Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Chapter 2 Summarizing and Graphing Data 2-1 Review and Preview 2-2 Frequency Distributions 2-3 Histograms 2-4 Statistical Graphics 2-5 Critical Thinking: Bad Graphs Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Section 2-1 Review and Preview In the last chapter we discussed collecting and analyzing data sets. Data sets are usually very large and nobody wants to look at tables of numbers… => Data has to be represented graphically. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Frequency Distributions Section 2-2 Frequency Distributions Copyright © 2010 Pearson Education Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Definition Frequency Distribution (or Frequency Table) shows how a data set is partitioned among all of several categories (or classes) by listing all of the categories along with the number of data values in each of the categories. Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Pulse Rates of Females and Males Original Data Simple random samples of 40 females and 40 males. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Frequency Distribution Pulse Rates of Females The frequency for a particular class is the number of original values that fall into that class. Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Frequency Distributions Definitions Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Lower Class Limits Lower Class Limits are the smallest numbers that can actually belong to different classes Lower Class Limits Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Upper Class Limits Upper Class Limits are the largest numbers that can actually belong to different classes Upper Class Limits Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education

Class Boundaries Class Boundaries are the numbers used to separate classes, but without the gaps created by class limits 59.5 69.5 79.5 89.5 99.5 109.5 119.5 129.5 Class Boundaries Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Class Midpoints Class Midpoints are the values in the middle of the classes and can be found by adding the lower class limit to the upper class limit and dividing the sum by two 64.5 74.5 84.5 94.5 104.5 114.5 124.5 Class Midpoints Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education

Class Width Class Width is the difference between two consecutive lower class limits or two consecutive lower class boundaries Class Width 10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education

Relative Frequency Distribution includes the same class limits as a frequency distribution, but the frequency of a class is replaced with a relative frequencies (a proportion) or a percentage frequency ( a percent) relative frequency = class frequency sum of all frequencies percentage frequency class frequency sum of all frequencies  100% = Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education

Relative Frequency Distribution * Sum of relative frequencies should total to 1 or 100% Total Frequency = 40 * 12/40  100 = 30% Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Cumulative Frequency Distribution Cumulative Frequencies = sum of frequencies for that class and all previous classes Cumulative Frequencies Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education

Data->Data Analysis->Histogram 1st install Data Analysis Toolpack (Excel Options, Data Analysis, press Go) File FHEALTH Data->Data Analysis->Histogram Select data Data->Data Analysis->Histogram To install Data Analysis toolpack go to Excel Options and choose Add-Ins Do not forget to click on GO Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Critical Thinking Interpreting Frequency Distributions In later chapters, there will be frequent reference to data with a normal distribution. One key characteristic of a normal distribution is that it has a “bell” shape. The frequencies start low, then increase to one or two high frequencies, then decrease to a low frequency. The distribution is approximately symmetric, with frequencies preceding the maximum being roughly a mirror image of those that follow the maximum. Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Public and Private 2 and 4 year colleges. Table 2.5, p 50 IQ scores of 1000 adults – looks normal Graphs above are for quantitative data, but they can also be used for qualitative data. Ex 6, p 52, Table 2.9 Distribution of undergraduate college student enrollment among Public and Private 2 and 4 year colleges. My example: Degree of Agreement Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

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Section 2-3 Histograms Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Histogram Basically a graphic version of a frequency distribution. STATDISK Data->Hystogram Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Relative Frequency Histogram Has the same shape and horizontal scale as a histogram, but the vertical scale is marked with relative frequencies instead of actual frequencies – better for comparison with other histograms. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

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Critical Thinking Interpreting Histograms Objective is not to construct a histogram (computer job), but rather to understand something about the data (center, variation, distribution, outliers, etc…) For example, when graphed, a normal distribution has a “bell” shape. Characteristic of the bell shape are: (1) The frequencies increase to a maximum, and then decrease, and (2) symmetry, with the left half of the graph roughly a mirror image of the right half. The histogram on the next slide illustrates this. Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Statdisk-generated histogram of 1000 IQ’s Table 2.5 p 50 When graphed, a normal distribution has a “bell” shape. Characteristic of the bell shape are The frequencies increase to a maximum, and then decrease symmetry, with the left half of the graph roughly a mirror image of the right half. Statdisk-generated histogram of 1000 IQ’s Table 2.5 p 50 Statdisk-generated histogram of Datasets -> Female Health Exam file, 2nd column - height Copyright © 2010 Pearson Education Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Section 2-4 Statistical Graphics Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Frequency Polygon Uses line segments connected to points directly above class midpoint values Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Relative Frequency Polygon Uses relative frequencies (proportions or percentages) for the vertical scale. It is crucial to use relative frequencies when comparing data sets Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Ogive A line graph that depicts cumulative frequencies - Useful for determining the number of values below some particular value class boundaries Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Bar Graph Uses bars of equal width to show frequencies of categories of qualitative data. Vertical scale represents frequencies or relative frequencies. Horizontal scale identifies the different categories of qualitative data. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Multiple Bar Graph A multiple bar graph has two or more sets of bars, and is used to compare two or more data sets. Median Income of Males and Females Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Scatter Plot (or Scatter Diagram) A plot of paired (x,y) data with a horizontal x-axis and a vertical y-axis. Used to determine whether there is a relationship between the two variables Temperature vs number of cricket chirps per minute. There appears to be a relationship. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

More scatter plot examples: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Using StatDisk height(inches) vs weight (lbs) for Female Health Exam data Excel Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

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Time-Series Graph Data that have been collected at different points in time: time-series data Example: Dow Jones Industrial Average vs Time: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

More examples of time series We describe time series by looking for an overall pattern and for striking deviations from that pattern. In a time series: A trend is a rise or fall that persists over time, despite small irregularities. A pattern that repeats itself at regular intervals of time is called seasonal variation. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Retail price of fresh oranges over time Time is on the horizontal, x axis. The variable of interest—here “retail price of fresh oranges”— goes on the vertical, y axis. This time plot shows a regular pattern of yearly variations. These are seasonal variations in fresh orange pricing most likely due to similar seasonal variations in the production of fresh oranges. There is also an overall upward trend in pricing over time. It could simply be reflecting inflation trends or a more fundamental change in this industry. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Critical Thinking: Bad Graphs Section 2-5 Critical Thinking: Bad Graphs Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Nonzero Axis Are misleading because one or both of the axes begin at some value other than zero, so that differences are exaggerated. Ex: Left – bar graph of CNN poll regarding the case of Terry Schiavo. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Another example of graph not starting at 0: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Pictographs -- are drawings of objects. Three-dimensional objects - money bags, stacks of coins, army tanks (for army expenditures), people (for population sizes), barrels (for oil production), and houses (for home construction) are commonly used to depict data. These drawings can create false impressions that distort the data. If you double each side of a square, the area does not merely double; it increases by a factor of four; If you double each side of a cube, the volume does not merely double; it increases by a factor of eight. Pictographs using areas and volumes can therefore be very misleading. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Pictographs Part (b) is designed to exaggerate the difference by increasing each dimension in proportion to the actual amounts of oil consumption. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Annual Incomes of Groups with Different Education Levels -- not misleading because the bars have same width. But somewhat too busy. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Annual Incomes of Groups with Different Education Levels Misleading. Depicts one-dimensional data with three-dimensional boxes. Last box is 64 times as large as first box, but income is only 4 times as large. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Annual Incomes of Groups with Different Education Levels Fair, objective, no distracting features. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

What is wrong with these graphs? Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.