Wk2 DQ - Data Analysis & Business Intelligence

Lind_18e_Chap004_PPT.pptx

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Describing Data: Displaying and Exploring Data

Chapter 4

4-1

This chapter continues our study of descriptive statistics with dot plots, stem-and-leaf displays, percentiles, and box plots. These charts and statistics will give us additional insight about where the data is concentrated as well as the general shape of the data. Later, bivariate data is examined. Studying this chapter will give us experience with business analytics.

Learning Objectives

LO4-1 Construct and interpret a dot plot

LO4-2 Identify and compute measures of position

LO4-3 Construct and analyze a box plot

LO4-4 Compute and interpret the coefficient of skewness

LO4-5 Create and interpret a scatter diagram

LO4-6 Compute and interpret the correlation coefficient

LO4-7 Develop and explain a contingency table

4-2

Dot Plots Example

Use dot plots to compare the two data sets like these of the number of vehicles serviced last month for two different dealerships

4-3

Dot plots are useful for comparing two different data sets. Here are tables with the number of vehicles serviced by two of the dealerships owned by the Applewood Auto Group. We will use dot plots to show the difference in location and dispersion of the observations. To develop a dot plot, we display a dot for each observation along a horizontal number line. If there are identical values, the dots are piled on top of each other.

Dot Plots Example (2 of 2)

Minitab provides dot plots and summary statistics

4-4

These dot plots of data from two dealerships owned by the Applewood Auto Group show the difference in location and dispersion of the observations. We can clearly see the number of vehicles serviced at the Sheffield dealership is more widely dispersed and has a larger mean than at the Tionesta dealership. Notice the identities of the individual values have not been lost.

Measures of Position

Measures of location also describe the shape of the distribution and can be expressed as percentiles

Quartiles divide a set of observations into four equal parts

The interquartile range is the difference between the third quartile and the first quartile

Deciles divide a set of observations into 10 equal parts

Percentiles divide a set of observations into 100 equal parts

4-5

The standard deviation is the most widely used measure of dispersion, but another method of describing dispersion is with measures of location. On this slide is the formula that can be used to find any percentile where p represents percentile and n is the number of values in the data set. To find the first quartile, Q1, use 25 as the percentile, and to find the third quartile, use 75. Use this formula to find the median, or Q2, using 50 as the percentile. The data must be ordered from smallest to largest.

Quartiles, deciles and percentiles are commonly used measures of location. 25% of the data is less than the 1st quartile, 50% of the data is less than the 2nd quartile, and 75% of the data is less than the 3rd quartile. The interquartile range covers the middle 50% of the data.

Measures of Position Example

Morgan Stanley is an investment company with offices located throughout the United States. Listed below are the commissions earned last month by a sample of 15 brokers

First, sort the data from smallest to largest

4-6

Measures of Position Example (2 of 2)

Next, find the median

L50 = (15+1)*50/100 = 8

So the median is $2,038, the value at position 8

$1,460	$1,471	$1,637	$1,721	$1,758	$1,787	$1,940	$2,038
2,047	2,054	2,097	2,205	2,287	2,311	2,406

4-7

The median is the same as the 50th percentile.

The first quartile, Q1, L25 = (15+1)*25/100 = 4 so Q1=$1,721, the 4th value.

The third quartile, Q3 is the 75th percentile, L75 = (15+1)*75/100 = 12, so Q3 is $2,205, the 12th value.

Box Plots

The interquartile range is Q3 – Q1

Outliers are values that are inconsistent with the rest of the data and are identified with asterisks in box plots

4-8

BOX PLOT A graphic display that shows the general shape of a variable’s distribution. It is based on five descriptive statistics: the maximum and minimum values, the first and third quartiles, and the median.

Begin by drawing a number line that will accommodate the minimum and maximum values. Then draw a vertical line above the median, Q1, and Q3. Enclose the regions between the first and third quartile, creating the box. Next, draw dotted line segments from the third quartile to the largest value and from the first quartile to the smallest value. This shows the largest 25% and smallest 25% respectively. The interquartile range is the middle 50% of the data.

Box Plot Example

Alexander’s Pizza offers free delivery of its pizza within 15 miles. How long does a typical delivery take? Within what range will most deliveries be completed?

Using a sample of 20 deliveries, Alexander determined the following:

Minimum value = 13 minutes

Q1 = 15 minutes

Median = 18 minutes

Q3 = 22 minutes

Maximum value = 30 minutes

Develop a box plot for delivery times

4-9

Box Plot Example Continued

Begin by drawing a number line using an appropriate scale

Next, draw a box that begins at Q1 (15 minutes) and ends at Q3 (22 minutes)

Draw a vertical line at the median (18 minutes)

Extend a horizontal line out from Q3 to the maximum value (30 minutes) and out from Q1 to the minimum value (13 minutes)

4-10

The box plot reveals the data is positively skewed since the dashed line to the right of the box (from 22 minutes to 30 minutes) is longer than the dashed line to the left of the box (from 15 minutes to 13 minutes) and since the median is not in the center of the box.

Common Shapes of Data

4-11

There are four common shapes of data, as we see here. Symmetric when mean = median. Positively skewed or skewed to the right when mean > median. Negatively skewed or left skewed when the data values extend further to the left and the mean < median. A bimodal shape has two or more peaks and may indicate two or more populations have been combined.

Skewness

The coefficient of skewness is a measure of the symmetry of a distribution

Two formulas for coefficient of skewness

4-12

Professor Karl Pearson developed the simplest formula for calculating skewness, which is based on the difference between the mean and the median.

The textbook also shows the software method of calculating skewness that is based on the cubed deviations from the mean. See Formula 4-3 for more information.

Skewness (2 of 2)

The coefficient of skewness can range from -3 to +3

A value near -3 indicates considerable negative skewness

A value of 1.63 indicates moderate positive skewness

A value of 0 means the distribution is symmetrical

4-13

Professor Karl Pearson developed the simplest formula for calculating skewness, which is based on the difference between the mean and the median.

The textbook also shows the software method of calculating skewness that is based on the cubed deviations from the mean. See Formula 4-3 for more information.

Skewness Example

Following are the earnings per share for a sample of 15 software companies for the year 2018. The earnings per share are arranged from smallest to largest.

Begin by finding the mean, median, and standard deviation. Find the coefficient of skewness.

What do you conclude about the shape of the distribution?

4-14

Skewness Example (2 of 2)

What do you conclude about the shape of the distribution?

4-15

The distribution is moderately positively skewed.

Describing the Relationship Between Two Variables

Both variables are measured with interval or ratio level scale

If the scatter of points moves from the lower left to the upper right, the variables under consideration are directly or positively related

If the scatter of points moves from the upper left to the lower right, the variables are inversely or negatively related

4-16

SCATTER DIAGRAM Graphical technique used to show the relationship between two variables measured with interval or ratio scales.

When studying the relationship between two variables the data is referred to as bivariate. When studying just one variable, the data is univariate.

Scatter Diagrams

4-17

To graph a scatter diagram scale, put one of the variables, the independent variable, on the horizontal axis, and the dependent variable on the vertical axis. In the first graph, there is a positive relationship between the age of the buses and their maintenance cost: as buses increases in age, the maintenance cost increases. The middle graph displays little or no relationship between Batting Average and Home Runs. The graph on the right shows a negative relationship between Days on the Market and the Price of a home, that is, as the Days on the Market increases the Price of the home decreases.

Correlation Coefficient

A statistic called the correlation coefficient can be calculated to measure the direction and strength of the relationship between two variables

Can range from -1.0 to +1.0

The closer the coefficient is to −1.0 or +1.0, the stronger the relationship

If r is close to 0.0, we can say that there is no relationship between the variables

4-18

Correlation Coefficient (2 of 2)

4-19

Based on the value of the correlation coefficient, it can complement the interpretation of scatter diagrams. For example, if r = −1.0 the relationship between the two variables is a perfectly negative; if r = +1.0 the relationship is perfectly positive.

Contingency Tables

A contingency table is used to classify nominal scale observations according to two characteristics

It is a cross-tabulation that simultaneously summarizes two variables of interest

Both variables need only be nominal or ordinal

4-20

CONTINGENCY TABLE A table used to classify observations according to two identifiable characteristics.

If your data is interval or ratio level, it needs to be converted to nominal or ordinal.

Contingency Table Example

Applewood Auto Group’s profit comparison

90 of the 180 cars sold had a profit above the median and half below. This meets the definition of median.

The percentage of profits above the median are Kane 48%, Olean 50%, Sheffield 42% , and Tionesta 60%.

4-21

Compute the median profit for all sales last month and then classify profit from sales data as being above or below the median.

Chapter 4 Practice Problems

4-22

Question 3

4-23

Consider the following chart.

What is this chart called?

How many observations are in the study?

What are the maximum and the minimum values?

Around what values do the observations tend to cluster?

LO4-1

Question 7

4-24

The Thomas Supply Company Inc. is a distributor of gas-powered generators. As with any business, the length of time customers take to pay their invoices is important. Listed below, arranged from smallest to largest, is the time, in days, for a sample of the Thomas Supply Company Inc. invoices.

Determine the first and third quartiles.

Determine the second decile and the eighth decile.

Determine the 67th percentile.

LO4-2

Question 9

4-25

The box plot below shows the amount spent for books and supplies per year by students at four-year public colleges.

Estimate the median amount spent.

Estimate the first and third quartiles for the amount spent.

Estimate the interquartile range for the amount spent.

Beyond what point is a value considered an outlier?

Identify any outliers and estimate their values.

Is the distribution symmetrical or positively or negatively skewed?

LO4-3

Question 15

4-26

Listed below are the commissions earned ($000) last year by the 15 sales representatives at Furniture Patch Inc.

Determine the mean, median, and the standard deviation.

Determine the coefficient of skewness using Pearson’s method.

Determine the coefficient of skewness using the software method.

LO4-4

Question 17

4-27

Create a scatter diagram and compute a correlation coefficient. How would you describe the relationship between the values?

LO4-5,6

Question 19

4-28

The Director of Planning for Devine Dining Inc. wishes to study the relationship between the time of day a customer dined and whether the guest orders dessert. To investigate the relationship, the manager collected the following information on 200 recent customers.

What is the level of measurement of the two variables?

What is the above table called?

Does the data suggest that customer are more likely to order dessert? Explain why.

Does the data suggest that customers at lunch time are more likely to order dessert? Explain why.

Does the data suggest that customers at dinner time are more likely to order dessert? Explain why.

LO4-7

L25 = (15+1) 25

100 = 4 L75 = (15+1)

75 100

=12

Therefore, the first and third quartiles are located at the 4th and 12th positions, respectively: L25 = $1, 721;L75 = $2, 205

=(15+1)

100

=4 L

=(15+1)

100

=12

Therefore, the first and third quartiles are located at the 4th and 12th

positions, respectively: L

=$1,721;L

=$2,205

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