Office: E409 Tel: 3620623(office), 3620620(T.A.) Statistics for Business Instructor: Dr. Peng Xiaoling T.A. : Miss Li Jianxia Email: [email protected].

Office: E409

Tel: 3620623(office), 3620620(T.A.)

Statistics for Business

Instructor: Dr. Peng Xiaoling T.A. : Miss Li Jianxia

Email: [email protected] (Instructor) [email protected] (T.A.) Website: www.uic.edu.hk/~xlpeng

What is Statistics?What is Statistics?

Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting

numerical data to assist in making more effective decisions.

Statistics is very useful to your future career

Government officials use Government officials use conclusions drawn from the latest conclusions drawn from the latest data on unemployment and data on unemployment and inflation to make policy decisions.inflation to make policy decisions.

Financial planners use recent Financial planners use recent trends in stock market prices to trends in stock market prices to make investment decisions.make investment decisions.

General Information

(Textbook) Bowerman and O’Connell, Business Statistics in Practice, McGraw-Hill.

(Software) SPSS 17.0

(Assessment grade system)

A and A- (Not more than 10%) ,

A, A-, B+, B, B- (Not more than 65%).

Assessment

~ Continuous Assessment (50%)~ Homework (30%) Six or more sets of home works and several quizzes on class

~ Mid-term test (20%)

~ Final examination (50%)

Outline of the course

Introduction to Statistics Descriptive Statistics Probability and Random Variables Sampling Distributions Confidence Interval and Hypothesis Testing Statistical Inferences Based on Two Samples Some Optional Topics (Linear Regression)

Chapter 1An introduction to Business Statistics

Populations and Samples Populations and Samples

Sampling a Population of Existing UnitsSampling a Population of Existing Units

Sampling a ProcessSampling a Process

An Introduction to Survey SamplingAn Introduction to Survey Sampling

Populations and Samples

Population A set of existing units (people, objects, or events)

Sample A selected subset of the units of a population

Examples of Population:

All UIC graduates.All UIC graduates. All Lincoln Town Cars that were produced last year.All Lincoln Town Cars that were produced last year.

Population Size, N

Sample Size, n

Population vs. Sample

a b c d

e f g h i j k l m n

o p q r s t u v w

x y z

Population Sample

b c

g i n

o r u

y

Measures used to describe a population are called variables.

Measures computed from sample data are called statistics.

An examination of the entire population of An examination of the entire population of measurements.measurements.

Census(Census( 普查普查 ))

Note: Census usually too expensive, too time consuming, and Note: Census usually too expensive, too time consuming, and too much effort for a large population.too much effort for a large population.

A selected subset of the units of a population.A selected subset of the units of a population.SampleSample

Population Sample

For example, a university graduated 8,742 studentsFor example, a university graduated 8,742 studentsa.a. This is too large for a census.This is too large for a census.b.b. So, we select a sample of these graduates and learn So, we select a sample of these graduates and learn

their annual starting salaries.their annual starting salaries.

Questions:What’s the population & sample?

A pollster asks 30 adults at the mall about their shopping preferences.

Fox News does a poll, and reports the opinions of the 2500 people who called in.

Researchers test out a new cancer drug on 100 men with lung cancer.

Chapter 1 Introduction

A measurable characteristic of the population.

VariableVariable

We carry out a We carry out a measurementmeasurement to assign a to assign a valuevalue of a of a variable to each population unit. variable to each population unit.

Type of variables: Quantitative (numerical) Qualitative (Categorical)

The variable is said to be The variable is said to be quantitative or quantitative or numerical::Measurements that represent quantities (for example,Measurements that represent quantities (for example,““how much” or “how many”). For example, how much” or “how many”). For example, annualannualstarting salary starting salary is quantitative, is quantitative, age and number of age and number of childrenchildren is also quantitative is also quantitative

Number of children in a family

Balance in your bank account

Minutes remaining in class

The variable is said to be The variable is said to be qualitativequalitative or or categoricalcategorical: A : A descriptive category to which a population unit belongs. descriptive category to which a population unit belongs. For example, For example, a person’s gender a person’s gender and and whether a person whether a person who purchases a product is satisfied with the productwho purchases a product is satisfied with the product are are qualitative. qualitative.

G ender E yeC olor

T ype of carState of

B irth


NominativeNominative:: Identifier or nameIdentifier or name Unranked categorizationUnranked categorization

Example: gender, car colorExample: gender, car color Ordinal (can be compared)Ordinal (can be compared)::

All characteristics of nominative plus…All characteristics of nominative plus… Rank-order categoriesRank-order categories Ranks are relative to each otherRanks are relative to each other

Example: Low (1), moderate (2), or high (3) Example: Low (1), moderate (2), or high (3) riskrisk

There are two types of qualitative variables: There are two types of qualitative variables:

Descriptive Statistics Descriptive Statistics Vs.Vs. Inferential Statistics Inferential Statistics

Types of Statistics


For example, for a set of annual starting salaries, we For example, for a set of annual starting salaries, we want to know:want to know:

How much to expectHow much to expectWhat is a high versus low salaryWhat is a high versus low salaryHow much the salaries differ from each otherHow much the salaries differ from each other

If the population is small enough, could take a census If the population is small enough, could take a census and not have to sample and make any statistical inferencesand not have to sample and make any statistical inferences But if the population is too large, then ……….But if the population is too large, then ……….

The science of describing the important aspects of a set of The science of describing the important aspects of a set of measurementsmeasurements

Descriptive statisticsDescriptive statistics

Descriptive Statistics

Collect datae.g. Survey

Present datae.g. Tables and graphs

Characterize datae.g. Sample mean =

iX

n


Statistical InferenceStatistical InferenceThe science of using a sample of measurements to The science of using a sample of measurements to make generalizations about the important aspects of make generalizations about the important aspects of a population of measurements.a population of measurements. For example, use a sample of starting salaries to For example, use a sample of starting salaries to

estimate the important aspects of the population estimate the important aspects of the population of starting salariesof starting salaries

There is a criteria on how to choose a sample: the There is a criteria on how to choose a sample: the information contained in a sample is to accurately reflect information contained in a sample is to accurately reflect the population under study. the population under study.

Inferential Statistics

Estimatione.g.: Estimate the

population mean weight using the sample mean weight

Hypothesis testinge.g.: Test the claim that the

population mean weight is 120 pounds

Drawing conclusions and/or making decisions concerning a population based on sample results.

Inferential Statistics

Analysis of relationship e.g.: Does the rate of

growth of the money supply influence the inflation rate?

Forecastinge.g.: Prediction of future

interest rates

1.2 Sampling a Population of Existing Units

For example, randomly pick two different people from a group of 15:For example, randomly pick two different people from a group of 15: Number the people from 1 to 15; and write their numbers on 15 Number the people from 1 to 15; and write their numbers on 15

different slips of paperdifferent slips of paper Thoroughly mix the papers and randomly pick two of themThoroughly mix the papers and randomly pick two of them The numbers on the slips identifies the people for the sampleThe numbers on the slips identifies the people for the sample

Each population unit has the same chance of being selected as Each population unit has the same chance of being selected as every other unitevery other unit Each possible sample (of the same size) has the same chance Each possible sample (of the same size) has the same chance

of being selectedof being selected

A random sample is a sample selected from a population so that:A random sample is a sample selected from a population so that:

Random sampleRandom sample


Guarantees a sample of different unitsGuarantees a sample of different unitsEach sampled unit contributes different informationEach sampled unit contributes different informationSampling without replacement is the usual and customary Sampling without replacement is the usual and customary sampling methodsampling method

A sampled unit is withheld from possibly being A sampled unit is withheld from possibly being selected again in the same sampleselected again in the same sample

Sample without replacementSample without replacement

The unit is placed back into the population for possible reselection However, the same unit in the sample does not contribute new information

Replace each sampled unit before picking next unitReplace each sampled unit before picking next unitSample with replacementSample with replacement


Example 1.1Example 1.1 The Cell Phone Case: Estimating Cell Phone The Cell Phone Case: Estimating Cell Phone CostsCosts

The bank has 2,136 employees on a 500-minute-per-The bank has 2,136 employees on a 500-minute-per-month plan with a monthly cost of $50. The bank will month plan with a monthly cost of $50. The bank will estimate its cellular cost per minute for this plan by estimate its cellular cost per minute for this plan by examining the number of minutes used last month by each examining the number of minutes used last month by each of 100 randomly selected employees on this 500-minute of 100 randomly selected employees on this 500-minute plan.plan.

According to the cellular management service, if the According to the cellular management service, if the cellular cost per minute for the random sample of 100 cellular cost per minute for the random sample of 100 employees is over 18 cents per minute, the bank should employees is over 18 cents per minute, the bank should benefit from automated cellular management of its calling benefit from automated cellular management of its calling plans. plans.


In order to randomly select the sample of 100 cell In order to randomly select the sample of 100 cell

phone users, the bank will make a numbered list of the phone users, the bank will make a numbered list of the

2,136 users on the 500-munite plan. This list is called 2,136 users on the 500-munite plan. This list is called a a

frameframe. .

The bank can use The bank can use a random number tablea random number table, such as , such as

Table 1.1(a), or a computer software package, such as Table 1.1(a), or a computer software package, such as

Table 1.1 (b), to select the needed sample. Table 1.1 (b), to select the needed sample.

The 100 cellular-usage figures are given in Table 1.2.The 100 cellular-usage figures are given in Table 1.2.




Approximately Random SamplesApproximately Random Samples

Sometimes it is not possible to list and thus number all Sometimes it is not possible to list and thus number all the units in a population. In such a situation we often the units in a population. In such a situation we often select select a systematic samplea systematic sample, which approximates a random , which approximates a random sample. sample.

A Systematic SampleA Systematic Sample

Randomly enter the population and systematically samplRandomly enter the population and systematically sample every e every kkthth unit. unit.


Example 1.2Example 1.2 The Marketing Research Case: The Marketing Research Case: Rating a New Bottle DesignRating a New Bottle Design

To study consumer reaction to a new design, the brand group will To study consumer reaction to a new design, the brand group will use “mall intercept method” in which shoppers at a large use “mall intercept method” in which shoppers at a large metropolitan shopping mall are intercepted and asked to participate metropolitan shopping mall are intercepted and asked to participate in in a consumer surveya consumer survey. The questionnaire are shown in Figure 1.1. . The questionnaire are shown in Figure 1.1. Each shopper will be exposed to the new bottle design and asked to Each shopper will be exposed to the new bottle design and asked to rate the bottle image using a 7-point “Likert scale.”rate the bottle image using a 7-point “Likert scale.”

We select a systematic sample. To do this, every 100We select a systematic sample. To do this, every 100 thth shopper shopper passing a specified location in the mall will be invited to participate passing a specified location in the mall will be invited to participate in the survey. During a Tuesday afternoon and evening, a sample of in the survey. During a Tuesday afternoon and evening, a sample of 60 shoppers is selected by using the systematic sampling process. 60 shoppers is selected by using the systematic sampling process. The 60 composite scores are given in Table 1.3. From this table, we The 60 composite scores are given in Table 1.3. From this table, we can estimate that 95 percent of the shoppers would give the bottle can estimate that 95 percent of the shoppers would give the bottle design a composite score of at least 25.design a composite score of at least 25.



Voluntary response sampleVoluntary response sample

Participants select themselves to be in the sampleParticipants select themselves to be in the sample

Participants “self-select”Participants “self-select” For example, calling in to vote on For example, calling in to vote on American IdolAmerican Idol Commonly referred to as a “non-scientific” sampleCommonly referred to as a “non-scientific” sample

Usually not representative of the populationUsually not representative of the population Over-represent individuals with strong opinionsOver-represent individuals with strong opinions

Usually, but not always, negative opinionsUsually, but not always, negative opinions

Another Sampling Method


1.3 Sampling a Process

ProcessA sequence of operations that takes inputs (labor, raw materials, methods, machines, and so on) and turns them into outputs (products, services, and the like)

Inputs Process Outputs


Cars will continue to be made over timeCars will continue to be made over time

For example, all automobiles of a For example, all automobiles of a particular make and model, for particular make and model, for instance, the Lincoln Town Carinstance, the Lincoln Town Car

The “population” from a process is all output The “population” from a process is all output produced in the past, present, and the yet-to-occur produced in the past, present, and the yet-to-occur future.future.

Processes produce output over time


Example 1.3Example 1.3 The Coffee Temperature Case:The Coffee Temperature Case: Monitoring Coffee TemperaturesMonitoring Coffee Temperatures

This case concerns coffee temperatures at a fast-food This case concerns coffee temperatures at a fast-food restaurant. To do this, the restaurant personnel measure restaurant. To do this, the restaurant personnel measure the temperature of the coffee being dispensed (in degrees the temperature of the coffee being dispensed (in degrees F) at half-hour intervals from 10 A.M. to 9:30 P.M. on a F) at half-hour intervals from 10 A.M. to 9:30 P.M. on a given day. Data is list on Table 1.7. given day. Data is list on Table 1.7. A process is in statistical control if it does not exhibit any unusual process variations.To determine if a process is in control or not, sample To determine if a process is in control or not, sample the process often enough to detect unusual variationsthe process often enough to detect unusual variationsA A runs plotruns plot is a graph of individual process is a graph of individual process measurements over time. Figure 1.3 shows a runs plot measurements over time. Figure 1.3 shows a runs plot of the temperature data. of the temperature data.



Figure 1.3 Runs Plot of Coffee Temperatures: The Process Figure 1.3 Runs Plot of Coffee Temperatures: The Process is in Statistical Control.is in Statistical Control.


Over time, temperatures appear to have a fairly constant amount of variation around a fairly constant level The temperature is expected to be at the constant level

shown by the horizontal blue line Sometimes the temperature is higher and

sometimes lower than the constant level About the same amount of spread of the values (data

points) around the constant level The points are as far above the line as below it The data points appear to form a horizontal band

So, the process is in statistical control Coffee-making process is operating “consistently”

Results Results


Because the coffee temperature has been and is presently Because the coffee temperature has been and is presently in control, it will likely stay in control in the futurein control, it will likely stay in control in the future If the coffee making process stays in control, then If the coffee making process stays in control, then

coffee temperature is predicted to be between 152coffee temperature is predicted to be between 152oo and 170and 170oo F F

In general, if the process appears from the runs plot to In general, if the process appears from the runs plot to be in control, then it will probably remain in control in be in control, then it will probably remain in control in the futurethe future The sample of measurements was approximately The sample of measurements was approximately

randomrandom Future process performance is predictableFuture process performance is predictable

RemarkRemark


1.4 An Introduction to Survey Sampling

Already know some sampling methodsAlso called sampling designs, they are: Random sampling

The focus of this book Systematic sampling Voluntary response sampling

But there are other sample designs: Stratified random sampling Cluster sampling

Systematic Sampling


More on Systematic Sampling

Want a sample containing n units from a population containing N units

Take the ratio N/n and round down to the nearest whole number Call the rounded result k

Randomly select one of the first k elements from the population list

Step through the population from the first chosen unit and select every kth unit

This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering


Stratified Random Sample Divide the population into non-overlapping groups,

called strata, of similar units Separately, select a random sample from each and

every stratum Combine the random samples from each stratum to

make the full sample

Population

Divided

into 4

strataSample

Appropriate when the population consists of two or more different groups so that:

The groups differ from each other with respect to the variable of interest

Units within a group are similar to each other For example, divide population into strata by

age, gender, income, etc

Stratified Random Sample


Cluster Sampling “Cluster” or group a population into subpopulations

Cluster by geography, time, and so on… Each cluster is a representative small-scale version of the

population (i.e. heterogeneous group) A simple random sample is chosen from each cluster Combine the random samples from each cluster to make the

full sample

Population divided into 16 clusters. Randomly selected

clusters for sample

There are different sections or regions in the area with respect to the variable of interest

A random sample of the cluster

Cluster Sampling

Appropriate for populations spread over a large geographic area so that…


Sampling Problems Random sampling should eliminate bias But even a random sample may not be representative

because of: Under-coverage

Too few sampled units or some of the population was excluded

Non-response When a sampled unit cannot be contacted or

refuses to participate Response bias

Responses of selected units are not truthful

Chapter 2Descriptive StatisticsDescribing the Shape of a DistributionDescribing the Shape of a Distribution

Describing Central TendencyDescribing Central Tendency

Measures of VariationMeasures of Variation

Percentiles, Quartiles, and Box-and-Percentiles, Quartiles, and Box-and-Whiskers DisplaysWhiskers Displays

Describing Qualitative DataDescribing Qualitative Data

Using Scatter Plots*Using Scatter Plots*

Weighted Means and Grouped DataWeighted Means and Grouped Data

Chapter 2 Descriptive Statistics

2.1 Describing the Shape of a Distribution

To know what the population looks like, find the To know what the population looks like, find the “shape” of its distribution“shape” of its distribution

Picture the distribution graphically by any of the Picture the distribution graphically by any of the following methods:following methods: Stem-and-leaf displayStem-and-leaf display Frequency distributionsFrequency distributions HistogramHistogram Dot plotDot plot


Stem-and-leaf Display The purpose of a stem-and-leaf display is to see the The purpose of a stem-and-leaf display is to see the

overall pattern of the data, by grouping the data into overall pattern of the data, by grouping the data into classesclasses To see:To see:

the variation from class to classthe variation from class to class the amount of data in each classthe amount of data in each class the distribution of the data within each classthe distribution of the data within each class

Best for small to moderately sized data distributionsBest for small to moderately sized data distributions


Example 2.1Example 2.1 The Car Mileage Case The Car Mileage Case

In this case study, we consider a tax credit offered by In this case study, we consider a tax credit offered by the federal government to automakers for improving the federal government to automakers for improving the fuel economy of midsize cars. the fuel economy of midsize cars.

To find the combined city and highway mileage To find the combined city and highway mileage estimate for a particular car model, the EPA tests a estimate for a particular car model, the EPA tests a sample of cars. sample of cars.

Table 2.1 presents the sample of 49 gas mileages that Table 2.1 presents the sample of 49 gas mileages that have been obtained by the new midsize model. have been obtained by the new midsize model.


30.830.8 30.930.9 32.032.0 32.332.3 32.632.6

31.731.7 30.430.4 31.431.4 32.732.7 31.431.4

30.130.1 32.532.5 30.830.8 31.231.2 31.831.8

31.631.6 30.330.3 32.832.8 30.630.6 31.931.9

32.132.1 31.331.3 32.032.0 31.731.7 32.832.8

33.333.3 32.132.1 31.531.5 31.431.4 31.531.5

31.331.3 32.532.5 32.432.4 32.232.2 31.631.6

31.031.0 31.831.8 31.031.0 31.531.5 30.630.6

32.032.0 30.430.4 29.829.8 31.731.7 32.232.2

32.432.4 30.530.5 31.131.1 30.630.6

Table 2.1 A sample of 49 mileages


The stem-and-leaf display of car mileages:The stem-and-leaf display of car mileages:

29 829 830 1344566688930 1344566688931 0012334445556677788931 0012334445556677788932 000112234455678832 000112234455678833 333 3

29 + 0.8 = 29.829 + 0.8 = 29.8

33 + 0.3 = 33.333 + 0.3 = 33.3


Another display of the same data using more classes

Starred classes (*) extend from 0.0 to 0.4

Unstarred classes extend from 0.5 to 0.9

29 830* 134430 566688931* 00123344431 5556677788932* 000112234432 55678833* 3


Looking at the last stem-and-leaf display, the Looking at the last stem-and-leaf display, the distribution appears almost “distribution appears almost “symmetricalsymmetrical”” The upper portion of the display…The upper portion of the display…

Stems 29, 30*, 30, and 31*Stems 29, 30*, 30, and 31* … … is almost a mirror image of the lower portion of is almost a mirror image of the lower portion of

the displaythe display Stems 31, 32*, 32, and 33*Stems 31, 32*, 32, and 33*

But not exactly a mirror reflectionBut not exactly a mirror reflection Maybe slightly more data in the lower Maybe slightly more data in the lower

portion than in the upper portionportion than in the upper portion Later, we will call this a slightly “left-Later, we will call this a slightly “left-

skewed” distributionskewed” distribution


Constructing a Stem-and-Leaf DisplayConstructing a Stem-and-Leaf Display

1.1. Decide what units will be used for the stems and the Decide what units will be used for the stems and the leaves. As a general rule, choose units for the stems so leaves. As a general rule, choose units for the stems so that there will be somewhere between 5 and 20 stems.that there will be somewhere between 5 and 20 stems.

2.2. Place the stems in a column with the smallest stem at Place the stems in a column with the smallest stem at the top of the column and the largest stem at the bottom.the top of the column and the largest stem at the bottom.

3.3. Enter the leaf for each measurement into the row Enter the leaf for each measurement into the row corresponding to the proper stem. The leaves should be corresponding to the proper stem. The leaves should be single-digit numbers (rounded values).single-digit numbers (rounded values).

4.4. If desired, rearrange the leaves so that they are in If desired, rearrange the leaves so that they are in increasing order from left to right.increasing order from left to right.


Example 2.2Example 2.2 The Payment Time Case: Reducing Payment Times

In order to assess the effectiveness of the system, the In order to assess the effectiveness of the system, the consulting firm will study the payment times for invoices consulting firm will study the payment times for invoices processed during the first three months of the system’s processed during the first three months of the system’s operation. operation.

During this period, 7,823 invoices are processed using During this period, 7,823 invoices are processed using the new system. To study the payment times of these the new system. To study the payment times of these invoices, the consulting firm numbers the invoices from invoices, the consulting firm numbers the invoices from 0001 to 7823 and uses random numbers to select a 0001 to 7823 and uses random numbers to select a random sample of 65 invoices. The resulting 65 payment random sample of 65 invoices. The resulting 65 payment times are given in Table 2.2times are given in Table 2.2


2222 2929 1616 1515 1818 1717 1212 1313 1717 1616 1515

1919 1717 1010 2121 1515 1414 1717 1818 1212 2020 1414

1616 1515 1616 2020 2222 1414 2525 1919 2323 1515 1919

1818 2323 2222 1616 1616 1919 1313 1818 2424 2424 2626

1313 1818 1717 1515 2424 1515 1717 1414 1818 1717 2121

1616 2121 2525 1919 2020 2727 1616 1717 1616 2121

Table 2.2 A Sample of Payment Times (in Days) for 65 Randomly Selected Invoices.


1 10 0 2 11 0 4 12 00 7 13 000 11 14 0000 18 15 0000000 27 16 000000000 (8) 17 00000000 30 18 000000 24 19 00000 19 20 000 16 21 000 13 22 000 10 23 00 8 24 000 5 25 00 3 26 0 2 27 0 1 28 1 29 0

Shorter tailL

onger tail

The leftmost column of The leftmost column of numbers are the numbers are numbers are the numbers are the amounts of values in the amounts of values in each stemeach stem

• The number 8 in The number 8 in parentheses indicates that parentheses indicates that there are 8 payments in there are 8 payments in the stem for 17 daysthe stem for 17 days

• The number 27 (no The number 27 (no parentheses) indicates that parentheses) indicates that there are 27 payments there are 27 payments made in 16 or less daysmade in 16 or less days


Looking at this display, we see that all of the sampled Looking at this display, we see that all of the sampled payment times are substantially less than the 39-day payment times are substantially less than the 39-day typical payment time of the former billing system. typical payment time of the former billing system.

The stem-and-leaf display do not appear symmetrical. The stem-and-leaf display do not appear symmetrical. The “tail” of the distribution consisting of the higher The “tail” of the distribution consisting of the higher payment times is longer than the “tail” of the payment times is longer than the “tail” of the distribution consisting of the smaller payment times.distribution consisting of the smaller payment times.

We say that the distribution is skewed with a tail to the We say that the distribution is skewed with a tail to the right.right.


Frequency Distribution and Histogram

A A frequency distributionfrequency distribution is a list of data classes is a list of data classes with the count or “frequency” of values that belong with the count or “frequency” of values that belong to each classto each class

• “ “Classify and count”Classify and count”• The frequency distribution is a tableThe frequency distribution is a table

Show the frequency distribution in a Show the frequency distribution in a histogramhistogram• The histogram is a picture of the frequency The histogram is a picture of the frequency distributiondistribution

See Examples 2.2, The Payment Time CaseSee Examples 2.2, The Payment Time Case


Constructing a Frequency Distribution

Steps in making a frequency distribution:Steps in making a frequency distribution:

1.1. Determine the number of classes Determine the number of classes KK

2.2. Determine the class lengthDetermine the class length

3.3. Set the starting value for the classes, that is, the Set the starting value for the classes, that is, the distribution “floor”distribution “floor”

4.4. Calculate the class limitsCalculate the class limits

5.5. Setup all the classesSetup all the classes Then tally the data into the Then tally the data into the KK classes and record the classes and record the

frequenciesfrequencies


Number of Classes K

Group all of the n data into K number of classes K is the smallest whole number for which

2K n

In Examples 2.2 , n = 65 For K = 6, 26 = 64, < n For K = 7, 27 = 128, > n So use K = 7 classes


Class Length Class length L is the step size from one to the next

In Examples 2.2, The Payment Time Case, the largest value is 29 days and the smallest value is 10 days, so

Arbitrarily round the class length up to 3 days/class

KL

value smallest - value Largest

days/class 71432classes 7

days 19

classes 7

days 10 - 29.L


Starting the Classes The classes start on the smallest data value

This is the lower limit of the first class The upper limit of the first class is

smallest value + (L – 1) In the example, the first class starts at 10 days and goes

up to 12 days The second class starts at the upper limit of the first class +

1 and goes up (L – 1) more The second class starts at 13 days and goes up to 15

days And so on


Tallies and Frequencies:Example 2.2

Classes (days) Tally Frequency

10 to 12 ||| 3

13 to 15 |||| 14

16 to 18 ||| 23

19 to 21 || 12

22 to 24 ||| 8

25 to 27 |||| 4

28 to 30 | 1

65

||||||||

|||||||| ||||||||

||||||||

||||

Check: All frequencies must sum to Check: All frequencies must sum to nn


The relative frequency of a class is the proportion or fraction of data that is contained in that class Calculated by dividing the class frequency by the

total number of data values Relative frequency may be expressed as either a

decimal or percent A relative frequency distribution is a list of all

the data classes and their associated relative frequencies

Relative Frequency


Relative Frequency: Example 2.2

Classes (days) Frequency Relative Frequency

10 to 12 3 3/65 = 0.0462

13 to 15 14 14/65 = 0.2154

16 to 18 23 0.3538

19 to 21 12 0.1846

22 to 24 8 0.1231

25 to 27 4 0.0615

28 to 30 1 0.0154

65 1.0000

Check: All relative frequencies must sum to 1Check: All relative frequencies must sum to 1


Histogram

A graph in which rectangles represent the A graph in which rectangles represent the classesclasses

The base of the rectangle represents the class The base of the rectangle represents the class lengthlength

The height of the rectangle represents The height of the rectangle represents the frequency in a frequency histogram, orthe frequency in a frequency histogram, or the relative frequency in a relative frequency the relative frequency in a relative frequency

histogramhistogram

Histogram : Daily High Tem perature

0

1

2

3

4

5

6

7

5 15 25 35 45 55 65

Fre

qu

ency

Class Midpoints

Histogram Example

(No gaps between bars)

Class

10 but less than 20 15 3





FrequencyClass

Midpoint


Example 2.2: The Payment Times CaseExample 2.2: The Payment Times Case

Frequency HistogramFrequency Histogram Relative Frequency HistogramRelative Frequency Histogram

As with the earlier stem-and-leaf display, the tail on the As with the earlier stem-and-leaf display, the tail on the right appears to be right appears to be longerlonger than the tail on the left. than the tail on the left.


The Normal CurveSymmetrical and bell-shaped Symmetrical and bell-shaped curve for a normally distributed curve for a normally distributed populationpopulationThe height of the normal over The height of the normal over any point represents the relative any point represents the relative proportion of values near that pointproportion of values near that point

Example 2.1, The Car Mileages Example 2.1, The Car Mileages CaseCase


SkewnessSkewed distributions are not symmetrical about their center. Rather, they are lop-sided with a longer tail on one side or the other.• A population is distributed according to its relative

frequency curve• The skew is the side with the longer tail

Right SkewedRight SkewedLeft SkewedLeft Skewed SymmetricSymmetric


Dot Plots

On a number line, each data value is represented by a dot placed above the corresponding scale value

Scores on Exams 1 and 2Scores on Exams 1 and 2

Unusually low score, so an “Unusually low score, so an “outlieroutlier””


2.2 Describing Central Tendency

Population Parameters

A population parameter is a number calculated from all the population measurements that describes some aspect of the population

The population mean, denoted , is a population parameter and is the average of the population measurements


Point Estimates and Sample Statistics

A point estimate is a one-number estimate of the value of a population parameter

A sample statistic is a number calculated using sample measurements that describes some aspect of the sample Use sample statistics as point estimates of the population Use sample statistics as point estimates of the population

parametersparameters

The sample mean, denoted x, is a sample statistic and is the average of the sample measurements The sample mean is a point estimate of the population The sample mean is a point estimate of the population

meanmean


Measures of Central Tendency

MeanMean, , : The average or expected value : The average or expected value

MedianMedian, M, Mdd:: The value of the middle point of tThe value of the middle point of t

he ordered measurementshe ordered measurements

ModeMode, M, Moo: The most frequent value: The most frequent value


The Mean

Population X1, X2, …, XN

Population Mean

N

X

N

=1ii

Sample x1, x2, …, xn

Sample Mean

n

x x

n

=1ii

x


The Sample Mean

For a sample of size For a sample of size nn, the , the sample meansample mean is defined as is defined as

n

xxx

n

xx n

n

ii

...211

and is a point estimate of the population mean

• It is the value to expect, on average and in the long run


Example: Car Mileage Case

Sample mean for first five car mileages from Table 2.1

30.8, 31.7, 30.1, 31.6, 32.1

5554321

5

1 xxxxxx

x ii

26.315

3.156

5

1.326.311.307.318.30

x


Example: Car Mileage Case Continued

Sample mean for all the car mileages from Table 2.1Sample mean for all the car mileages from Table 2.1

5531.3149

1.1546

49

49

1 i

ixx

Based on this calculated sample mean, the point estimate of mean mileage of all cars is 31.5531 mpg


The MedianThe population or sample median Md is a value such that 50% of all measurements, after having been arranged in numerical order, lie above (or below) it

The median Md is found as follows:

1. If the number of measurements is odd, the median is the middlemost measurement in the ordered values

2. If the number of measurements is even, the median is the average of the two middlemost measurements in the ordered values


Example: Sample Median

Internist’s Yearly Salaries (x$1000)Internist’s Yearly Salaries (x$1000)

127 132 138 141 144 146 127 132 138 141 144 146 152152 154 165 171 177 192 241 154 165 171 177 192 241

Because Because nn = 13 (odd,) then the median is the middlemost = 13 (odd,) then the median is the middlemost or 7or 7thth value of the ordered data, so value of the ordered data, so

MMdd=152=152

An annual salary of $180,000 is in the high end, well An annual salary of $180,000 is in the high end, well above the median salary of $152,000above the median salary of $152,000

• In fact, $180,000 a very high and competitive In fact, $180,000 a very high and competitive salarysalary

Example 2.3Example 2.3


The ModeThe mode Mo of a population or sample of measurements is the measurement that occurs most frequently

• Modes are the values that are observed “most typically”

• Sometimes higher frequencies at two or more values

• If there are two modes, the data is bimodal

• If more than two modes, the data is multimodal

• When data are in classes, the class with the highest frequency is the modal class

• The tallest box in the histogram


Example 2.4Example 2.4 DVD Recorder SatisfactionDVD Recorder Satisfaction

Satisfaction rankings on a scale of 1 (not satisfied) to 10 Satisfaction rankings on a scale of 1 (not satisfied) to 10 (extremely satisfied), arranged in increasing order(extremely satisfied), arranged in increasing order

1 3 5 5 7 8 8 8 8 8 8 9 9 9 9 9 10 10 10 10 1 3 5 5 7 8 8 8 8 8 8 9 9 9 9 9 10 10 10 10

Because Because nn = 20 (even,) then the median is the average of = 20 (even,) then the median is the average of two middlemost ratings; these are the 10two middlemost ratings; these are the 10 thth and 11 and 11thth values. Both of these are 8 (circled), so values. Both of these are 8 (circled), so

MMdd = 8 = 8

Because te rating 8 occurs with the highest rating, Because te rating 8 occurs with the highest rating,

MMoo = 8 = 8


Relationships Among Mean,

Median and Mode


Comparing Mean, Median & Mode

Bell-shaped distribution: Mean = Median = Mode

Right skewed distribution: Mean > Median > Mode

Left-skewed distribution: Mean < Median < Mode Also: The median is not affected by extreme values

• “Extreme values” are values much larger or much smaller than most of the data

• The median is resistant to extreme values The mean is strongly affected by extreme values

• The mean is sensitive to extreme values


Payment Time Case

Mean=18.108 daysMean=18.108 daysMedian=17.000 daysMedian=17.000 daysMode=16.000 daysMode=16.000 daysSo:So:Expect the mean payment time to be 18.108 Expect the mean payment time to be 18.108 daysdaysA long payment time would be > 17 days and a A long payment time would be > 17 days and a short payment time would be < 17 daysshort payment time would be < 17 daysThe typical payment time is 16 daysThe typical payment time is 16 days

Office: E409 Tel: 3620623(office), 3620620(T.A.) Statistics for Business Instructor: Dr. Peng Xiaoling T.A. : Miss Li Jianxia Email: [email protected].

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