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Describing Data Visually Chapter Contents3.1 Stem-and-Leaf Displays and Dot Plots3.2 Frequency Distributions and Histograms3.3 Excel Charts3.4 Line Charts3.5 Bar Charts3.6 Pie Charts3.7 Scatter Plots3.8 Tables3.9 Deceptive Graphs

Chapter 3

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Chapter Learning Objectives LO3-1: Make a stem-and-leaf or dot plot by hand or by computer.

LO3-2: Create a frequency distribution for a data set.

LO3-3: Make a histogram with appropriate bins.

LO3-4: Identify skewness, modal classes, and outliers in a histogram.

LO3-5: Make an effective line chart using Excel.

Chapter 3

Describing Data Visually

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Chapter Learning Objectives LO3-6: Know the rules for effective bar charts and pie charts.

LO3-7: Make and interpret a scatter plot using Excel.

LO3-8: Make simple tables and pivot tables.

LO3-9: Recognize deceptive graphing techniques.

Chapter 3

Describing Data Visually

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• Methods of organizing, exploring and summarizing data include:

- Visual (charts and graphs) provides insight into characteristics of a data set without showing all the numbers.

- Numerical (statistics or tables) provides insight into characteristics of a data set showing numbers.

Chapter 3

3.1 Stem-and-Leaf Displays and Dot Plots

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• Begin with univariate data (a set of n observations on one variable) and consider the following:

Chapter 3

3.1 Stem-and-Leaf Displays and Dot Plots

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MeasurementLook at the data and visualize how they were collected and measured. This data set has high quality data. It is numeric, continuous, and has a ratio scale since there is a meaningful zero.

SortingSort the data and then summarize in a graphical display. Here are the sorted P/E ratios (random sample of 44 companies; values from Table 3.2).

Chapter 3

3.1 Stem-and-Leaf Displays and Dot Plots(Example: Price/Earnings Ratios)

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The type of graph you use to display your data is dependent on the type of data you have. Some charts are better suited for quantitative data, while others are better for displaying categorical data.

One simple way to visualize small data sets is a stem-and-leaf plot. The stem-and-leaf plot is a tool of exploratory data analysis (EDA) that seeks to reveal essential data features in an intuitive way. A stem-and-leaf plot is basically a frequency tally, except that we use digits instead of tally marks. For two-digit or three-digit integer data, the stem is the tens digit of the data, and the leaf is the ones digit.

Chapter 3

3.1 Stem-and-leaf Displays and Dot Plots

Stem-and-Leaf PlotLO3-1: Make a stem-and-leaf or dot plot by hand or by computer.

LO3-1

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For the 44 P/E ratios, the stem-and-leaf plot is given below.

Chapter 3

3.1 Stem-and-Leaf Displays and Dot Plots

For example, the data values in the fourth stem are 31, 37, 37, 38. We always use equally spaced stems (even if some stems are empty). The stem-and-leaf can reveal central tendency (24 of the 44 P/E ratios were in the 10–19 stem) as well as dispersion (the range is from 7 to 59). In this illustration, the leaf digits have been sorted, although this is not necessary. The stem-and-leaf has the advantage that we can retrieve the raw data by concatenating a stem digit with each of its leaf digits. For example, the last stem has data values 50 and 59. This plot is best used for data that lies within a relatively narrow range. How many companies in the sample had a P/E ratio of 10?

LO3-1

10’s digit Unit digit

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• Steps in Making a Dot Plot

• A dot plot is the simplest graphical display of n individual values of numerical data. - Easy to understand. - It reveals dispersion, central tendency, and the shape of the distribution.

1. Make a scale that covers the data range.

2. Mark the axes and label them.

3. Plot each data value as a dot above the scale at its approximate location.

Note: If more than one data value lies at about the same axis location, the dots are stacked vertically.

Chapter 3

Dot Plots

LO3-1 3.1 Stem-and-Leaf Displays and Dot Plots

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• The range is from 7 to 59.• All but a few data values lie between 10 and 25.• A typical “middle” data value would be around 17 or 18.• The data are not symmetric due to a few large P/E ratios.

Chapter 3

LO3-1 3.1 Stem-and-Leaf Displays and Dot Plots

Discussion: Which plot is better?

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Comparing Groups• A stacked dot plot compares two or more groups using a common

X-axis scale.

Chapter 3

3.1 Stem-and-Leaf Displays and Dot Plots

LO3-1

Stem and leaf plots good for small amounts of data. Dot plots OK for small to medium amounts of data. In class Example: 3.1 (a)

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Bins and Bin Limits

• A frequency distribution is a table formed by classifying n data values into k classes (bins).

• Bin limits define the values to be included in each bin. Widths must all be the same except when we have open-ended bins.

• Frequencies are the number of observations within each bin.

• Express as relative frequencies (frequency divided by the total) or percentages (relative frequency times 100).

Chapter 3

3.2 Frequency Distributions and HistogramsLO3-2

LO3-2: Create a frequency distribution for a data set

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- Herbert Sturges proposed the following rule: Constructing a Frequency Distribution

Chapter 3

3.2 Frequency Distributions and HistogramsLO3-2

2k-1 = n

How many bins? In other words, what is the value of k here?

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Chapter 3

LO3-23.2 Frequency Distributions and Histograms

Bin width = (xmax – xmin )/k = (59-7)/6 = 8.67 which is rounded to 10.

In general lower limit is included, and upper limit is excluded for the bin. However, Excel histograms include upper limit and exclude lower limit.In class Exercise: 3.4 frequency distribution only.

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Histograms

• A histogram is a graphical representation of a frequency distribution.

Y-axis shows frequency within each bin.

X-axis ticks shows end points of each bin.

Chapter 3

LO3-23.2 Frequency Distributions and Histograms

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• Consider 3 histograms for the P/E ratio data with different bin widths. What do they tell you?

Chapter 3

3.2 Frequency Distributions and HistogramsLO3-3

LO3-3: Make a histogram with appropriate bins.

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• Choosing the number of bins and bin limits in creating histograms

requires judgment.

• One can use software programs to create histograms with different bins. These include software such as:

• Excel• MegaStat• Minitab

In class Example: 3.4 histogram only.

Chapter 3

LO3-3: Make a histogram with appropriate bins.

3.2 Frequency Distributions and HistogramsLO3-3

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Modal ClassA histogram bar that is higher than those on either side.

• Unimodal – a single modal class.

• Bimodal – two modal classes.

• Multimodal – more than two modal classes.

• Modal classes may be artifacts of the way bin limits are chosen.

Chapter 3

3.2 Frequency Distributions and HistogramsLO3-3

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Shape• A histogram may suggest the shape of the population.

• Skewness – indicated by the direction of the longer tail of the histogram.

• It is influenced by the number of bins and bin limits.

Left-skewed – (negatively skewed) a longer left tail.

Right-skewed – (positively skewed) a longer right tail.

Symmetric – both tail areas are the same.

Chapter 3

LO3-4: Identify skewness, modes, and outliers in a histogram.

LO3-43.2 Frequency Distributions and Histograms

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Chapter 3

3.2 Frequency Distributions and HistogramsLO3-4

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Chapter 3

Frequency Polygons and Ogives

• A frequency polygon is a line graph that connects the midpoints of the histogram intervals, plus extra intervals at the beginning

and end so that the line will touch the X-axis. It serves the same purpose as a histogram, but is attractive when you

need to compare two data sets (since more than one frequency polygon can be plotted on the same scale).

• An ogive (pronounced “oh-jive”) is a line graph of the cumulative frequencies.

It is useful for finding percentiles or in comparing the shape of the sample with a known benchmark such as the normal

distribution (we will be seeing in the normal distribution in the next chapter).

3.2 Frequency Distributions and Histograms

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Chapter 3

3.2 Frequency Distributions and Histograms

Frequency Polygons and Ogives

Slope of ogive = percent value of frequency polygon

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Chapter 3

3.3 Excel Charts

This section describes how to use Excel to create charts. Please refer to the text.

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• Used to display a time series or spot trends, or to compare time periods.

• Can display several variables at once.

Simple Line Charts

Chapter 3

LO3-5: Make an effective line chart using Excel.

3.4 Line ChartsLO3-5

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• Two-scale line chart – used to compare variables that differ in magnitude or are measured in different units.

Simple Line Charts

Chapter 3

3.4 Line ChartsLO3-5

Should we do the above using two separate line charts instead?

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Log Scales• Arithmetic scale – distances on the Y-axis are proportional to the

magnitude of the variable being displayed.

• Logarithmic scale – (ratio scale) equal distances represent equal ratios.

• Use a log scale for the vertical axis when data vary over a wide range, say, by more than an order of magnitude.

• This will reveal more detail for smaller data values.

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LO3-5 3.4 Line Charts

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Log ScalesA log scale is useful for time series data that might be expected to grow at a compound annual percentage rate (e.g., GDP, the national debt, or yourfuture income). It reveals whether the quantity is growing at an increasing percent (concave upward), US Trade Data setconstant percent (straight line), or declining percent (concave downward)

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3.4 Line ChartsLO3-5

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Line Chart Rules

• Use line charts for time series data, not for cross sectional data.

• Time units usually go on the X-axis, increasing from left to right.

• Use a zero origin on the Y axis for arithmetic scales

In Class Exercise: 3.13

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• Most common way to display attribute data in business. - Bars represent categories or attributes. - Lengths of bars represent frequencies.

Simple Bar Charts

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3.5 Bar ChartsLO3-6: Know the rules for effective bar charts and pie charts.

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• Special type of bar chart used in quality management to display the frequency of defects or errors of different types.

• Categories are displayed in descending order of frequency.

• Focus on significant few (i.e., few categories that account for most defects or errors).

Pareto Charts

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3.5 Bar ChartsLO3-6

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• Column height is the sum of several subtotals. Areas may be compared by color to show patterns in the subgroups and total.

Stacked Column Chart

Chapter 3

3.5 Bar ChartsLO3-6

Discuss Vail Example: showing that pricing was important

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Tips for Bar Charts

• Category labels on X axis usually, while numerical levels on Y axis

• If a time series is displayed then put years or months or days on X- axis with increasing time from left to right

• Zero origin is usually recommended. This allows each bar length to be proportional to its value.

• Try and put numerical values at the top of each bar or in each bar if stacked.

In Class Exercise: 3.16 (a)

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• A pie chart can only convey a general idea of the data.• Pie charts should be used to portray data which sum to a total

(e.g., percent market shares). Must represent slices of a whole pie.

• A pie chart should only have a few (i.e., 2 to 5) slices.• Each slice can be labeled with data values or percents.

An Oft-Abused Chart

Chapter 3

3.6 Pie ChartsLO3-6: Know the rules for effective bar charts and pie charts.

LO3-6

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• Consider the following charts used to illustrate an article from the Wall Street Journal. Which type appears to be better?

An Oft-Abused Chart

Chapter 3

3.6 Pie ChartsLO3-6

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• Exploded and 3-D pie charts add strong visual impact. But the slice sizes are harder to assess.

Pie Chart Options

Chapter 3

3.6 Pie ChartsLO3-6

In Class Example: 3.19

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• Scatter plots can convey patterns in data pairs that would not be apparent from a table. Useful for bivariate data.

Chapter 3

3.7 Scatter PlotsLO3-7: Make and interpret a scatter plot using Excel.

LO3-7

Refer to the text for EXCEL outputs.

In class Example: 3.21

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• Tables are the simplest form of data display.

• Arrangement of data is in rows and columns to enhance meaning.

• A compound table is a table that contains time series data down the columns and variables across the rows.

• The data can be viewed by focusing on the time pattern (down the columns) or by comparing the variables (across the rows).

Example: School Expenditures

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3.8 Tables

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• Units of measure are stated in the footnote.• Note merged headings to group columns.• See text for “Tips for Effective Tables.” Tables”.

Example: School Expenditures

Chapter 3

3.8 Tables

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Chapter 3

3.8 Tables

Here are some tips for creating effective tables:

1. Keep the table simple, consistent with its purpose. Put summary tables in the main body of the written report and

detailed tables in an appendix. 2. Display the data to be compared in columns rather than rows. 3. For presentation purposes, round off to three or four significant

digits.4. Physical table layout should guide the eye toward the

comparison you wish to emphasize. 5. Row and column headings should be simple yet descriptive.6. Within a column, use a consistent number of decimal digits.

LO3-8: Make simple tables and Pivot tables

LO3-8

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• A nonzero origin will exaggerate the trend.

Deceptive Correct

Error 1: Nonzero Origin

Chapter 3

LO3-9: Recognize deceptive graphing techniques.

LO3-9 3.9 Deceptive Graphs

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• Keep the aspect ratio (width/height) below 2.00 so as not to exaggerate the graph. By default, Excel uses an aspect ratio of 1.68.

Error 2: Elastic Graph Proportions

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LO3-9 3.9 Deceptive Graphs

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• Can make trends appear to dwindle into the distance or loom towards you.

Error 4: 3-D and Novelty Graphs

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3.9 Deceptive GraphsLO3-9

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• Can make trends appear to dwindle into the distance or loom towards you.

Error 5: 3-D and Rotated Graphs

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3.9 Deceptive GraphsLO3-9

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• Avoid if possible. Keep your main objective in mind. Break graph into smaller parts.

Error 8: Complex Graphs

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LO3-9 3.9 Deceptive Graphs

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• As figure height increases, so does width, distorting the graph.

Error 11: Area Trick

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3.9 Deceptive GraphsLO3-9

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• Other deceptive graphing techniques.

• Error 3: Dramatic Title and Distracting Pictures• Error 6: Unclear Definitions or Scales• Error 7: Vague Sources• Error 9: Gratuitous Effects• Error 10: Estimated Data

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LO3-9 3.9 Deceptive Graphs

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Homework

• 3.2 (a) (use stems with values from 8 to 19 for the 10’s place)• 3.6• 3.12• 3.32