Lesson 002

IBS Statistics Year 1Dr. Ning DING

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Table of content• Review

• Learning Goals

• Chapter 3-A

• Summary

ReviewChapter 1: What is Statistics?

What is the level of measurement for these items related to the newspaper business?

a.The number of papers sold each Sunday during 2006.

b.The departments, such as editorial, advertising , sports, etc.

c.A summary of the number of papers sold by county.

d.The number of years with the paper for each employee.

RatioRatio

RatioRatio

NominalNominal

RatioRatio

P14. N.2 Ch.1

RatioRatioNominalNominal OrdinalOrdinal IntervalInterval

ReviewChapter 1: What is Statistics?

For the follow questions, would you collect information using a sample or a population?

a.Statistics 201 is a course taught at a university. Professor A has taught nearly 1,500 students in the course over the past 5 years. You would like to know the average grade for the course

b.You are looking forward to graduation project and your first job as a salesperson for one of five large corporations. Planning for your interviews, you will need to know about each company’s mission, profitability, products, and markets.

SampleSample

PopulationPopulationP16. N.8 Ch.1

SampleSample PopulationPopulation

ReviewChapter 2: Describing Data

Qualitative DataQualitative Data

Bar Chart

Bar Chart

Pie Chart

Pie Chart


Quantitative DataQuantitative Data

Histogram

Histogram Polygon

Polygon

Cumulative Frequency

Distribution


Distribution


Polygon

Polygon


Distribution


Distribution


P34. N.10 Ch.2

A set of data contains 53 observations. The lowest value is 43 and the largest is 129. The data are to be organized into a frequency distribution.

a. How many classes would you suggest?

25 = 32, 26 = 64, suggests 6 classes

i > ≈ 15130 - 42

6

Use interval of 15And start first class at 40

b. What would you suggest as class interval & the lower limit of the first class?

Learning Goals• Chapter 3:

– Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean

– Explain the characteristics, uses, advantages, and disadvantages of each measure of location

– Identify the position of the mean, median, and mode for both symmetric and skewed distributions

– Compute and interpret the range, mean deviation, variance, and standard deviation

– Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion

– Understand Chebyshev’s theorem and the Empirical Rule as they relate to a set of observations

1. Population MeanChapter 3: Describing Data

Parameter: a numerical characteristic of a population.

Example: The fraction of U. S. voters who support Sen. McCain for President is a parameter.

Statistic: A statistic is a numerical characteristic of a sample.Example: If we select a simple random sample of n = 1067 voters from the population of all U. S. voters, the fraction of people in the sample who support Sen. McCain is a statistic.

Summary: Parameter & StatisticsChapter 3: Describing Data


N

X=μ

∑

Population mean = Sum of all the values in the population

Number of values in the population


N

X=μ

∑

Example:

2. Sample MeanChapter 3: Describing Data

Sample mean = Sum of all the values in the sample

Number of values in the sample

nΣX

=X


nΣX

=X

Example:


nΣX

=X

Example: A sample of five executives received the following bonus last year ($000):14.0, 15.0, 17.0, 16.0, 15.0

15.4=577

=5

15.0+...+14.0=

nΣX

=X

3. Properties of the Arithmetic MeanChapter 3: Describing Data

1. Every set of interval- or ratio-level data has a mean2. All the values are included in computing the mean3. The mean is unique.4. The sum of the deviations of each value from the

mean is zero.Example: Consider the set of values: 3, 8, and 4. The mean is 5. Illustrating the fifth property:

0=5)(4+5)(8+5)(3=)X -Σ(X ---

4. The Weighted MeanChapter 3: Describing Data

Weighted Mean: a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula:

)n21

nn2211w ...w+w+(w

)Xw+...+Xw+X(w=X

4. The Weighted MeanChapter 3: Describing Data

Example: During a one hour period on a hot Saturday afternoon, Julie served fifty lemon drinks. She sold five drinks for $0.50, fifteen for $0.75, fifteen for $0.90, and fifteen for $1.10. Compute the weighted mean of the price of the drinks.

$0.89=50

$44.50=

15+15+15+515($1.15)+15($0.90)+15($0.75)+5($0.50)

=Xw

Frequency counts

Exercise

P62. N.14 Ch.3

The Bookstall sold books via internet. Paperbacks are $1.00 each, and hardcover books are $3.50. Of the 50 books sold on last Tuesday, 40 were paperback and the rest were hardcover. What was the weighted mean price of a book?

Chapter 3: Describing Data

)n21

nn2211w ...w+w+(w

)Xw+...+Xw+X(w=X

50.1$50

$3.50)*10+$1.00*(40=Xw

40 paperback$1.00

40 paperback$1.00

10 hardcover$3.50

10 hardcover$3.50

6. The ModeChapter 3: Describing Data

Mode:There is one situation in which the mode is the only measure of central tendency that can be used – when we have categorical, or non-numeric data. In this situation, we cannot calculate a mean or a median. The mode is the most typical value of the categorical data.

Example: Suppose I have collected data on religious affiliation of citizens of the U.S. The modal, or most Typical value, is Roman Catholic, since The RomanCatholic Church is the largest religious organization in the U.S.


Mode:

The value of the observation that appears most frequently.


Mode:

The value of the observation that appears most frequently.

Example: The exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87.

Because the score of 81 occurs the most often, it is the mode.

7. The MedianChapter 3: Describing Data

Median: the midpoint of the values after they have been ordered from the smallest to the largest.

Example: The ages for a sample of five college students are:

21, 25, 19, 20, 22

Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21.


For an even set of values, the median will be the arithmetic average of the two middle numbers.

Example: The heights of four basketball players, in inches, are: 76, 73, 80, 75

Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75.5


72 68 65 70 75 79 73

Example:

Finding the median

65 68 70 72 73 75 79

65 68 70 72 73 75 79 79

72.5

65 68 70 72 73 75 79 79,000

72.5

Exercise

P65. N.18 & 22 Ch.3

Determine the mean, median, mode41 15 39 54 31 15 33


Mean= 32.57; Median=33; Mode=15

List below are the total automobile sales (in millions of dollars) for the last 14 years. What was the median number of automobiles sold? What is the mode?

41 15 39 54 31 15 33

Median = 9.2 (millions of dollars)Modes are 8.2, 8.5 and 10.3 (millions of dollars)

Exercise

P69. N.26 Ch.3


Mean Mode MedianCity - - -Wind direction

- Southwest -

Temperature

91 o F 92 o F 92 o F

Pavement - Wet & Dry Trace

5-B The Arithmetic Mean of Grouped Data


Example:

32

Recall in Chapter 2, we constructed a frequency distribution for the vehicle selling prices. The information is repeated below. Determine the arithmetic mean vehicle selling price.

The Arithmetic Mean of Grouped Data -Example

32



33


33


5-B The Arithmetic Mean of Grouped Data


33


31

The Arithmetic Mean of Grouped Data

Exercise 5-B

P87. N.58 Ch.3

Determine the mean of the following frequency distribution.


31


31


X=380/30=12.67

6-B. The Mode for Grouped DataChapter 3: Describing Data

Example:

Finding the mode for grouped data

Step 1:

Modal class with the highest frequency

32



Step 2:

Midpoint of the modal class is the mode

19.519.5

7-B. The Median for Grouped DataChapter 3: Describing Data

Example:


Step 1:

Modal class with the highest frequency

32



Step 2:

Midpoint of the modal class is the mode

19.519.5


Example:


Step 1:

Cumulative Frequency Distribution


Step 1:


Step 2:

Determine the position of the median and the median class


Step 1:


Step 2:

Determine the position of the median and the median class

Step 3:

Draw two lines (value & position)

Median – 100 150 - 100

300.5 – 201388 - 201=

AA

BB

Value: 100 Median 150

Position: 201 300.5 388

Median = 300.5 – 201388 - 201 * 50 + 100 = 126.60 (dollars)

SummaryChapter 3: Describing Data

• Chapter 3: – Calculate the arithmetic mean, weighted mean, median,

mode, and geometric mean– Explain the characteristics, uses, advantages, and

disadvantages of each measure of location– Identify the position of the mean, median, and mode for

both symmetric and skewed distributions– Compute and interpret the range, mean deviation,

variance, and standard deviation– Understand the characteristics, uses, advantages, and

disadvantages of each measure of dispersion– Understand Chebyshev’s theorem and the Empirical Rule

as they relate to a set of observations

8. The Relative Positions of the Mean, Median, and Mode


skewed



Zero skewness

mode=median=mean

Zero skewness

mode=median=mean



positive skewness

Mode median mean

positive skewness

Mode median mean< <



negative skewness

Mode median mean

negative skewness

Mode median mean> >



9. The Geometric MeanChapter 3: Describing Data

Geometric mean (GM) :a set of n numbers is defined as the nth root of the product of the n

numbers.

The formula is:

The geometric mean is used to average percents, indexes, and relatives.

The geometric mean is not applicable when some numbers are negative.

n n321 ))...(X)(X)(X(X=GM


n n321 ))...(X)(X)(X(X=GM

Example: Suppose you receive a 5 percent increase in salary this year and a 15 percent increase next year. The average annual percent increase is 9.886, not 10.0. Why is this so? We begin by calculating the geometric mean.

098861151051 . ).)(.(GM

Not understand percentage?Click here


n n321 ))...(X)(X)(X(X=GM

Example: The return on investment earned by Atkins construction Company for four successive years was: 30 percent, 20 percent, -40 percent, and 200 percent. What is the geometric mean rate of return on investment?

..).)(.)(.)(.(GM 2941808203602131 44


1period) of beginningat (Value

period) of endat (Value=GM n -

Geometric mean (GM) :Another use of the geometric mean is to determine the percent increase in

sales, production or other business or economic series from one time period

to another.

http://www2.goldmansachs.com/


1period) of beginningat (Value

period) of endat (Value=GM n -

Example: The total number of females enrolled in American colleges increased from

755,000 in 1992 to 835,000 in 2000.

.0127=1 - 755,000

835,000=GM 8

That is, the geometric mean rate of increase is 1.27%.


Example: A banker wants to get an annual return of 100% on its loan in credit card

business. What monthly interest rate should he charge?

.059=1 - 100

200=GM 12

A monthly interest rate of 5.9%.


Example: The Chinese government claimed in 1990 that their GDP will double in 20

years.

What must the annual GDP growth rate be for this dream to come true?

A annual GDP growth of 3.5%.

.035=1 - 100

200=GM 20


Example: The 2006 population size of Duval County was 837,964. The population grew by7.6% between 2000 and 2006. We want to project the size of the population in2030, assuming that the growth rate remains the same; i.e., 7.6% every 6 years.

The Projected population size in 2030 is (1.0764 X 837,964) = 1123245. Theaverage growth rate over the 24 years is found by calculating the geometric mean:

The average growth rate is just what we expect.

076.1076.1076.1076.1076.14 GM

Exercise

P71. N.32 Ch.3

In 1976 the nationwide average price of a gallon of unleaded gasoline at a self-serve pump was $0.605. By 2005 the average price had increased to $2.57. What was the geometric mean annual increase for the period?


5.11% found by -1292.570.605


P27. N.4 Ch.2

Qualitative DataQualitative Data

Two thousand frequent mIdwestern business travelers are asked which Midwest city they prefer: Indianapolis, Saint Louis, Chicago, or Milwaukee.

The results were 100 liked Indianapolis best, 450 liked Saint Louis, 1,300 liked Chicago, and the remainder preferred Milwaukee. Develop a frequency table and a relative frequency table to summarize this information.


P34. N.12 Ch.2

The daily number of oil changes at the Oak Streek outlet in the past 20 days are:

The data are to be organzied into a frequency distribution. a. How many classes would you recommend?

24 = 16, 25 = 32, suggests 5 classes

i > ≈ 1099 - 51

5

Use interval of 10

b. What class interval would you suggest?


P34. N.12 Ch.2


The data are to be organzied into a frequency distribution.

c. What lower limit would you recommend for the first class?

start first class at 50


P34. N.12 Ch.2


d. Organize the number of oil changes into a frequency distribution.


P34. N.12 Ch.2


e. Comment on the shape of the frequency distribution. Also determine the relative frequency distribution.

The fewest number is about 50, the highest about 100.

The greatest concentration is in classes 60 up to 70 and 70 up to 80.

Exercise

P65. N.20 Ch.3


Determine the mean, median, mode12 8 17 6 11 14 8 17 10 8

Mean=11.10; Median=10.50; Mode=8

Exercise

P60. N.2 Ch.3

a. Compute the mean of the following population values: 7, 5, 7, 3, 7, 4


μ = 5.5 found by (7+5+7+3+7+4)/6

Exercise

P60. N.4 Ch.3

Compute the mean of the following sample values: 1.3 7.0 3.6 4.1 5.0

b. Show that Σ(X - X)=0


(1.3-4.2)+(7.0-4.2)+(3.6-4.2)+(4.1-4.2)+(5.0-4.2)=0

X = 4.2 found by 21/5

More Information

Source: Keller, Statistics for Management and Economics, 2005

More InformationPercentage

We added memory to our computer system. We had 96 MB of main memory and now with our new addition, we have 256 MB of main memory. I would like to figure out what percent increase this represents. If you go from 100 MB of memory to 200 MB then you've increased it by 100 percent, because the amount of the increase (100 MB) is 100% of the original amount (100 MB).

That is... if you double your memory then you've increased it by 100 percent. If you add another 100 MB, you're adding another 100% of the original amount, so you have a 200% increase, from 100 MB to 300 MB.

In this case, you have gone from about 100 to about 250. Since 250 is halfway between 200 MB and 300 MB, you could guess that the answer is about 150 percent. Does this make sense? Now let's find the actual value.

I'm going to do a simple example first so you see how percentages work. If I go from 100 MB to 105 MB, what is the percent increase? In this case, the numbers are straightforward: the increase (5 MB) is 5 percent of the original amount (100 MB). But we can use a method that will work even when the numbers aren't this tidy: I ask: 100 times what number will give me 105? 100 * x = 105 x = 105 / 100 x = 1.05 Then I ask: What increase is that over 100%? x - 1 = 1.05 - 1 = 0.05 = 5/100 = 5% So I have an increase of 5%. Now let's do the same thing with your numbers: 1) 96 * x = 256 x = 256 / 96 x = 2.67 2) x - 1 = 2.67 - 1 = 1.67 = 167/100 = 167% which is pretty close to the original estimate of 150%. That gives us some confidence that we have the right answer.

http://images.clipartof.com/small/24876-Clipart-Illustration-Of-A-White-Character-Pushing-Up-A-Blue-Percentage-Symbol-Symbolizing-Percentage-Rates-For-Loans.jpg

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Lesson 002

Education