© 2003 Prentice-Hall, Inc. Basic Business Statistics Chapter 3 Numerical Descriptive Measures Chapters Objectives: Learn about Measures of Center. How

© 2003 Prentice-Hall, Inc.

Basic Business Statistics Chapter 3

Numerical Descriptive Measures

Chapters Objectives:Learn about Measures of Center.

How to calculate mean, median and midrangeLearn about Measures of Spread

Learn how to calculate Standard Deviation, IQR and Range

Learn about 5 number summariesLearn how to create Box Plots


Chapter Topics

Measures of Central Tendency Mean, Median, Midrange, Geometric Mean

Quartile Measure of Variation

Range, Interquartile Range, Variance and Standard Deviation, Coefficient of Variation

Shape Symmetric, Skewed, Using Box-and-Whisker

Plots


Chapter Topics

The Empirical Rule and the Bienayme-Chebyshev Rule

Coefficient of Correlation Pitfalls in Numerical Descriptive

Measures and Ethical Issues

(continued)


Summary Measures

Central Tendency

MeanMedian

Mode

Quartile

Geometric Mean

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of Variation

Range


The Second Step of Data Analysis

We looked at displaying data using graphs. But numeric summaries help us to compare variables

and to talk about relationships among variables in precise (exact) ways.

After drawing the graph, it is usual to calculate summary values.

This Lesson considers a different numerical summaries.


Summaries of Center and Spread

Can you think of a way of estimating the Center of this histogram or How wide it is (Spread or Variance)

Measure of Center

Measureof Spread

Mean Standard Deviation

Media IQR

Midrange Range0

5

10

15

20

25

30

30-40 40-50 50-60 60-70 70-80 80-90 90-100

•The shape that appears over the histogram is called the normal distribution shape

•This shape is used to estimate the shape of the histogram


Finding the Central Value

We summarize the distribution with mean median and midrange

The mean median and midrange have different idea of what is the center, and behaves differently

The Mode is also used however this is not a good measure of the central value

Geometric Mean will not be covered in this course


Measures of Central Tendency

Central Tendency

Mean Median Midrange

Geometric Mean1

1

n

ii

N

ii

XX

n

X

N

1/

12

n

Gn XXXX


Mean (Arithmetic Mean)

Mean (Arithmetic Mean) of Data Values Sample mean

Population mean

1 1 2

n

ii n

XX X X

Xn n

1 1 2

N

ii N

XX X X

N N

Sample Size

Population Size


Mean (Arithmetic Mean)

The Most Common Measure of Central Tendency

Affected by Extreme Values (Outliers)

(continued)

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6


Approximating the Arithmetic Mean Used when raw data are not available

Mean (Arithmetic Mean)(continued)

1

sample size

number of classes in the frequency distribution

midpoint of the th class

frequencies of the th class

c

j jj

j

j

m f

Xn

n

c

m j

f j


How to compute the Mean?Step 1: Add all the data points. This is called the SUM or the Total

Step 2: Count the number of Data points. This is sometimes called the count

Step 3: Divide The Total by the Number of Data points

Calculate the mean for the following data points. 5,20,15,3,7

Revision: 5.1 Know how to compute the mean of a collection of data values.


Median Robust Measure of Central Tendency Not Affected by Extreme Values

In an Ordered Array, the Median is the ‘Middle’ Number If n or N is odd, the median is the middle

number If n or N is even, the median is the average of

the 2 middle numbers

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5


Example: Find median of the following numbers

14.1 3.2 25.3 2.8 -17.5 13.9 45.8

Step 1 Order the data1 2 3 4 5 6 7

-17.5 2.8 3.2 13.9 14.1 25.3 45.8

Step 2 There are 7 valuesn is oddStep 3 a Median is the middle number

Median is 13.9If we added 35.7 to the above numbers (data set) . What would the median now be?

14.1 3.2 25.3 2.8 -17.5 13.9 45.8 35.7

Step 3 b Median is average of the two middle numbers5 position 14.14 position 13.9

Median = '(13.9+14.1)/2= 14

1 2 3 4 5 6 7 8-17.5 2.8 3.2 13.9 14.1 25.3 35.7 45.8

Step 1 Order the data

n is even

Step 2 There are 8 values

Calculate the median for the following data points. 5,20,15,3,7

Calculate the median for the following data points. 5,20,15,3,7,10

MEDIAN

MEDIAN between 13.9 and 14.1


Mode

A Measure of Central Tendency Value that Occurs Most Often Not Affected by Extreme Values There May Not Be a Mode There May Be Several Modes Used for Either Numerical or Categorical

Data

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode


Why the mode is not really a measure of center

Often listed among measures of center. Is it really a central value?

1. The mode of a categorical variable is the category with the highest frequency.

It is not a center in because the categories of a categorical variable have no specific order

we can choose to place the modal category in a bar chart on the right, on the left, or in the middle..

2. When a continuous distribution is unimodal and symmetric, the single, central mode is often near to other measures of center.

3. Modes of data measured on quantitative variables are rarely useful as measures of center.

It is a coincidence when two quantitative measurements agree exactly,

counting such occurrences tells us little about the variable. As a consequence, statistics packages rarely compute or report modes.


Geometric Mean

Useful in the Measure of Rate of Change of a Variable Over Time

Geometric Mean Rate of Return Measures the status of an investment over

time

1/

1 2

n

G nX X X X

1/

1 21 1 1 1n

G nR R R R


Example

An investment of $100,000 declined to $50,000 at the end of year one and rebounded back to $100,000 at end of year two:

1 20.5 (or 50%) 1 (or 100% )R R

1/ 2

1/ 2 1/ 2

Average rate of return:

( 0.5) (1)0.25 (or 25%)

2Geometric rate of return:

1 0.5 1 1 1

0.5 2 1 1 1 0 (or 0%)

G

R

R


How to compute the Midrange

The range of the data is defined as the difference between the Maximum and Minimum (Range = Max – Min)

What would the Range be in the previous example?

What would the midrange be? (max + min)/2

DISADVANTAGE: - one extreme value can make the midrange very large, so it does not really represent the data.

If you deleted 45.8 and added 1000 to the previous example what would the midrange be?

63.3

14.15

491.25

Calculate the midrange and range for the following data points. 5,20,15,3,7


Summary Measures of center

Measures of center are the most commonly used (and misused) summaries of data.

If you understand how they differ and what they really say you can avoid misunderstanding data or being fooled by inappropriate summaries.

5.1 Check your knowledge of centers.


Quartiles Split Ordered Data into 4 Quarters

Position of i-th Quartile

Q1 and Q3 are Measures of Noncentral Location Q2= Median, a Measure of Central Tendency The calculation will be covered in the IQR slides.

25% 25% 25% 25%

1Q 2Q 3Q

1

4i

i nQ

Do not use this formulaDo not use this formula


Know the differing properties of the mean, median, and

midrange

The most common summary value describing a distribution of data values is some measure of a typical or central value.

This can be the mean, median or midrange

Revision 5.1 Know the differing properties of the mean, median, and midrange of a set of data values

You have calculate the 3 measures of center (mean, median and midrange) for the following data points. 5,20,15,3,7.

1. Change 15 to 19 and recalculate the mean, median and midrange.

Which measures of center have changed?

2. Change 20 to 200 and recalculate the mean, median and midrange.

Which measures of center have changed?

The median is not effected by outliers like the midrange and Mean

The mean is effected by very small changes to data


Why Measures Spread? Spread measures how often something varies. Imagine you go to a restaurant and received such very

good meal, you return for the same meal the next day. However, this time the food tastes very bad. You never

go back. The standard of quality has varied. The restaurant management needs to be able to control

the quality of food. A key aspect of management is to reduce variability,

and provide a consistent quality of service or product The first step to control variability is to measure it. Then you can work out what causes the variability Then work out how to reduce variability.

5.2 Compare ways to find the spread of a distribution

© 2003 Prentice-Hall, Inc. Days of Week

The following shows the number of Cakes made by a Bakery in Phnom Penh.It has two bakeries producing identical cakes

Day 1 Day 2 Day 3 Day 4 Day 5Center (Mean)

Spread (Std Dev)

Bakery A 550 750 800 750 500 670 135Bakery B 625 670 650 675 630 650 23Total 1175 1420 1450 1425 1130

The bakery has to produce at least 1200 cakes to meet daily customer demandThe bakery has to produce can not produce more than 1400 cakes because customers will not buy them and they will not be eaten

500

550

600

650

700

750

800

Day 1 Day 2 Day 3 Day 4 Day 5

No

Cak

es b

aked

Bakery A

Bakery B

Mean A

Mean B

Spread A is 135. Variance is very high

Spread B is 23

The two bakeries have been baking too many or too few cakes each day. Why?


Different Summaries of Spread

Summaries of Spread include Range Interquartile Range (IQR) and Standard Deviation

These summaries highlight different aspects of the distribution and have different values

But they all summarize how the values are spread out.

We have already seen that Range = Max - Min


Measures of Variation

Variation

Variance Standard Deviation Coefficient of Variation

PopulationVariance

Sample

Variance

PopulationStandardDeviationSample

Standard

Deviation

Range

Interquartile Range


Range

Measure of Variation Difference between the Largest and the

Smallest Observations:

Ignores How Data are Distributed

Largest SmallestRange X X

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5


Measure of Variation Also Known as Midspread

Spread in the middle 50% Difference between the First and Third

Quartiles

Not Affected by Extreme Values3 1Interquartile Range 17.5 12.5 5Q Q

Interquartile Range

Data in Ordered Array: 11 12 13 16 16 17 17 18 21

1317 4


5 Steps to calculate the Interquartile Range (IQR)

Step 1Put the data in Order Step 2 Divide the dataset in two. Dataset 1 and Dataset 2. NOTE: If n is odd the median should be included in both data sets Step 3Calculate Q1 (25th Percentile)

If n of Dataset 1 is odd, the Q1 is the middle number of Dataset A

If n of Dataset 1 is even, the median is the average of the 2 middle numbers

Step 4Calculate Q3 (75th percentile) As per step 3

Step 5 Find The Interquartile Range (IQR) IQR= Q3-Q1


5 Steps to Calculate Q1, Q3 and IQR when n is odd

In order to calculate IQR we need to to first of all calculate Q1 and Q3.We are going to calculate the Q1, Q3 and IQR for the following dataset: 7,8,25,11,13,5,6We require to know: Median (Q2 or 50th percentile), Q1 (25th Percentile), Q3 (75th percentile)

Step 1 Order Data

12 3 4 5 6 7

5 6 7 8 11 13 25


Step 2 Divide the dataset in twoDivide the above data set into Dataset 1 and Dataset 2.NOTE: If n is odd the median should be included in both data sets Dataset 1 is used to calculate Q1 and Dataset 2 is used to calculate Q3

Dataset 1 For Q1 Data Set 2 For Q3

1 2 3 4 1 2 3 4

5 6 7 8 8 11 13 25

Step 3 Calculate Q1 (25th Percentile)N is even so the Q1 is the average of the 2 middle numbers which is position 2 and 3Q1 = (6+7)/2 =6.5(This is the same method used to calculate the median for a even number of data points)

5 Steps Calculate Q1, Q3 and IQR when n is odd (Cont)


Step 4 Calculate Q3 (75th percentile)N is even for dataset 2, the median is the average of the 2 middle numbers which is position 2 and 3

Thus, Q3 = (11+13)/2 =12

Step 5 Find The Interquartile Range (IQR)IQR= Q3-Q1 = 12-6.5 =5.5

5 Steps to Calculate Q1, Q3 and IQR when n is odd(Cont)

1 2 3 4

8 11 13 25

Calculate the Q1, Q3 and IQR for the following data points. 5,20,15,3,7,16,4


5 Steps to Calculate Q1, Q3 and IQR when n is EVEN

When n is even we we use almost the same process to calculate Q1 Q3 and IQR. In order to calculate IQR we need to to first of all calculate Q1 and Q3.We are going to calculate the Q1, Q3 and IQR for the following dataset: 7,8,25,11,13,5,6 and 30We require to know: Q1 (25th Percentile), Q3 (75th

Step 1 Order Data

12 3 4 5 6 7 8

5 6 7 8 11 13 25 30


Step 2 Divide the dataset in twoDivide the above data set into Dataset 1 and Dataset 2.Dataset 1 is used to calculate Q1 and Dataset 2 is used to calculate Q3

Dataset 1 For Q1 Dataset 2 For Q3

1 2 3 4 1 2 3 4

5 6 7 8 11 13 25 30

Step 3 Calculate Q1 (25th percentile)N is even so the Q1 is the average of the 2 middle numbers which is position 2 and 3Q1 = (6+7)/2 =6.5(This is the same method used to calculate the median for a even number of data points)

5 Steps: Calculate Q1, Q3 and IQR when n is even (Cont)


Step 4 Calculate Q3 (75th percentile)N is even so the Q1 is the average of the 2 middle numbers which is position 2 and 3

Thus, Q3 = (13+25)/2 =19

Step 5 Find The Interquartile Range (IQR)IQR= Q3-Q1 =19-6.5=12.5

5 Steps to Calculate Q1, Q3 and IQR when n is even (Cont)

1 2 3 4

11 13 25 30

Calculate the Q1, Q3 and IQR for the following data points. 5,20,15,3,7,16,4,30


2

2 1

N

ii

X

N

Important Measure of Variation Shows Variation about the Mean

Sample Variance:

Population Variance:

2

2 1

1

n

ii

X XS

n

Variance


Standard Deviation Most Important Measure of Variation Shows Variation about the Mean Has the Same Units as the Original Data

Sample Standard Deviation:

Population Standard Deviation:

2

1

1

n

ii

X XS

n

2

1

N

ii

X

N


Approximating the Standard Deviation Used when the raw data are not available

and the only source of data is a frequency distribution

Standard Deviation(continued)

2

1

1sample size

number of classes in the frequency distribution

midpoint of the th class

frequencies of the th class

c

j jj

j

j

m X f

Sn

n

c

m j

f j


Standard Deviation

•The sum of the deviations from the mean, divided by the count minus 1 is the variance

•The square root of the variance is a measure of spread called the Standard Deviation

5.2 Know how to calculate the Standard Deviation


How to Calculate the Standard Deviation?

Step 1 Step 2 Step 3 Step 4

Count Data (y) Mean (y-Mean)1 3.4 33.35 -29.95 8972 20 33.35 -13.35 1783 63 33.35 29.65 8794 47 33.35 13.65 186

Total 133.4 2141 Step 5Mean = SUM/Count =133.4/4=33.35 33.35

Step 6 Total Step 5/( Count -1)=2141/(4-1) = 713.6

Step 7 Square root of 713.6= 26.7

Steps to Calc Standard Deviation1: Calculate the Mean =Sum/Count

Steps to Calc Standard Deviation1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table

Steps to Calc Standard Deviation1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table3: Take the each data point from the mean

Steps to Calc Standard Deviation1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table3: Take the each data point from the mean 4: Square the answer to Step 3 (Multiply each data point by each other)

Steps to Calc Standard Deviation1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table3: Take the each data point from the mean 4: Square the answer to Step 3 (Multiply each data point by each other) 5: Sum the all the data points6: Divide the total for Step 5 by (Count -1)7: Calculate the square root of Step 6

Steps to Calc Standard Deviation1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table3: Take the each data point from the mean 4: Square the answer to Step 3 (Multiply each data point by each other) 5: Sum the all the data points6: Divide the total for Step 5 by (Count -1)

Steps to Calculate Standard Deviation1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table 3: Take the each data point from the mean 4: Square the answer to Step 3 (Multiply each data point by each other) 5: Sum the all the data points

Calculate the Standard Deviation for the following data points. 5,20,15,3,7,16,4


Histograms to display measures of spread

When the distribution of data is unimodal and symmetric, the middle 38% of the distribution is about one standard deviation wide.

This rule of thumb provides a way to display the standard deviation, but it only applies to unimodal, symmetric distributions.

We can display common measures of spread in a Histogram.

5.2 Learn a way to display the spread in a Histogram.

5.2 Check your knowledge of measures of Spread


Comparing Standard Deviations

Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5 s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 s = 4.57

Data C


Coefficient of Variation

Measure of Relative Variation Always in Percentage (%) Shows Variation Relative to the Mean Used to Compare Two or More Sets of Data

Measured in Different Units

Sensitive to Outliers

100%S

CVX


Comparing Coefficientof Variation

Stock A: Average price last year = $50 Standard deviation = $2

Stock B: Average price last year = $100 Standard deviation = $5

Coefficient of Variation: Stock A:

Stock B:

$2100% 100% 4%

$50

SCV

X

$5100% 100% 5%

$100

SCV

X


Shape of a Distribution

Describe How Data are Distributed Measures of Shape

Symmetric or skewed

Mean = Median =Mode Mean < Median < Mode Mode < Median < Mean

Right-SkewedLeft-Skewed Symmetric


Describing Distribution:Shape, Center and Spread

Start by displaying the distribution of your data. You should describe: Symmetry vs. skewness * (Look at the tails)

Single vs. multiple modes

Possible outliers separated from the main body of the data.

... But remember that not all distributions have simple shapes.

Symmetry Skewness

Unimodel Bimodel multimodal Uniform

Outliers

4.3 Recognize the important attributes of distribution shape in a histogram or stem-and-leaf display.


Identifying Distribution Shape

When looking at the distribution of a quantitative variable, consider:

Symmetry, How many modes it has, Whether it has any outliers or gaps.

4.4 Practice describing distribution shapes


Summary: Describing a Distribution

You should always begin analyzing data by graphing the distributions of the variables.

We describe the shape of the distribution of a quantitative variable by noting whether it is symmetric or skewed, unimodal, bimodal, or multimodal.

Following the general rule that we first describe a pattern and then look for deviations from that pattern, we look at distributions to see if there are straggling outliers or gaps in the distribution shape.

Each of these descriptions is quite general and vague, so people may disagree on the "close calls." Nevertheless, we usually begin describing data by summarizing the distribution shapes.


Exploratory Data Analysis

Box-and-Whisker NOTE: Different from Basic Business Statistics Book!

Graphical display of data using 5-number summary

Median( )

4 6 8 10 12

Top of Whiskers

Bottom of Whiskers

1Q 3Q2Q


Distribution Shape & Box-and-Whisker

Right-SkewedLeft-Skewed Symmetric

1Q 1Q 1Q2Q 2Q 2Q3Q 3Q3Q


Box Plots

Whenever we have a five-number summary of a (QUANTITIVE) variable we can produce a graph (display) called a box plot

NOTE: Basic Business Statistics Book Does not Include Outliers!

1. High Outliers (Above Upper Fence)2. Top of Whiskers3. Q34. Median5. Q16. Bottom of Whiskers7. Low Outliers (Below Lower Fence)


Box Plot: Good way of Comparing Categories

Category 1

EG Males

Category 2

EG Females

Outliers

Whiskers do not extend further than 1.5 x IQR

5.4 Know how to make boxplots

Top of Whiskers

Bottom of Whiskers


5 Steps to Create a Box PlotDraw Box Plots for the dataset: 7,8,25,11,13,5,6We require to know: Median, Q1 (25% Percentile), Q3 (75% Percentile, Top of Whiskers, Bottom of Whiskers and Outliers

Step 1 Order Data

12 3 4 5 6 7

5 6 7 8 11 13 25

Step 2 Calculate MedianN is odd so data point at position is the middle value therefore Median = 8


5 Steps to Create a Box Plot (Cont).

Step 3 Calculate Q1 and Q3Divide the above data set into Data set 1 and Data Set 2.Because n is odd include the median in both setsData set 1 is used to calculate Q1 and Data set 1 is used to calculate Q3

Dataset 1 For Q1 Dataset 2 For Q3

1 2 3 4 1 2 3 4

5 6 7 8 8 11 13 25

For Q1: N for dataset 1 is even so Q1 is the average of the two middle at position 2 and 3

Q1 = (6+7)/2 =6.5(This is the same method used to calculate the median for a even number of data points)


5 Steps to Create a Box Plot (Cont).For Q3: N for dataset 2 is even so Q3 is the average of the two middle at positions 2 and 3Thus, Q3 = (11+13)/2 =12

Step 4 Find if there are any OutliersIQR= Q3-Q1 =5.5Any data points above Upper Fence (UF) are outliers? UF =Q3+1.5*IQR =12+1.5*5.5=20.25Any data points below Lower Fence (LF) are outliers? LF =Q1- 1.5*IQR = 6.5-1.5*5.5=-1.75

1 2 3 4 5 6 7

5 6 7 8 11 13 25

OutliersOutliers


That 25 is the only data point that is an outlier.

1 2 3 4 5 6 7

5 6 7 8 11 13 25

OutliersOutliers

Upper Fence (UF) =20.25Lower Fence (LF) =-1.75

Bottom of Whisker is 5

Top of Whisker of is 13


Step 5: Create the Box Plot (Output from SPSS)

7N =

VAR00001

30

25

20

15

10

5

0

3

1 2 3 4 5 6 7

5 6 7 8 11 13 25Outlier

Outlier = 25

Top of Whisker = 13Bottom

of Whisker

= 5


Examples

Calculate box plots for the following data points.

1. 5,20,15,3,7,16,4

2. 49,22,22,23,25,2

3. 4,11,10,12,15

4. 4,11,11,9,10,12,18


Box Plot can be used to compare two categories. Exam Results by

Class

MA1 MA2 MBA8E MBA8W MDM6E MDM6W

Batch

20

40

60

80

100

Pe

rc

en

ta

ge

PHON PHEUY PHEUY

PRUM CHANVIBOL

MENG VICHEKA

Serey ManopCopied


When comparing groups with boxplots

Compare the medians. Which group has a larger center? Compare the IQR's. Which group is more spread out? Compare the difference between the medians to the IQR's. On the scale of the IQR's are the medians very different? Check for possible outliers. Identify them if you can. Does

knowing which values are outliers tell you anything about the data, the groups, or how they compare?

Numeric comparisons of groups often consider the difference between the centers of the groups.

This difference should be compared to the size of the spreads of the groups.

Data such as the rat lifetimes data that have outliers are better summarized with medians and quartiles than with means and standard deviations.


Box plots explore the relationships among several

groups.

Easy to focus on center spread and symmetry

Look for patterns of increasing spread and symmetry

Lookout for outliers


Examples

Do examples…..


The Empirical Rule

For Most Data Sets, Roughly 68% of the Observations Fall Within 1 Standard Deviation Around the Mean

Roughly 95% of the Observations Fall Within 2 Standard Deviations Around the Mean

Roughly 99.7% of the Observations Fall Within 3 Standard Deviations Around the Mean


The Bienayme-Chebyshev Rule

The Percentage of Observations Contained Within Distances of k Standard Deviations Around the Mean Must Be at Least Applies regardless of the shape of the data set At least 75% of the observations must be

contained within distances of 2 standard deviations around the mean

At least 88.89% of the observations must be contained within distances of 3 standard deviations around the mean

At least 93.75% of the observations must be contained within distances of 4 standard deviations around the mean

21 1/ 100%k


Coefficient of Correlation

Measures the Strength of the Linear Relationship between 2 Quantitative Variables

1

2 2

1 1

n

i ii

n n

i ii i

X X Y Yr

X X Y Y


Features of Correlation Coefficient

Unit Free Ranges between –1 and 1 The Closer to –1, the Stronger the

Negative Linear Relationship The Closer to 1, the Stronger the

Positive Linear Relationship The Closer to 0, the Weaker Any Linear

Relationship


Scatter Plots of Data with Various Correlation

CoefficientsY

X

Y

X

Y

X

Y

X

Y

X

r = -1 r = -.6 r = 0

r = .6 r = 1


Scatter Plots Best way to see relationships between two Quantitative Variables If there is a relationship it is called an association. Scatter plots

are the best way to look for associations

Relationship of Class Attended and Statitstics Score

2025303540455055606570

4 6 8 10 12 14

Number Classes Attended

Sc

ore


Scatter Plots Only Quantitative Variables can be used The following is the relationship between the Mango Taste

Score and the Number of Weeks it grown for

Weeks Grown on Tree

Time taken to grow the best Mango

0

20

40

60

80

100

6 8 10 12 14 16 18

Ta

ste

Sc

ore


Direction: Positive

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6

Scatter Plots : DIRECTION and FORM

Negtive Direction

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6

•Direction: +Ve

•Form: Linear (Straight Line)

•Direction: +Ve

•Form: Linear (Straight Line)


Form describes the shape

Describe the Form if there is one. EG Straight, or arched or curved

A Curve

0

2

4

6

8

10

12

14

16

18

20

0 1 2 3 4 5 6

An Arch

0

20

40

60

80

100

6 8 10 12 14 16 18

Ta

ste

Sc

ore


Scatter: The following Scatter Plots have no scatter

0

50

100

150

200

250

0 5 10 15 20

0

5

10

15

20

25

30

0 2 4 6 8 10 12

•The third thing to look for is scatter.

•These graphs have no Scatter because the dots follow each other in a line.


Scatter: The following Scatter Plots have scatter

•These graphs have Scatter because the dots are like a cloud.

•Always look for something you do not expect to find

•One Graph has an outlier which are very important unexpected points that may occur

0

2

4

6

8

10

12

0 10 20 30 40 500

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60


How to Measure Scatter: Correlation Coefficient, r

I X Y X-Mean X Y-Mean X Product1 2 5 -4 -10 402 4 10 -2 -5 103 6 15 0 0 04 8 20 2 5 105 10 25 4 10 40

Total 30 75 100Mean 6 15Standard Deviation 3.162278 7.905694

Sx = 3.162278 Sy = 7.905694

1905694.7*162278.3*)15(

100

)1(

)(*)(

SxSyn

yyxxCorrCoef


Plot Chart and describe scatter plot. Calculate Correlation Coefficient

0102030

0 5 10 15X

Y

These data points go in straight line with no scatter in a positive direction (positively correlated)

1905694.7*162278.3*)15(

100

)1(

)(*)(

SxSyn

yyxxCorrCoef


Example 2

For the following data, plot a scatter plot and describe it. Calculate Correlation Coefficient

n X Y X-Mean X Y-Mean X Product1 4 22 -2 7.6 -15.22 5 23 -1 8.6 -8.63 6 12 0 -2.4 04 7 13 1 -1.4 -1.45 8 2 2 -12.4 -24.8

Total 30 72 -50Mean 6 14.4Standard Deviation 1.581139 8.561542


Answer: Example 2

n X Y X-Mean X Y-Mean X Product1 4 22 -2 7.6 -15.22 5 23 -1 8.6 -8.63 6 12 0 -2.4 04 7 13 1 -1.4 -1.45 8 2 2 -12.4 -24.8

Total 30 72 -50Mean 6 14.4Standard Deviation 1.581139 8.561542

These data points go in straight line with some scatter in a negative direction (negatively correlated)

0

5

10

15

20

25

3 4 5 6 7 8 9

x

y

92.01214.9*5811.1*)15(

50

)1(

)(*)(

SxSyn

yyxxCorrCoef


Example 3

For the following data, plot a scatter plot and describe it. Calculate Correlation Coefficient

n X Y X-Mean X Y-Mean X Product1 1 5 -2 -10.2 20.42 2 21 -1 5.8 -5.83 3 25 0 9.8 04 4 19 1 3.8 3.85 5 6 2 -9.2 -18.4

Total 15 76 0Mean 3 15.2Standard Deviation 1.581139 9.121403


Answer: Example 3n X Y X-Mean X Y-Mean X Product1 1 5 -2 -10.2 20.42 2 21 -1 5.8 -5.83 3 25 0 9.8 04 4 19 1 3.8 3.85 5 6 2 -9.2 -18.4

Total 15 76 0Mean 3 15.2Standard Deviation 1.581139 9.121403

0102030

0 2 4 6X

Y

These data points are in an arched shape with a little scatter.

01214.9*5811.1*)15(

0

)1(

)(*)(

SxSyn

yyxxCorrCoef


Example 4 For the following data, plot a scatter

plot and describe it. Calculate Correlation Coefficient

n X Y X-Mean X Y-Mean X Product1 1 17 -4.4 5.8 -25.522 3 1 -2.4 -10.2 24.483 6 22 0.6 10.8 6.484 7 3 1.6 -8.2 -13.125 10 13 4.6 1.8 8.28

Total 27 56 0.6Mean 5.4 11.2Standard Deviation 3.507136 9.011104


Answer: Example 4

n X Y X-Mean X Y-Mean X Product1 1 17 -4.4 5.8 -25.522 3 1 -2.4 -10.2 24.483 6 22 0.6 10.8 6.484 7 3 1.6 -8.2 -13.125 10 13 4.6 1.8 8.28

Total 27 56 0.6Mean 5.4 11.2Standard Deviation 3.507136 9.011104

0102030

0 5 10 15X

Y

0.0059.011*3.5071*1)(5

6.0

1)SxSy(n

)y(y*)x(xCorrCoef

These data points have a lot of scatter and they are not correlated


Pitfalls in Numerical Descriptive Measures and

Ethical Issues Data Analysis is Objective

Should report the summary measures that best meet the assumptions about the data set

Data Interpretation is Subjective Should be done in a fair, neutral and clear manner

Ethical Issues Should document both good and bad results Presentation should be fair, objective and neutral Should not use inappropriate summary measures to

distort the facts


Chapter Summary

Described Measures of Central Tendency Mean, Median, Mode, Geometric Mean

Discussed Quartiles Described Measures of Variation

Range, Interquartile Range, Variance and Standard Deviation, Coefficient of Variation

Illustrated Shape of Distribution Symmetric, Skewed, Using Box-and-Whisker

Plots


Chapter Summary

Described the Empirical Rule and the Bienayme-Chebyshev Rule

Discussed Correlation Coefficient Addressed Pitfalls in Numerical

Descriptive Measures and Ethical Issues

(continued)

Documents

© 2003 Prentice-Hall, Inc. Basic Business Statistics Chapter 3 Numerical Descriptive Measures Chapters Objectives: Learn about Measures of Center. How