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Category: $mathematics$ mathematics

Statistics for business and economics. Chapter

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CHAPTER 3
STATISTICS FOR BUSINESS AND ECONOMICS 13e Anderson,
David R 2017

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Descriptive Statistics
MEASURE OF
LOCATION
MEASURES OF
VARIABILITY
MEASURES OF
DISTRIBUTION
SUMMARIES
AND BOX PLOTS
MEASURES OF
ASSOCIATION

Statistics in Practice
■ Toy and accessory company, that designs and imports products for infants
– Product lines: teddy bears, mobiles, musical toys, rattles and security blankets
– Using high quality color, texture and sound
■ Uses independent representative to sell products. Represented in more than 1000
retail outlets in US
Biggest Challenge, Cash Flow Management

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Statistics in Practice
■ Company sets the following goals
– The average age for outstanding invoices should not exceed 45 days
– And the dollar value of invoices more than 60 days old should not exceed 5%
of the dollar value of all accounts receivable
■ Company can use Descriptive Statistics tools to achieve its goals
– Mean
40 days
– Median
35 Days
– Mode
31 days
■ What do these results mean?

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Statistics in Practice
■ Requirement 1, outstanding invoices should not exceed 45 days
– Average age of an invoice is 40 days
– Median shows, that half of the invoices remain outstanding 35 days or more
– Mode of 31 days indicates, that most common length of time an invoice is
outstanding is 31 days
■ Statistical summary also showed that only 3% of the dollar value of all accounts
receivable was more than 60 days old (barrier - 5%)
■ In this chapter we will learn how to compute and interpret some of the statistical
measured used in this example

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1. Measure of Location
Mean
■ Average value for a variable

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1. Measure of Location
Mean
■ Example

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1. Measure of Location
Mean
■ Let’s visualize

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1. Measure of Location
Mean
■ Problem – mean is highly effected by extreme values
■ For example, lets replace 54 by 114. new mean is 56. What is the problem?

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1. Measure of Location
Mean
■ Let’s do another example
■ Mean is $3940
■ However, do every data points have same “weight”?

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1. Measure of Location
Weighted Mean
■ Giving each observation a weight that reflects its relative importance
■ Computed as follows

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1. Measure of Location
Weighted Mean
■ Example
■ What is the average price of meat?

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1. Measure of Location
Median
■ Value in the middle when the data are arranged in ascending order
■ With an odd number of observations, median is middle value
■ With even number of observations, average of two middle observations
■ For example
■ Or

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1. Measure of Location
Geometric Mean
■ Measure of location that is calculated by finding the nth root of the product of n
values
■ Often used in analyzing growth rates in financial data. Where arithmetic mean or
average value will provide misleading results

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1. Measure of Location
Geometric Mean
■ Example
■ If we start with $100
■ End of 1 year our balance is $77.90
■ Beginning of 2 year, starting balance is $77.90. If growth rate is 28.7%. Then ending
balance of year 2 is $100.26
■ Etc

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1. Measure of Location
Geometric Mean
■ example
■ At the end of year 10 ending balance can be calculated by using Growth Factor:
■ Therefore growth factor is 1,334493
■ Geometric mean is:

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1. Measure of Location
Geometric Mean
■ Example
■ Using geometric mean, at the end of year 10, average growth rate is 2,93%
■ However, if we use arithmetic mean, growth rate is 5.04%. Using this result can be
misleading
■ If average growth rate is 5,04%, at the end of year 10 our initial investment of 100$
would have been $163.51. When we have calculated that it is $133,45

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1. Measure of Location
Mode
■ It is the value that occurs with greatest frequency
■ Provide information about how the data is spread over the interval from the smallest
value to the largest
■ Our previous example
■ Mode is $3880

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1. Measure of Location
Percentile
■ Provides information about how the data are spread over the interval from the
smallest value to the largest value
■ Pth percentile divides the data into two parts: approximately p% of the observations
are less than the pth percentile, and approximately (100-p)% of the observations are
greater than the pth percentile
■ For example the test results (SAT)
■ Student received 630 points. Is it a bad or good result?
■ We can't say anything if we don’t know other results to compare it to
■ If we know 630 points corresponds to the 82nd percentile, we know that
approximately 82% of the applicants scored lower than this individual, and
approximately 18% of the applicants scored higher than this individual

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1. Measure of Location
Percentile
■ Formula
■ If we want to calculate 80th percentile for the data provided below

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1. Measure of Location
Percentile
■ 80the percentile would be
■ 10th position value is 4050. plus .4 or 40% of the value to the next one. That is:

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1. Measure of Location
Quartiles
■ It is often desired to divide a data in set of four parts, each containing 25% of the
data
■ These division points are referred to as the quartiles
– Q1 = first quartile, or 25th percentile
– Q2 = second quartile, or 50th percentile (also the median)
– Q3 = third quartile, or 75th percentile
■ Example:

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2. Measurement of Variability
■ In addition to measures of location, we need to take into account the measures of
variability or dispersion
■ Let’s assume that we have two different suppliers
■ After several months of operation, we find out that mean number of daus required
to fill orders is 10 days for bot of suppliers
■ Which supplier is better?
■ Let’s look at histogram

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2. Measurement of Variability
■ What can we say now?

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2. Measurement of Variability
■ We need to measure the variability

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2. Measurement of Variability
Range
■ Simplest measure of variability
■ Let’s refer to our data on starting salaries
– Largest starting salary is 4325
– Smallest one is 3710
– Range 4325-3710 = 615

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2. Measurement of Variability
Interquartile Range
■ A measure of variability that overcomes the dependency on extreme values
■ Interquartile range is the range for the middle 50% of the data
■ For our data
– Q3 is 4000
– Q1 is 3865
– IQR is 135

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2. Measurement of Variability
Variance
■ A measure of variability that utilizes all the data
■ Measure by the difference between the value of each observation and the mean
■ Population
■ Sample

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2. Measurement of Variability
Variance
■ Example
■ Variance is

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2. Measurement of Variability
Standard Variance
■ Positive square root of the variance
■ In our case it is:

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2. Measurement of Variability
Coefficient of Variation
■ Indicates how large the standard deviation is relative to the mean
■ Formula
■ In our example
– (165,65/3940)*100 = 4,2%
■ Therefore sample standard deviation is 4.2% of the value of the sample mean

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3. Measures of Distribution Shape, Relative
Location and Detecting Outliers
■ Aside from the measures of
location and variability for the data,
we also care for a measure of the
shape of a distribution
■ An important numerical measure of
the shape of the distribution is
called skewness

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3. Measures of Distribution Shape, Relative
Location and Detecting Outliers
Distribution Shape
■ Formula itself is complicated
■ But histogram helps us to inspect it
visually

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3. Measures of Distribution Shape, Relative
Location and Detecting Outliers
Distribution Shape
■ For e symmetric distribution, the mean
and the median are equal
■ When it is skewed positively, the mean
will usually be greater than the median
■ When negative, the mean will usually
be less than the median
■ Panel D, the mean is $77.60 and the
median is $59.70

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3. Measures of Distribution Shape, Relative
Location and Detecting Outliers
Z-scores
■ In addition to measures of location, variability and the shape, we are also interested
in the relative location of values within a data set
■ It will help us determine how far a particular value is from the mean
■ Using both mean and standard deviation, we can determine the relative location of
any observation
■ Formula:

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3. Measures of Distribution Shape, Relative
Location and Detecting Outliers
Z-scores
■ Example
■ Z-score of 1.2 means, indicates, that xi is 1.2 standard deviations greater than the
sample mean

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3. Measures of Distribution Shape, Relative
Location and Detecting Outliers
Chebyshev’s Theorem
■ Enables us to make statements about the proportion of data values that must be within
a specified number of standard deviations of the mean
■ Formula:
■ Implications being:
– At least 75% of data values must be within z=2 standard deviations of the mean
– At least 89% of data values must be within z=3 standard deviations of the mean
– At least 94% of data values must be within z=4 standard deviations of the mean

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3. Measures of Distribution Shape, Relative
Location and Detecting Outliers
Chebyshev’s Theorem
■ Example
■ Test scores for 100 sudents indicate, that mean is 70 points and standard deviation
of 5
■ What portion of students should be between 60-80 points?
■ Answer:
– 60 is 2 standard deviations from the mean
– Therefore 75%

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3. Measures of Distribution Shape, Relative
Location and Detecting Outliers
Empirical Rule
■ Chebyshev’s theorem can be applied to any data set regardless of the shape of the
distribution
■ However, when data are believed to be approximate to normal or bell-shaped
distribution, then empirical rule can be used to determine the percentage of data
values that must be within a specified number of standard deviations of the mean

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3. დისტრიბუციის გაზომვა
ემპირიული კანონი

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3. Measures of Distribution Shape, Relative
Location and Detecting Outliers
Detecting Outliers
■ Sometimes a data set will have one or more observations with unusually large or
unusually small values
■ We need to identify outliers and review them carefully
■ Outlier may be
– Value that has been incorrectly recorded
– Incorrectly included in the data set
– It may be recoded correctly and may belong to the data set, but may be
unusual data value
■ Standardized values (z-scores) can be used to identify outliers

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3. Measures of Distribution Shape, Relative
Location and Detecting Outliers
Detecting Outliers
■ Using z-scores
■ It is recommended to treat a value as outlier if it’s z-score is less than -3 or greater
than +3
■ Why?
■ Another method is based upon the values of first and third quartiles and the
interquartile range (IQR)

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4. Five-Number Summaries and Box
Plots
■ Based on summary statistics
■ We will use
– Five-number summary
– And box plots

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4. Five-Number Summaries and Box
Plots
Five-Number Summary
■ We need:
– Smallest value
– First quartile (Q1)
– Median (Q2)
– Third quartile (Q3)
– Largest value

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4. Five-Number Summaries and Box
Plots
Five-Number Summary
■ Example
■ We know that: Q1=3857, Q2=3905, Q3=4025
■ Therefore five-number summary results are:
■ Starting salary minimum is 3710 and largest one is 4325 შორის
■ Median, or the middle value is 3905
■ Q1 and Q3 show that approximately 50% of the starting salaries are between 3857,5
and 4025

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4. Five-Number Summaries and Box
Plots
Box Plot
■ Graphical representation of five-number summary

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4. Five-Number Summaries and Box
Plots
Box Plot
■ Example

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5. Measures of Association between
Two Variables
■ Thus far we have examined numerical methods used to summarize the data for one
variable at a time
■ Often we need to make a decision based on the relationship between two variables

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5. Measures of Association between
Two Variables
Covariance
■ Descriptive measure of the linear association between two variables
■ Let’s look at the relationship between TV commercials and sales at the store the
following week

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5. Measures of Association between
Two Variables
Covariance

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5. Measures of Association between
Two Variables
Covariance
■ Let’s look at scatter diagram
■ It is divided into four quadrants
■ I quadrants, corresponds xi greater
than xbar and yi greater than ybar
(positive covariance)
■ II, xi less than xbar and yi less than
ybar (negative covariance)
■ etc

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5. Measures of Association between
Two Variables
Covariance

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5. Measures of Association between
Two Variables
Correlation Coefficient
■ If all points in a data set fall on a positively sloped straight line, the value of the
sample correction coefficient is +1
■ If all points in a data set fall on a negatively sloped straight line, the value of the
sample correction coefficient is -1