bcor 1020 business statistics lecture 3 – january 24, 2008

BCOR 1020Business Statistics

Lecture 3 – January 24, 2008

Overview• Chapter 3 – Describing Data Visually…

– Visual Description– Dot Plots– Frequency Distribution and Histograms– Simple Line Charts & Bar Charts– Scatter Plots– Tables– Pie Charts– Maps and Pictograms– Deceptive Graphs

Chapter 3 – Visual DescriptionMethods of organizing, exploring and summarizing data include:

• Visual (charts and graphs) – provides insight into characteristics of a data set without using mathematics.

• Numerical (statistics or tables) – provides insight into characteristics of a data set using mathematics.

Chapter 3 – Visual DescriptionBeginning with univariate data (a set of n observations on one variable), consider the following:

• Measurement – What are the units of measurement? Are the data integer or continuous? Any missing observations? Any concerns with accuracy or sampling methods?

• Central Tendency – Where are the data values concentrated? What seem to be typical or middle data values?

• Dispersion – How much variation is there in the data? How spread out are the data values? Are there unusual values?

• Shape – Are the data values distributed symmetrically? Skewed? Sharply peaked? Flat? Bimodal?

Chapter 3 – Visual Description• Example: Price/Earnings Ratios:

• P/E ratios are current stock price divided by earnings per share in the last 12 months. For example:

Chapter 3 – Visual DescriptionMeasurement – Look at the data and visualize how it was collected and measured.

Sorting – Sort the data and then summarize in a graphical display.

– Here are the sorted P/E ratios:

– Sorting allows you to observe central tendency, dispersion and shape as well as minimum, maximum and range.

88 1010 1010 1010 1313 1313 1414 1414 1515 1515

1616 1616 1717 1818 1919 1919 2020 2020 2121 2222

2323 2626 2626 2727 2929 2929 3434 4848 5555 6868

Chapter 3 – Dot PlotsA dot plot is the simplest graphical display of n individual values of numerical data.

– Easy to understand.– Not good for large samples (e.g., > 5,000).

Steps in Making a Dot Plot: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. If more than one data value lies at about the same axis location, the dots are piled up vertically.

* Figure 3.4 in your text details the MegaStat menus for creating a dotplot.

Chapter 3 – Dot Plots

• Range of data shows dispersion.

• Can add annotations (text boxes) to call attention to specific features.

• Clustering shows central tendency. • Dot plots do not tell much of shape of distribution.

Chapter 3 – Frequency Distributions and Histograms

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.• Frequencies are the number of observations within

each bin.• Express as relative frequencies (frequency divided by

the total) or percentages (relative frequency times 100).


Constructing a Frequency Distribution:1. Sort data in ascending order (e.g., P/E ratios)

2. Choose the number of bins (k).– k should be much smaller than n. – Too many bins results in sparsely populated bins, too few and

dissimilar data values are lumped together.

– Herbert Sturges proposes the following rule: k = 1 + log2(n)

Sample Size Sample Size (n)(n)

Number of Bins Number of Bins (k)(k)

1616 55

3232 66

6464 77

128128 88

Sample Size Sample Size (n)(n)

Number of Bins Number of Bins (k)(k)

256256 99

512512 1010

10241024 1111


Constructing a Frequency Distribution:

3. Set the bin limits: Bin width max minX X

k

In our example, we will use k = 7 bins to get convenient bin limits. The approximate bin width is:

68 8 608.57

7 7

Bin width

To obtain “nice” limits, we round the width to 10 and start the first bin at 0 to get bin limits:

0, 10, 20, 30, 40, 50, 60, 70


Constructing a Frequency Distribution:4. Put the data values in the appropriate bin.

– In general, the lower limit is included in the bin while the upper limit is excluded.

5. Create the table, you can include: – Frequencies – counts for each bin– Relative frequencies – absolute frequency divided by total

number of data values.– Cumulative frequencies – accumulated relative frequency

values as bin limits increase.

Example: Back to the P/E ratio data…


What are the bin limits for the P/E ratio data?

Bin Range FrequencyRelative

Frequency

Cumulative Relative

Frequency

0<P/E Ratio<10 1 0.0333 0.0333

10<P/E Ratio<20 15 0.5000 0.5333

20<P/E Ratio<30 10 0.3333 0.8666

30<P/E Ratio<40 1 0.0333 0.8999

40<P/E Ratio<50 1 0.0333 0.9332

50<P/E Ratio<60 1 0.0333 0.9665

60<P/E Ratio<70 1 0.0333 0.9998


Histograms:• A histogram is a graphical representation of a frequency distribution.• A histogram is a bar chart.

– X-axis ticks shows end points of each bin.

– Y-axis shows frequency (or relative/cumulative frequency) within each bin.Consider 3 histograms for the P/E ratio data with different bin widths. Do they give you different impressions of the data?

k = 4 k = 7 k = 13

* Figures 3.8 & 3.9 in your text details the MegaStat menus for creating a histogram.


Modal Class – a histogram bar that is higher than those on either side:• Monomodal – a single modal class.• Bimodal – two modal classes.• Multimodal – more than two modal classes.

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


Shape:• A histogram suggests the shape of the

population.• Skewness – indicated by the direction of the

longer tail of the histogram.– Left-skewed – (negatively skewed) a longer left tail.– Right-skewed – (positively skewed) a longer right tail.– Symmetric – both tail areas approximately the same.

Some examples…

ClickersConsider the histogram of the P/E ratio data that was displayed earlier in this lecture. How would you describe the skewness of this histogram?

A = symmetric

B = left-skewed

C = right-skewed

Chapter 3 – Simple Line Charts

Simple Line Charts – Used to display a time

series or spot trends, or to compare time periods.

• Can display several variables at once.

Chapter 3 – Simple Line ChartsTwo-scale line chart –used to compare variables that differ in magnitude or are measured in different units.

Grid Lines – A line graph usually has no vertical grid lines. Horizontal lines can be added to make it easier to establish the y value.

Which is easier to read?


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 small data values.

– Log scale is only suited for positive data values.– Reveals whether the quantity is growing at an

increasing percent (concave upward), constant percent (straight line), or declining percent (concave downward)

Example…Consider the following graphs illustrating U.S. Trade from 1959 to 2002. What does the log scale graph tell you about growth rate for both series?

Arithmetic scale Log scale


When to Use Log Scales:• Useful for…

• time series data that might be expected to grow at a compound annual percentage rate (e.g., GDP, national debt, future income)

• financial charts that cover long periods of time-data that grow rapidly (e.g., revenues)

Chapter 3 – Simple Line ChartsTips for Effective Line Charts:1. Line charts are used for time series data (never

for cross-sectional data).2. Y-axis shows numerical variable while X-axis

shows time units with time increasing left to right.

3. Use a zero origin on the Y-axis unless more detail is needed.

4. Omit numerical labels on a line chart to avoid clutter. Use gridlines if needed.

5. Use data markers (squares, triangles, circles) if they don’t clutter the graph.

6. Don’t make lines too thick.

Chapter 3 – Bar Charts

Plain Bar Charts – Most common way to display attribute data.• Bars represent categories or attributes.• Lengths of bars represent frequencies.

Chapter 3 – Bar ChartsPareto Charts – 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).


Stacked Bar Chart – Bar height is the sum of several subtotals. Areas may be compared by color to show patterns in the subgroups and total.


Bar Charts for Time Series Data – Bar charts can be (and often are) used for time series data although it may be harder to compare trends.


Tips for Effective Bar Charts:1. Show the numerical variable of interest with

vertical bars on the Y-axis, category labels on the X-axis.

2. For time series quantities, display the category labels on the horizontal X-axis with time increasing from left to right.

3. The height or length of each bar should be proportional to the quantity displayed.

4. Put numerical values at the top of each bar, except if too cluttered.

Chapter 3 – Scatter Plots

Example: Aircraft Fuel Consumption:• Consider five observations on flight time and fuel

consumption for a twin-engine Piper Cheyenne aircraft.

Trip Trip LegLeg

Flight Flight Time Time

(hours)(hours)

Fuel Fuel Used Used

(pounds)(pounds)

11 2.32.3 145145

22 4.24.2 258258

33 3.63.6 219219

44 4.74.7 276276

55 4.94.9 283283

• A causal relationship is assumed since a longer flight would consume more fuel.

Chapter 3 – Scatter Plots

• Example: Aircraft Fuel Consumption:• Here is the scatter plot with flight time on the

X-axis and fuel use on the Y-axis.

• Is there an association

between variables?

* Figure 3.31 in your text details the Excel menus for creating a scatter plot.

Chapter 3 – Scatter PlotsDegree of Association/Correlation:

Very strong association Strong association

Moderate association Little or no association

ClickersConsider the scatter plot (below) comparing birthrates and life expectancies in several countries.

True or False: This graph shows a strong association between these two variables.

A = True

B = False

Chapter 3 – TablesTables are the simplest form of data display. A compound table is a table that contains time series data down the columns and variables across the rows.Example: School Expenditures

• Arrangement of data is in rows and columns to enhance meaning.• The data can be viewed by focusing on the time pattern (down the columns) or

by comparing the variables (across the rows).• Units of measure are stated in the footnote.• Note merged headings to group columns.

Elementary and SecondaryElementary and Secondary Colleges and UniversitiesColleges and Universities

YearYear All SchoolsAll Schools TotalTotal PublicPublic PrivatePrivate TotalTotal PublicPublic PrivatePrivate

19601960 142.2142.2 99.699.6 93.093.0 6.66.6 42.642.6 23.323.3 19.319.3

19701970 317.3317.3 200.2200.2 188.6188.6 11.611.6 117.2117.2 75.275.2 41.941.9

19801980 373.6373.6 232.7232.7 216.4216.4 16.216.2 140.9140.9 93.493.4 47.447.4

19901990 526.1526.1 318.5318.5 293.4293.4 25.125.1 207.6207.6 132.9132.9 74.774.7

20002000 691.9691.9 418.2418.2 387.8387.8 30.330.3 273.8273.8 168.8168.8 105.0105.0

Source: U.S. Census Bureau, Source: U.S. Census Bureau, Statistical Abstract of the United States: 2002Statistical Abstract of the United States: 2002, p. 133. Note: All , p. 133. Note: All figures are in billions of constant 2000/2001 dollars.figures are in billions of constant 2000/2001 dollars.

Chapter 3 – TablesTips for Effective Tables:1. Keep the table simple, consistent with its purpose.

– Summary tables go in the main body.– Detailed tables go in an appendix.– In a slide show, main point of table should be clear within 10 seconds,

otherwise, break up table.

2. Display the data to be compared in columns.3. Round off data to 3 or 4 significant figures.4. Table layout should guide the eye towards the desired

comparison.– Use spaces or shading to separate rows or columns.– Use lines sparingly.

5. Keep row and column headings simple yet descriptive.6. Use a consistent number of decimal digits within a

column.– Right-justify or decimal align the data.

Chapter 3 – Pie Charts

An Oft-Abused Chart:• 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).– If frequency counts are important, use a bar chart or

histogram.

• A pie chart should only have a few (i.e., 2 or 3) slices.

• Each slice should be labeled with data values or percents.

Chapter 3 – Maps and Pictograms

Spatial Variation and GIS:• Maps can be used for displaying many kinds of

data.– Appropriate when patterns of variation across space are

of interest.– Self-explanatory and revealing.– Assess patterns based on geography.

• GIS (geographic information systems) combines statistics, geography and graphics.

Chapter 3 – Maps and PictogramsExample: U.S. population change by county, 1990/2000U.S. population change by county, 1990/2000

Chapter 3 – Maps and PictogramsExample: U.S. presidential election results, 2004U.S. presidential election results, 2004

On election night 2004 and in the months and years since then, we have seen many maps that look like this.

The amount of red on the map is skewed because there are a lot of large states (geographically) in which a majority voted Republican.

One possible way to allow for this, suggested by Robert Vanderbei at Princeton University, is to use not just two colors on the map, red and blue, but instead to use red, blue, and shades of purple to indicate percentages of voters. Here is what the normal map looks like if you do this.

Source: http://www-personal.umich.edu/~mejn/election/

http://www.princeton.edu/~rvdb/JAVA/election2004/

Chapter 3 – Maps and PictogramsExample: U.S. presidential election results, 2004U.S. presidential election results, 2004

We can also correct for this by making use of a cartogram, a map in which the sizes of states have been rescaled according to their population. That is, states are drawn with a size proportional not to their sheer topographic acreage -- which has little to do with politics -- but to the number of their inhabitants, states with more people appearing larger than states with fewer, regardless of their actual area on the ground.

Source: http://www-personal.umich.edu/~mejn/election/

Chapter 3 – Maps and PictogramsPictograms – A visual display in which data values are replaced by pictures.

• Although entertaining, they can create visual distortion. What do you think?

Chapter 3 – Deceptive Graphs

• A nonzero origin will exaggerate the trend.

Error 1: Nonzero Origin


• Keep the aspect ratio (width/height) below 2.00 so as not to exaggerate the graph. By default, Excel uses an aspect ratio of 1.8.

Error 2: Elastic Graph Proportions


• Keep short and grab readers attention.

• Avoid so as not to distract readers or impart an emotional slant.

• Can use pictures of authority figures to impart credibility to self-serving commercial claims.

Error 3: Dramatic Title

Error 4: Distracting Pictures

Error 5: Authority Figures


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

Deceptive Correct

Error 6: 3-D and Rotated Graphs


• If tick marks are missing, you cannot identify individual data values.

• Missing or unclear units of measurement can render a chart useless.

• May indicate lost citation, unknown source, or mixed data sources. Use complete source citations.

Error 7: Missing Axis Demarcations

Error 8: Missing Measurement Units or Definitions

Error 9: Vague Source


• Avoid if possible. Keep your main objective in mind. If necessary, break graph into smaller parts.

Error 10: Complex Graphs


• Avoid too many annoying special effects when using slide shows.

• Estimated points should be noted when used or avoided if possible.

Error 11: Gratuitous Effects

Error 12: Estimated Data


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

Error 13: Area Trick

ClickersConsider the graph given below. What error is present that makes this a deceptive graph?

A = Non-Zero Origin

B = Dramatic Title

C = 3-D or Rotated

D = Complex Graph

Avg. Annual Tuition at Colorado 4-Year Universities

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$6,000

$7,000

1997

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1998

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1999

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2000

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2001

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2002

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3

2003

-200

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2004

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2005

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2006

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Chapter 3 – Appendix:Effective Excel Charts

Use the mouse Use the mouse to select to select (highlight) the (highlight) the data you want to data you want to plot.plot.

• Click on the Chart Wizard icon on the toolbar to open a sequence of pop-up menus to guide you through the steps of creatingcreating a chart.

• Step 1:Step 1: Select the Select the Chart typeChart type and then click and then click NextNext..

Effective Excel ChartsEffective Excel ChartsEffective Excel ChartsEffective Excel Charts

• Chart WizardChart Wizard

• Step 2:Step 2: Add labels for Add labels for years on the years on the XX-axis by -axis by selecting a data range selecting a data range (B4:B13). Click (B4:B13). Click NextNext..



• Step 3:Step 3: Embellish Embellish the chart by adding the chart by adding a title, axis labels, a title, axis labels, adjusting the adjusting the gridlines or gridlines or appending a data appending a data table to the graph by table to the graph by clicking on the clicking on the appropriate tab.appropriate tab.



• Step 4: Click Step 4: Click NextNext to display the finished chart. to display the finished chart.



• Charts created in Excel can be edited to:Charts created in Excel can be edited to:

- Improve the titles (main, - Improve the titles (main, XX-axis, -axis, YY-axis).-axis).

- Change the axis scales (minimum, maximum, - Change the axis scales (minimum, maximum, demarcations).demarcations).

- Display the data - Display the data values (on the top values (on the top of each bar).of each bar).

Fractional Shares, 1993-2002

110 158 285 548957

1551

2607

3834

4871

5827

0

1000

2000

3000

4000

5000

6000

7000

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

Series1


• Embellished ChartsEmbellished Charts


- Add a data table underneath the graph.- Add a data table underneath the graph.


0

2000

4000

6000

8000

Series1

Series1 110 158 285 548 957 1551 2607 3834 4871 5827

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002




- Change color or patterns in the plot or chart areas.- Change color or patterns in the plot or chart areas.


0

1000

2000

3000

4000

5000

6000

7000

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

Series1



• Charts created in Excel can be edited to:Charts created in Excel can be edited to:- Format the decimals (on the axes or data labels).- Format the decimals (on the axes or data labels).

- Edit the - Edit the gridlines gridlines (color, (color, dotted or dotted or solid, solid, patterns).patterns).

0

1000

2000

3000

4000

5000

6000

7000

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

Series1




- Alter the appearance of the bars (color, pattern, gap - Alter the appearance of the bars (color, pattern, gap width).width).


0

1000

2000

3000

4000

5000

6000

7000

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

Series1



• To alter a chart’s appearance, click on any chart object To alter a chart’s appearance, click on any chart object and then right-click to see a menu of properties that you and then right-click to see a menu of properties that you can change.can change.

• For example, right-click on the For example, right-click on the YY-axis scale and choose -axis scale and choose Format AxisFormat Axis..



• Be careful about over-embellishing your charts.Be careful about over-embellishing your charts.



Embellished bar chartEmbellished bar chart Over-embellished chartOver-embellished chart

• Excel offers many other types of specialized charts.Excel offers many other types of specialized charts.



Multiple bar chartMultiple bar chartArea (mountain) chartArea (mountain) chart

Data from http://peltiertech.com/Excel/ChartsHowTo/HowToBubble.html

• Other specialized Excel charts:Other specialized Excel charts:- - BubbleBubble chart displays three variables on a chart displays three variables on a 2-dimensional scatter plot. 2-dimensional scatter plot.

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10

Series1

- Note: bubble - Note: bubble size is size is proportional to proportional to third variable.third variable.



http://peltiertech.com/Excel/ChartsHowTo/HowToBubble.html

http://peltiertech.com/Excel/ChartsHowTo/HowToBubble.html

• Other specialized Excel charts:Other specialized Excel charts:

- - StockStock chart chart for high/lowfor high/low/close stock /close stock prices.prices.

Data fromhttp://finance.yahoo.com



http://finance.yahoo.com/

http://finance.yahoo.com/

• Other specialized Excel charts:

- Radar (or Spider) chart compares individual performance against abenchmark.

- Caution, data may be distorted by emphasized areas.



• Other specialized Excel charts:Other specialized Excel charts:

- Use - Use floating floating barbar charts to charts to show a range show a range of data.of data.



bcor 1020 business statistics lecture 3 – january 24, 2008

Documents

data set

data range

sort data

univariate data

summarizing data

data integer

n data values

middle data values