1. Business and Economics 7th Edition
Statistics for
Business and Economics 7th Edition
Chapter 1-2
Basics
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2. Dealing with Uncertainty
1.1
Everyday decisions are based on incomplete information
Consider:
Will the job market be strong when I graduate?
Will the price of Yahoo stock be higher in six months than it is now?
Will interest rates remain low for the rest of the year if the budget deficit is as high as predicted?
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3. Dealing with Uncertainty
(continued)
Numbers and data are used to assist decision making
Statistics is a tool to help process, summarize, analyze, and interpret data
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4. Key Definitions
1.2
A population is the collection of all items of interest or under investigation
N represents the population size
A sample is an observed subset of the population
n represents the sample size
A parameter is a specific characteristic of a population
A statistic is a specific characteristic of a sample
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5. Population vs. Sample Population Sample a b c d b c ef gh i jk l m n
o p q rs t u v w
x y z
b c
g i n
o r u y
Values calculated using population data are called parameters
Values computed from sample data are called statistics
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6. Examples of Populations
Names of all registered voters in Turkey
Incomes of all families living in Hawai
Annual returns of all stocks traded on the New York Stock Exchange
Grade point averages of all the students in your university
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7. Random Sampling Simple random sampling is a procedure in which
each member of the population is chosen strictly by chance,
each member of the population is equally likely to be chosen,
every possible sample of n objects is equally likely to be chosen
The resulting sample is called a random sample
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8. Descriptive and Inferential Statistics
Two branches of statistics:
Descriptive statistics
Graphical and numerical procedures to summarize and process data
Inferential statistics
Using data to make predictions, forecasts, and estimates to assist decision making
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9. Descriptive Statistics
Collect data
e.g., Survey
Present data
e.g., Tables and graphs
Summarize data
e.g., Sample mean =
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10. Inferential Statistics
Estimation
e.g., Estimate the population mean weight using the sample mean weight
Hypothesis testing
e.g., Test the claim that the population mean weight is 140 pounds
Inference is the process of drawing conclusions or making decisions about a population based on sample results
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11. Types of Data Examples: Marital Status Are you registered to vote?
Eye Color
(Defined categories or groups)
Examples:
Number of Children
Defects per hour
(Counted items)
Examples:
Weight
Voltage
(Measured characteristics)
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12. Describing Data Numerically
Central Tendency
Variation
Arithmetic Mean
Range
Median
Interquartile Range
Mode
Variance
Standard Deviation
Coefficient of Variation
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13. Measures of Central Tendency
2.1
Measures of Central Tendency
Overview
Central Tendency
Mean
Median
Mode
Arithmetic average
Midpoint of ranked values
Most frequently observed value
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14. Arithmetic Mean
The arithmetic mean (mean) is the most common measure of central tendency
For a population of N values:
For a sample of size n:
Population values
Population size
Observed values
Sample size
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15. Arithmetic Mean The most common measure of central tendency
(continued)
The most common measure of central tendency
Mean = sum of values divided by the number of values
Affected by extreme values (outliers)
Mean = 3
Mean = 4
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16. Median
In an ordered list, the median is the “middle” number (50% above, 50% below)
Not affected by extreme values
Median = 3
Median = 3
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17. Finding the Median The location of the median:
If the number of values is odd, the median is the middle number
If the number of values is even, the median is the average of the two middle numbers
Note that is not the value of the median, only the position of the median in the ranked data
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18. Mode A measure of central tendency Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may may be no mode
There may be several modes
No Mode
Mode = 9
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19. Review Example Five houses on a hill by the beach
House Prices: $2,000, , , , ,000
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20. Review Example: Summary Statistics
House Prices: $2,000,000
500, , , ,000
Sum 3,000,000
Mean: ($3,000,000/5)
= $600,000
Median: middle value of ranked data = $300,000
Mode: most frequent value = $100,000
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21. Which measure of location is the “best”?
Mean is generally used, unless extreme values (outliers) exist . . .
Then median is often used, since the median is not sensitive to extreme values.
Example: Median home prices may be reported for a region – less sensitive to outliers
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22. Measures of Variability
2.2
Variation
Range
Interquartile
Range
Variance
Standard Deviation
Coefficient of Variation
Measures of variation give information on the spread or variability of the data values.
Same center,
different variation
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23. Range = Xlargest – Xsmallest
Simplest measure of variation
Difference between the largest and the smallest observations:
Range = Xlargest – Xsmallest
Example:
Range = = 13
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24. Disadvantages of the Range
Ignores the way in which data are distributed
Sensitive to outliers
Range = = 5
Range = = 5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
Range = = 4
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = = 119
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25. Interquartile Range
Can eliminate some outlier problems by using the interquartile range
Eliminate high- and low-valued observations and calculate the range of the middle 50% of the data
Interquartile range = 3rd quartile – 1st quartile
IQR = Q3 – Q1
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26. Interquartile Range Example: Median (Q2) X X Q1 Q3 Interquartile range
maximum
minimum
25% % % %
Interquartile range
= 57 – 30 = 27
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27. Population Variance
Average of squared deviations of values from the mean
Population variance:
Where
= population mean
N = population size
xi = ith value of the variable x
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28. Sample Variance
Average (approximately) of squared deviations of values from the mean
Sample variance:
Where
= arithmetic mean
n = sample size
Xi = ith value of the variable X
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29. Population Standard Deviation
Most commonly used measure of variation
Shows variation about the mean
Has the same units as the original data
Population standard deviation:
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30. Sample Standard Deviation
Most commonly used measure of variation
Shows variation about the mean
Has the same units as the original data
Sample standard deviation:
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31. Calculation Example: Sample Standard Deviation
Sample Data (xi) :
n = Mean = x = 16
A measure of the “average” scatter around the mean
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32. Measuring variation Small standard deviation Large standard deviation
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33. Comparing Standard Deviations
Data A
Mean = 15.5
s = 3.338
Data B
Mean = 15.5
s = 0.926
Data C
Mean = 15.5
s = 4.570
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34. Advantages of Variance and Standard Deviation
Each value in the data set is used in the calculation
Values far from the mean are given extra weight (because deviations from the mean are squared)
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35. Coefficient of Variation
Measures relative variation
Always in percentage (%)
Shows variation relative to mean
Can be used to compare two or more sets of data measured in different units
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36. Comparing Coefficient of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Both stocks have the same standard deviation, but stock B is less variable relative to its price
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37. Using Microsoft Excel
Descriptive Statistics can be obtained from Microsoft® Excel
Select: data / data analysis / descriptive statistics
Enter details in dialog box
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38. Using Excel Select data / data analysis / descriptive statistics
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39. Using Excel Enter input range details Check box for summary statistics
Click OK
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40. Excel output Microsoft Excel descriptive statistics output,
using the house price data:
House Prices: $2,000,000
500, , , ,000
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41. The Sample Covariance
2.4
The covariance measures the strength of the linear relationship between two variables
The population covariance:
The sample covariance:
Only concerned with the strength of the relationship
No causal effect is implied
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42. Interpreting Covariance
Covariance between two variables:
Cov(x,y) > x and y tend to move in the same direction
Cov(x,y) < x and y tend to move in opposite directions
Cov(x,y) = x and y are independent
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43. Coefficient of Correlation
Measures the relative strength of the linear relationship between two variables
Population correlation coefficient:
Sample correlation coefficient:
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44. Features of Correlation Coefficient, r
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 positive linear relationship
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45. Scatter Plots of Data with Various Correlation CoefficientsY Y Y X X X
r = -1
r = -.6
r = 0 Y Y Y X X X
r = +1
r = +.3
r = 0
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46. Using Excel to Find the Correlation Coefficient
Select Data / Data Analysis
Choose Correlation from the selection menu
Click OK . . .
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47. Using Excel to Find the Correlation Coefficient
(continued)
Input data range and select appropriate options
Click OK to get output
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48. Interpreting the Result
There is a relatively
strong positive linear
relationship between
test score #1
and test score #2
Students who scored high on the first test tended to score high on second test
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