15161718192021Interest RateHouseholdBankCentralBankAverageBankCHOICEVISAGraphical Displays What could be wrong with this bar chart? What makes it misleading? For data to be useful, it often must be summarized either numerically or graphically, or both. Tables are an obvious way to summarize data, but are often hard to read. • We’ll learn other numerical summaries in Chapter 12 Consider the following two table summaries of unemployment data. This one is the unemployment rate for high school grads…..Chapters 10 and 11 Page 2 This one is the unemployement rate for college grads… Is it easy to see patterns or trends in these data?Chapters 10 and 11 Page 3 Average Unemployment Rate Blue = High School Grads Pink = College Grads In deciding which graphical display is appropriate for a data summary, we need to discuss different types of variables. Types of Variables Categorical (Category) • Nominal – Name of a Category – Ex. • Ordinal – Has a natural ordering – Ex. Quantitative (Quantity) • Discrete – distinct cutoffs between values – Ex. • Continuous – on a continuum – Ex.Chapters 10 and 11 Page 4 What type of Variable is choice of diet (vegetarian vs. non-vegetarian) A. Categorical & Nominal B. Categorical & Ordinal C. Quantitative & Discrete D. Quantitative & Continuous the number of Facebook “Friends” you have A. Categorical & Nominal B. Categorical & Ordinal C. Quantitative & Discrete D. Quantitative & Continuous The DISTRIBUTION of a variable tells us what values it takes and how often it takes these values. A distribution can be described by a table, graph, list, or function Good Graphical Displays should be clearly labeled with • Title – general subject, Label all variables, Units, Key (if necessary) Identify the source of the data. Do not over-label! Pie Chart Pie charts show how a whole is divided into parts Can be used to graphically represent the distribution of a categorical variable To make a pie chart, first draw a circle – Circle represents the whole (100% of the data) – Wedges w/in the circle represent the parts, with the angle spanned by each wedge in proportion to the size of that part. Example: Distribution of Education Level Among Persons Aged 30 – 34 in 2003Chapters 10 and 11 Page 5 Bar Graph A bar graph displays the number OR percentage of items in each category using proportionally-sized bars. Can be used to graphically represent a categorical variable or compare a quantitative variable across categories The bars do not touch in a bar graph! Called a “Frequency Bar Graph” when counts are used Called a “Relative Frequency Bar Graph” when percentages are used Which type of bar graph is this? A. Frequency B. Relative Frequency Which of the values can not be used on the vertical axis of a Relative Frequency Bar Chart? A. 0 B. 0.25 C. 50% D. 150Chapters 10 and 11 Page 6 Pictograms Beware of pictograms – these are generally considered to be poor displays of data since it is difficult to make a pictogram that is not misleading. A pictogram uses pictures instead of bars to represent quantities. Plus: Can be more visually appealing than a bar graph Minus: Can be misleading because our eyes respond to total area, not just height If we stretch the picture only in the vertical direction (which is how it should be done to represent the counts or relative frequencies in a category, then you distort the picture). Check out this pictogram displaying advertising dollars for three magazines. What is misleading about this graphic display? Line Graph To display change over time, make a line graph. This can be used to display a quantitative variable changing over time. A line graph of a variable plots each observation against the time at which it was measured. Time always goes on the horizontal scale. Variable you’re measuring always goes on the vertical scale. Connect the data points by lines to display the change over time.Chapters 10 and 11 Page 7 Average Price for Regular Unleaded Gasoline Bureau of Labor Statistics What should we look for? Overall pattern. – A trend is a long-term upward or downward movement over time. Striking deviations. Seasonal variation. – A pattern that repeats itself at known regular intervals of time is called seasonal variation. – Many series of regular measurements over time are seasonally adjusted. Histogram Histogram displays a quantitative variable across different groupings of values. One must be careful when choosing how to group together values: • All groupings must cover the same range so that the bars on the histogram are of equal width. • The height of each bin is how we compare the frequency of each range of values.Chapters 10 and 11 Page 8 Example Steps to create a frequency histogram: (1) Create equal width categories (groupings). (2) Count number of values in each category. (3) Draw histogram with a bar for each category. The height of a bar represents the count for that bar’s category. The bars touch since there are no gaps between categories. Be careful: (1) Number of categories can’t be too large or too small. (2) Don’t skip any categories. (3) Be clear about contents of each category Example Age at Time of First Oscar The groupings chosen here are [20,25), [25,30), [30,35), [35,40), [45,50), … If Jack Nicholson, age 70, wins for Best Actor, which category frequency will increase? A. [60,65) B.[65,70) C. [70,75) D. [75,80) Which graph type would be used to show the distribution of heights in the class?Chapters 10 and 11 Page 9 A. Line graph B. Pie Chart C. Histogram D. Bar Chart Let’s examine what it means to turn a frequency into a relative frequency by looking at the age at Oscar data. A relative frequency histogram depicts the relative frequency rather than the raw frequency (count) of categories. Changing frequencies to relative frequencies retains the shape of the distribution….Chapters 10 and 11 Page 10 Shapes of Distributions A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other. A distribution is skewed to the right if the right “tail” extends much farther out than the left tail. A distribution is skewed to the left if the left “tail” extends much farther out than the right tail. What is the shape of the distribution pictured here? A.