GRAPHICAL DESCRIPTIVE GRAPHICAL DESCRIPTIVE STATISTICS FOR STATISTICS FOR

QUANTITATIVE DATAQUANTITATIVE DATA

Graphical TechniquesQuantitative Data

• HistogramsHistograms– Consider

• Boundaries of measurement classesmeasurement classes– Choose values that are easy to read/understand– Number of classes will “fall out”

• Where to place data occurring at “break points”“break points”– Excel Excel – in the lower interval

• Inclusion of Outliers?Outliers? – unrepresentative data far above or far below most of the rest of the data

– Seek reason for outliers

• Relative Frequency (Percentage) HistogramsRelative Frequency (Percentage) Histograms• Cumulative Relative Frequency OgivesCumulative Relative Frequency Ogives

Frequency Distributions

• Take a Survey of Incomes of 200 High School Graduates 2 Years After Graduation

• Results: $31,500, $26,900, …., $26,100• Create Class Intervals So Data Can Convey

Information– Not too many– Not too few– Include All Data (?) --- Outliers– Intervals of Equal Size

Results

Income Frequency Rel. Freq.

$15,000-$20,000 9 9/200 = .045

$20,000-$25,000 34 34/200 = .170

$25,000-$30,000 91 91/200 = .455

$30,000-$35,000 61 61/200 = .305

$35,000-$40,000 5 5/200 = .025

Histogram

Incomes

0

20

40

60

80

100

17500 22500 27500 32500 37500

INCOME

Fre

qu

en

cy

RELATIVE FREQUENCY HISTOGRAM

Incomes

0

0.1

0.2

0.3

0.4

0.5

17500 22500 27500 32500 37500

INCOME

Re

lati

ve

Fre

qu

en

cy

Same shape as histogram – different scale on y-axis

EXCELDATA ANALYSIS

• Go to Tools Menu – Select Data Analysis

• What If Data Analysis Isn’t There?– Go to Tools Menu

• Select Add-Ins

Check Analysis Tool Pak

Check Analysis Tool Pak-VBA

Click OK

EXCELHistograms

• Basic Approach:– Put Data in a Column– Create Bins (Measurement Classes)– Go To ToolsTools

Select Data Analysis Data Analysis

Select Histogram Histogram

Check Chart OutputChart Output

Data put into a column

CREATE BINS

2.First entry should be less than lowest

value – actually the lower bound of the lower bound of the first measurement classfirst measurement class – this allows us to begin the histogram at a value >0.

3.Enter the upper bound of upper bound of

the first measurement classthe first measurement class

4.Highlight the first two

entries and drag down to the upper bound of the last

measurement class

++

1.

Enter label for X-Axis

TOOLS/DATA ANALYSIS/HISTOGRAM

Go to ToolsSelect Data Analysis

Select Histogram

Histogram Dialogue Box

1. Enter cells containing data including label

2. Entercells containing bins including label

3. CheckLabels

5. Enterwhere you want the output

4. CheckChart Output

Resizing

Grab Lower Corner and

drag to resize

Result of Resizing

Click and Delete

Click and Rename

Click in Grey Area and Delete

Change Numbers to Midpoint ValuesDelete first entry (15000)

and last entry (More)

To close gap width:Right mouse click on a barSelect Format Data SeriesFormat Data Series

Select OptionsOptionsSet Gap WidthGap Width = 0

Resulting Histogram

Relative Frequency

• Proportion of Data in a Particular Class

• Divide Frequencies by 200 gives these results:

Income Frequency Relative Frequency $15,000-$20,000 9 .045 $20,000-$25,000 34 .170 $25,000-$30,000 91 .455 $30,000-$35,000 61 .305 $35,000-$40,000 5 .025 200 1.000

Relative Frequency Histogram

• Change the numbers on the y-axis to percentages

• Can manipulate Excel Histogram– Numbers on Y-axis appear in column B– Somewhere create a cell with the formula =B2/200

(Say in cell B12)– Drag down until all relative frequencies are shown– Highlight this new set of numbers and press COPY– Then PASTE SPECIAL (Values) these numbers

back into cell B2– Erase numbers in cells B12 and below– Change Name in cell B1 and on Y-Axis to Relative

Frequency

Creating Relative Frequencies

3. Put cursor in cell B2

1. Enter =B2/200

Drag to B18

Then highlight B12:B18

2. Select Copy

4. Go to Edit

Select Paste Special

5. Select Values

Creating Relative Frequencies

6. Change to Relative Frequency

7. Highlight and delete

Cumulative Relative Frequencies

• Give the proportion of values that are less than the upper boundary point of the class

• Cumulative frequency for first class is the relative frequency

• For subsequent classes cumulative frequency = relative frequency + cumulative frequency of previous class

Cumulative Relative Frequencies

Income Frequency Relative Frequency Cumulative $15,000-$20,000 9 .045 .045 $20,000-$25,000 34 .170 .215 $25,000-$30,000 91 .455 .670 $30,000-$35,000 61 .305 .975 $35,000-$40,000 5 .025 1.000 200 1.000

The same as the

relative frequency.670 =

.455 + .215

Ogives

• Line graph of cumulative relative frequencies– Begin with y-value = 0 at $15,000 and draw line

to .045 at $20,000– Draw line from .045 at $20,000 to .215 at $25,000– Draw line from .215 at $25,000 to .670 at $30,000– Draw line from .670 at $30,000 to .975 at $35,000– Draw line from .975 at $35,000 to 1 at $40,000– Draw line flat at 1 (to infinity)

Result

Using Ogives to Approximate Prob (Income < $27,500)

27500

0.44

EXCEL Ogives

Check both

2.

Delete Legend

4.

Change Names

1.

Resize

6.

Delete More

3.

Delete Background

5.

Right Mouse Click

On any bar --Delete

RESULT

Review

• Frequency Distributions

• Frequency Histograms

• Relative Frequency Distributions

• Relative Frequency Histograms

• Cumulative Relative Frequencies

• Cumulative Relative Frequency Ogives