQR rule Describing Quantitative Data
60 Measuring Spread: The Standard Deviation The most common measure of spread looks at how far each observation is from the mean. This measure is called the standard deviation. Let s explore it! Consider the following data on the number of pets owned by a group of 9 children. 1) Calculate the mean. 2) Calculate each deviation. deviation = observation mean deviation: 1-5 = -4 deviation: 8-5 = 3 Describing Quantitative Data x = 5
61 Measuring Spread: The Standard Deviation 3) Square each deviation. 4) Find the average squared deviation. Calculate the sum of the squared deviations divided by (n-1) this is called the variance. 5) Calculate the square root of the variance this is the standard deviation. average squared deviation = 52/(9-1) = 6.5 x i (x i -mean) (x i -mean) = -4 (-4) 2 = = -2 (-2) 2 = = -1 (-1) 2 = = -1 (-1) 2 = = -1 (-1) 2 = = 0 (0) 2 = = 2 (2) 2 = = 3 (3) 2 = = 4 (4) 2 = 16 Sum=? Sum=? This is the variance. Describing Quantitative Data Standard deviation = square root of variance =
62 Measuring Spread: The Standard Deviation Definition: The standard deviation s x measures the average distance of the observations from their mean. It is calculated by finding an average of the squared distances and then taking the square root. This average squared distance is called the variance. variance = s x 2 (x 1 x )2 (x 2 x ) 2... (x n x ) 2 n 1 standard deviation = s x 1 (x i x ) 2 n 1 1 (x i x ) 2 n 1 Describing Quantitative Data
63 Choosing Measures of Center and Spread We now have a choice between two descriptions for center and spread Mean and Standard Deviation Median and Interquartile Range Choosing Measures of Center and Spread The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. Use mean and standard deviation only for reasonably symmetric distributions that don t have outliers. Describing Quantitative Data NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA!
64 Section 1.3 Describing Quantitative Data with Numbers Summary In this section, we learned that A numerical summary of a distribution should report at least its center and spread. The mean and median describe the center of a distribution in different ways. The mean is the average and the median is the midpoint of the values. When you use the median to indicate the center of a distribution, describe its spread using the quartiles. The interquartile range (IQR) is the range of the middle 50% of the observations: IQR = Q 3 Q 1.
65 Section 1.3 Describing Quantitative Data with Numbers Summary In this section, we learned that An extreme observation is an outlier if it is smaller than Q 1 (1.5xIQR) or larger than Q 3 +(1.5xIQR). The five-number summary (min, Q 1, M, Q 3, max) provides a quick overall description of distribution and can be pictured using a boxplot. The variance and its square root, the standard deviation are common measures of spread about the mean as center. The mean and standard deviation are good descriptions for symmetric distributions without outliers. The median and IQR are a better description for skewed distributions.
66 Looking Ahead In the next Chapter We ll learn how to model distributions of data Describing Location in a Distribution Normal Distributions
67 92 (a) The median is the average of the ranked scores in the middle two positions (the 15th and 16th ranked scores). The median is Half of the students scored less than and half scored more than Q1 is the score one-quarter up the list of ordered scores, 82. Q3 is the score three-quarters up the ordered list of scores, 93. IQR = =11 The middle 50% of the scores have a range of 11 points. Any observation above Q IQR = (11) = or below Q1-1.5 IQR = (11) = 65.5 is considered an outlier. Thus, the scores 43 and 45 are outliers.
Variable - what we are measuring Quantitative - numerical where mathematical operations make sense. These have UNITS Categorical - puts individuals into categories Numbers don't always mean Quantitative...
Exploratory data analysis (Chapter 2) Fall 2011 Data Examples Example 1: Survey Data 1 Data collected from a Stat 371 class in Fall 2005 2 They answered questions about their: gender, major, year in school,
Page 1 of 16 Chapter 2: Exploring Data with Graphs and Numerical Summaries Graphical Measures- Graphs are used to describe the shape of a data set. Section 1: Types of Variables In general, variable can
Exploratory Data Analysis Exploratory Data Analysis involves both graphical displays of data and numerical summaries of data. A common situation is for a data set to be represented as a matrix. There is
Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.
Chapter 3: Data Description Numerical Methods Learning Objectives Upon successful completion of Chapter 3, you will be able to: Summarize data using measures of central tendency, such as the mean, median,
Chapter 1: Exploring Data Chapter 1 Review 1. As part of survey of college students a researcher is interested in the variable class standing. She records a 1 if the student is a freshman, a 2 if the student
Learning objectives Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc To recognize different types of variables To learn how to appropriately explore your data How to display data using graphs How
Types of Variables Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Quantitative (numerical)variables: take numerical values for which arithmetic operations make sense (addition/averaging)
3: Summary Statistics Notation Let s start by introducing some notation. Consider the following small data set: 4 5 30 50 8 7 4 5 The symbol n represents the sample size (n = 0). The capital letter X denotes
Center: Finding the Median When we think of a typical value, we usually look for the center of the distribution. For a unimodal, symmetric distribution, it s easy to find the center it s just the center
Describing Data: Categorical and Quantitative Variables Population The Big Picture Sampling Statistical Inference Sample Exploratory Data Analysis Descriptive Statistics In order to make sense of data,
AP Statistics Chapter 1 Test - Multiple Choice Name: 1. The following bar graph gives the percent of owners of three brands of trucks who are satisfied with their truck. From this graph, we may conclude
A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that
Math 1530-017 Exam 1 February 19, 2009 Name Student Number E There are five possible responses to each of the following multiple choice questions. There is only on BEST answer. Be sure to read all possible
1.3 Measuring Center & Spread, The Five Number Summary & Boxplots Describing Quantitative Data with Numbers 1.3 I can n Calculate and interpret measures of center (mean, median) in context. n Calculate
Interpreting Data in Normal Distributions This curve is kind of a big deal. It shows the distribution of a set of test scores, the results of rolling a die a million times, the heights of people on Earth,
STATS8: Introduction to Biostatistics Data Exploration Babak Shahbaba Department of Statistics, UCI Introduction After clearly defining the scientific problem, selecting a set of representative members
A Few Sources for Data Examples Used Introduction to Environmental Statistics Professor Jessica Utts University of California, Irvine firstname.lastname@example.org 1. Statistical Methods in Water Resources by D.R. Helsel
Chapter 2- Problems to look at Use the given frequency distribution to find the (a) class width. (b) class midpoints of the first class. (c) class boundaries of the first class. 1) Height (in inches) 1)
NATIONAL MATH + SCIENCE INITIATIVE Mathematics American League AL Central AL West AL East National League NL West NL East Level 7 th grade in a unit on graphical displays Connection to AP* Graphical Display
Content DESCRIPTIVE STATISTICS Dr Najib Majdi bin Yaacob MD, MPH, DrPH (Epidemiology) USM Unit of Biostatistics & Research Methodology School of Medical Sciences Universiti Sains Malaysia. Introduction
AP * Statistics Review Descriptive Statistics Teacher Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production
Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction
Lecture 1: Review and Exploratory Data Analysis (EDA) Sandy Eckel email@example.com Department of Biostatistics, The Johns Hopkins University, Baltimore USA 21 April 2008 1 / 40 Course Information I Course
Chapter 2 Overview Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Classify as categorical or qualitative data. 1) A survey of autos parked in
Lecture 2: Descriptive Statistics and Exploratory Data Analysis Further Thoughts on Experimental Design 16 Individuals (8 each from two populations) with replicates Pop 1 Pop 2 Randomly sample 4 individuals
Descriptive statistics is the discipline of quantitatively describing the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics (or inductive statistics),
Northumberland Knowledge Know Guide How to Analyse Data - November 2012 - This page has been left blank 2 About this guide The Know Guides are a suite of documents that provide useful information about
Data Mining Part 2. and Preparation 2.1 Spring 2010 Instructor: Dr. Masoud Yaghini Introduction Outline Introduction Measuring the Central Tendency Measuring the Dispersion of Data Graphic Displays References
10-3 Measures of Central Tendency and Variation So far, we have discussed some graphical methods of data description. Now, we will investigate how statements of central tendency and variation can be used.
1 2-1 Overview Chapter 2: Learn the methods of organizing, summarizing, and graphing sets of data, ultimately, to understand the data characteristics: Center, Variation, Distribution, Outliers, Time. (Computer
Numerical Measures of Central Tendency Often, it is useful to have special numbers which summarize characteristics of a data set These numbers are called descriptive statistics or summary statistics. A
Using Your TI-NSpire Calculator: Descriptive Statistics Dr. Laura Schultz Statistics I This handout is intended to get you started using your TI-Nspire graphing calculator for statistical applications.
Lecture 1 1 Lecture I Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions. It is a process consisting of 3 parts. Lecture
13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers
CONDENSED L E S S O N 1.1 Bar Graphs and Dot Plots In this lesson you will interpret and create a variety of graphs find some summary values for a data set draw conclusions about a data set based on graphs
MODE The mode of the sample is the value of the variable having the greatest frequency. Example: Obtain the mode for Data Set 1 77 For a grouped frequency distribution, the modal class is the class having
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 5: Density Curves and Normal Distributions (I) 9/12/06 Lecture 5 1 A problem about Standard Deviation A variable
TYPES OF DATA Univariate data Examines the distribution features of one variable. Bivariate data Explores the relationship between two variables. Univariate and bivariate analysis will be revised separately.
Summarizing and Displaying Categorical Data Categorical data can be summarized in a frequency distribution which counts the number of cases, or frequency, that fall into each category, or a relative frequency
Statistics Chapter 2 Frequency Tables A frequency table organizes quantitative data. partitions data into classes (intervals). shows how many data values are in each class. Test Score Number of Students
Exploratory Data Analysis Johannes Schauer firstname.lastname@example.org Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction
Data Exploration Data Visualization What is data exploration? A preliminary exploration of the data to better understand its characteristics. Key motivations of data exploration include Helping to select
Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate
3. Since 4. The HOMEWORK 3 Due: Feb.3 1. A set of data are put in numerical order, and a statistic is calculated that divides the data set into two equal parts with one part below it and the other part
CH.6 Random Sampling and Descriptive Statistics Population vs Sample Random sampling Numerical summaries : sample mean, sample variance, sample range Stem-and-Leaf Diagrams Median, quartiles, percentiles,
Descriptive Statistics Understanding Data: Dataset: Shellfish Contamination Location Year Species Species2 Method Metals Cadmium (mg kg - ) Chromium (mg kg - ) Copper (mg kg - ) Lead (mg kg - ) Mercury
STAT355 - Probability & Statistics Instructor: Kofi Placid Adragni Fall 2011 Chap 1 - Overview and Descriptive Statistics 1.1 Populations, Samples, and Processes 1.2 Pictorial and Tabular Methods in Descriptive
Chapter 3: Central Tendency Central Tendency In general terms, central tendency is a statistical measure that determines a single value that accurately describes the center of the distribution and represents
GCSE Statistics Revision notes Collecting data Sample This is when data is collected from part of the population. There are different methods for sampling Random sampling, Stratified sampling, Systematic
Minitab Guide This packet contains: A Friendly Guide to Minitab An introduction to Minitab; including basic Minitab functions, how to create sets of data, and how to create and edit graphs of different
Exploratory Data Analysis Psychology 3256 1 Introduction If you are going to find out anything about a data set you must first understand the data Basically getting a feel for you numbers Easier to find
Week 1 Exploratory Data Analysis Practicalities This course ST903 has students from both the MSc in Financial Mathematics and the MSc in Statistics. Two lectures and one seminar/tutorial per week. Exam
Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research
Mathematics Box-and-Whisker Plots About this Lesson This is a foundational lesson for box-and-whisker plots (boxplots), a graphical tool used throughout statistics for displaying data. During the lesson,
A Correlation of to the South Carolina Data Analysis and Probability Standards INTRODUCTION This document demonstrates how Stats in Your World 2012 meets the indicators of the South Carolina Academic Standards
Statistics Basics Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion Part 1: Sampling, Frequency Distributions, and Graphs The method of collecting, organizing,
Graphical and Tabular Summarization of Data OPRE 6301 Introduction and Re-cap... Descriptive statistics involves arranging, summarizing, and presenting a set of data in such a way that useful information
1 Methods for Describing Data Sets.1 Describing Data Graphically In this section, we will work on organizing data into a special table called a frequency table. First, we will classify the data into categories.
Session 36 Five-Number Summary and Box Plots Interpret the information given in the following box-and-whisker plot. The results from a pre-test for students for the year 2000 and the year 2010 are illustrated
Chapter 3 Descriptive Statistics: Numerical Measures Slide 1 Learning objectives 1. Single variable Part I (Basic) 1.1. How to calculate and use the measures of location 1.. How to calculate and use the
Chapter 2 Descriptive Statistics 2.1 Descriptive Statistics 1 2.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Display data graphically and interpret graphs:
BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential
Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques
CHINHOYI UNIVERSITY OF TECHNOLOGY SCHOOL OF NATURAL SCIENCES AND MATHEMATICS DEPARTMENT OF MATHEMATICS MEASURES OF CENTRAL TENDENCY AND DISPERSION INTRODUCTION From the previous unit, the Graphical displays
Section 3.1 Measures of Central Tendency: Mode, Median, and Mean One number can be used to describe the entire sample or population. Such a number is called an average. There are many ways to compute averages,
How to interpret scientific & statistical graphs Theresa A Scott, MS Department of Biostatistics email@example.com http://biostat.mc.vanderbilt.edu/theresascott 1 A brief introduction Graphics:
Report of for Chapter 2 pretest Exam: Chapter 2 pretest Category: Organizing and Graphing Data 1. "For our study of driving habits, we recorded the speed of every fifth vehicle on Drury Lane. Nearly every
Chapter 3. The Normal Distribution Topics covered in this chapter: Z-scores Normal Probabilities Normal Percentiles Z-scores Example 3.6: The standard normal table The Problem: What proportion of observations
Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Algebra II/ Advanced Algebra Unit 7: Inferences & Conclusions from Data These materials are for nonprofit educational purposes only.
Introduction to Statistics for Psychology and Quantitative Methods for Human Sciences Jonathan Marchini Course Information There is website devoted to the course at http://www.stats.ox.ac.uk/ marchini/phs.html
Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav) Organize and Display One Quantitative Variable (Descriptive Statistics, Boxplot & Histogram) 1. Move the mouse pointer
MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain
Unit 21 Student s t Distribution in Hypotheses Testing Objectives: To understand the difference between the standard normal distribution and the Student's t distributions To understand the difference between
Practice#1(chapter1,2) Name Solve the problem. 1) The average age of the students in a statistics class is 22 years. Does this statement describe descriptive or inferential statistics? A) inferential statistics
Today: - Inter-Quartile Range, - Outliers, - Boxplots. Reading for today: Start Chapter 4. Quartiles and the Five Number Summary - The five numbers are the Minimum (Q0), Lower Quartile (Q1), Median (Q2),
Name: Date: 1. A study is conducted on students taking a statistics class. Several variables are recorded in the survey. Identify each variable as categorical or quantitative. A) Type of car the student
Lesson 13 Main Idea Describe a data distribution by its center, spread, and overall shape. Relate the choice of center and spread to the shape of the distribution. New Vocabulary distribution symmetric